Vapor–liquid equilibrium data for the nitrogen + n-octane system from (344.5 to 543.5) K and at pressures up to 50 MPa

Fluid Phase Equilibria 282 (2009) 3–10

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Vapor–liquid equilibrium data for the nitrogen + n-octane system from (344.5 to 543.5) K and at pressures up to 50 MPa Gaudencio Eliosa-Jiménez a , Fernando García-Sánchez a,∗ , Guadalupe Silva-Oliver b , Ricardo Macías-Salinas c a b c

Laboratorio de Termodinámica, Programa de Investigación en Ingeniería Molecular, Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, 07730 México, D.F., Mexico Departamento de Ingeniería Química Petrolera, ESIQIE, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, 07738 México, D.F., Mexico Departamento de Ingeniería Química, SEPI−ESIQIE, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, 07738 México, D.F., Mexico

a r t i c l e

i n f o

Article history: Received 26 November 2008 Accepted 19 April 2009 Available online 3 May 2009 Keywords: Vapor–liquid equilibria Enhanced oil recovery Equation of state Stability analysis Phase envelope Critical point

a b s t r a c t A static-analytical apparatus with visual sapphire windows and pneumatic capillary samplers has been used to obtain new vapor–liquid equilibrium data for the N2 + n-octane system over the temperature range from (344.5 to 543.5) K and at pressures up to 50 MPa. Equilibrium phase compositions and vapor–liquid equilibrium ratios are reported. The new results were compared with solubility data reported by other authors. The comparison showed that the solubility data reported in this work at 344.5 K are in good agreement with those determined by others at 344.3 K. The experimental data were modeled with the PR and PC-SAFT equations of state by using one-fluid mixing rules and a single temperature-independent interaction parameter. Results from the modeling effort showed that the PC-SAFT equation was superior to the PR equation in correlating the experimental data of the N2 + n-octane system. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Knowledge of N2 solubility in hydrocarbon compounds is very useful in relation to an optimal recovery strategy in enhanced oil recovery by gas N2 injection. Solubilities of N2 in paraffin, aromatic, and naphthenic compounds have already been studied (see Ref. [1] and references therein), but a lot of other mixtures remain to be investigated over wide ranges of temperature and pressure. From the theoretical point of view, the phase behavior prediction of systems containing N2 with hydrocarbons is a challenging task [2,3] due to that all binary N2 + hydrocarbon fluid mixtures develop, except for methane, type III phase diagrams [4], which exhibit two distinct critical curves: one starting at the critical point of the component with the higher critical temperature that goes to infinite pressures whereas the other critical curve starts at the critical point of the component with the lower critical temperature and meets a three-phase line liquid–liquid–vapor at an upper critical end point. Therefore, it is important to carry out additional experimental phase equilibrium studies at elevated temperatures and pressures of N2 -containing binary systems along with some modeling effort using thermodynamic models to give

∗ Corresponding author. Tel.: +52 55 9175 6574. E-mail address: [email protected] (F. García-Sánchez). 0378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2009.04.015

the right qualitative and quantitative description of the phase behavior of N2 –hydrocarbon mixtures and thus providing a better understanding of phase behavior patterns that hydrocarbon mixtures develop during an enhanced oil recovery process by N2 injection. In order to increase the experimental database of vapor–liquid equilibria for mixtures containing N2 and a hydrocarbon, we have undertaken a systematic study of the phase behavior of N2 + hydrocarbon mixtures at high pressures. Previously, we have reported vapor–liquid equilibrium data of N2 + n-alkane (n-pentane to n-heptane and n-nonane) mixtures over wide temperature ranges and pressures up to 50 MPa [5–8]. These studies are part of a research program where phase behavior is studied for enhanced oil recovery in selected Mexican fields by N2 injection. In this work, we report new vapor–liquid equilibrium measurements for the system N2 + n-octane over the temperature range from (344.5 to 543.5) K and at pressures up to 50 MPa. Five isotherms are reported in this study, which were determined in a high-pressure phase equilibrium apparatus of the static-analytical type using a sampling–analyzing process for determining the composition of the different coexisting phases. The experimental data obtained in our measurements were correlated using the PR [9] and PC-SAFT [10] equations of state. The mixing rules used for these equations were the classical one-fluid mixing rules. For both models, a single temperature-independent interaction parameter was fitted to all experimental data.

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G. Eliosa-Jiménez et al. / Fluid Phase Equilibria 282 (2009) 3–10

2. Experimental 2.1. Materials N2 and He (carrier gas) were acquired from Aga Gas (Mexico) and Infra (Mexico), respectively, both with a certified purity greater than 99.999 mol%. Octane normal was purchased from Aldrich (USA) with a GLC minimum purity of 99 mol%. They were used without any further purification except for careful degassing of the n-octane. 2.2. Apparatus and procedure The apparatus and procedures employed in its operation have been previously described [5–8] and will not be repeated in detail here. Briefly, the apparatus consists mainly of an equilibrium cell, a sampling–analyzing system, a pressure transducer, two platinum temperature sensors, a magnetic stirring device, a timer and compressed-air control device for each sampler, an analytical system, and feeding and degassing circuits. This apparatus is based on the static-analytical method with fluid phase sampling and can be used to determine the multiphase equilibrium of binary and multicomponent systems between (313 and 673) K and pressures up to 60 MPa. The equilibrium cell (made of titanium) has an internal volume of about 100 mL and holds two sapphire windows for visual observation. The cap of the cell, used for studies at high temperatures with up to three coexisting phases, holds the three sampling systems with capillaries of different lengths: the extremity of one capillary is at the top of the cell (vapor withdrawing), another one at the bottom of the cell (withdrawing of the dense liquid), and the third one in an intermediate position (light liquid phase withdrawing). The sampling–analyzing system is constituted of three RolsiTM capillary samplers [11] connected altogether on-line with a gas chromatograph, which makes the apparatus very practical and accurate for measurements at high temperatures and pressures. The equilibrium cell can be maintained within ±0.2 K for temperatures between (313 and 673) K by means of a high temperature regulating air thermostat. Temperature measurements in the equilibrium cell were monitored by using two Pt100 resistance thermometers, which are periodically calibrated against a 25- reference platinum resistance (Tinsley Precision Instruments). The resulting uncertainty of the two Pt100 probes is, usually, not higher than ±0.02 K; however, drift in the temperature of the air oven makes the uncertainty of the temperature measurements to be ±0.2 K. Pressure measurements were carried out by means of a pressure transducer (Sensonetics for temperatures up to 673 K), which is periodically calibrated against a dead-weight pressure balance. Pressure measurement uncertainties are estimated to be within ±0.02 MPa for pressures up to 50 MPa. The analytical work was carried out using a gas chromatograph (Varian 3800) equipped with a thermal conductivity detector (TCD), which is connected to a data acquisition system (Star GC Workstation, Ver. 5.3). The analytical column used is a Restek 3% OV-101 column (mesh 100/120 Silcoport-W, silcosteel tube, length: 3 m, diameter: 3.175 mm). The TCD was used to detect the n-octane and N2 compounds. It was calibrated by injecting known amounts of noctane and N2 through liquid-tight (5 ␮L) and gas-tight (1000 ␮L)

syringes, respectively. Calibration data were fitted to a quadratic polynomial, leading to an estimated mole fraction uncertainty less than 1% for liquid and vapor phases on the whole concentration range. Once the pressure transducer, platinum temperature probes, and chromatographic thermal conductivity detector have been calibrated, the system was preflushed with isopropyl alcohol and then dried under vacuum at 423 K. After drying under vacuum, the system was purged with N2 to ensure that the last traces of solvents were removed. During an experimental run, the liquid component, previously degassed according to the method of Battino et al. [12], is first introduced into the cell. The equilibrium cell and its loading lines are evacuated before filling it with the degassed liquid component. Once the desired temperature is reached and stabilized, the gaseous component, previously stored in a highpressure cell, is carefully introduced into the equilibrium cell until the pressure of measurement. Then the magnetic stirring device is activated to reach equilibrium, which is indicated by pressure stabilization. Pressure is adjusted by injecting again the gaseous component and acting the stirring device until reaching the desired pressure. After the equilibrium in the cell is achieved, measurements are performed using the capillary-sampling injectors, which are connected to the equilibrium cell by 0.1 mm internal diameter capillary tubes of different lengths. The samples are injected and vaporized directly into the carrier gas (He) stream of the gas chromatograph. For each equilibrium condition, at least 25 equilibrium samples are withdrawn using the pneumatic samplers and analyzed in order to check for measurement and repeatability. As the volume of the withdrawn samples is very small compared to the volume of the vapor or liquid present in the equilibrium cell, it is possible to withdraw many samples without disturbing the phase equilibrium. In order to avoid condensation and adsorption of the n-octane, the samplers and all of the lines for the gas stream are superheated to ensure that the whole of the samples is transferred to the chromatograph.

3. Experimental results The N2 + n-octane system has been previously studied by Llave and Chung [13] at different temperature and pressure conditions. The solubility data reported by these authors are, to the best of our knowledge, the only experimental solubility data at high pressures reported in the open literature for this system. Table 1 contains a summary of the earlier results, including those presented in this paper. The new measured phase equilibrium compositions for this binary system in terms of N2 liquid and vapor mole fractions, temperatures, and pressures, are tabulated in Table 2. Uncertainties in the phase compositions, due mainly to errors associated with sampling, are estimated to be ±0.003 in the mole fraction on the whole concentration range. Error calculations were performed in the following way: from Eq. (1) relating mole fractions zi (liquid or vapor) to chromatographic measurements zi =

ni

N

n j=1 j

=

ni , nT

zi = xi or yi

(1)

Table 1 Summary of vapor–liquid equilibria for the N2 + n-octane system. Reference Llave and Chung [13] This work

Temperature range (K) 322.0–344.3 344.5–543.5

Pressure range (MPa) 3.23–35.04 2.05–50.14

Number of points

Remarks

12 73

p–x data p–x–y data

G. Eliosa-Jiménez et al. / Fluid Phase Equilibria 282 (2009) 3–10

5

Table 2 Experimental vapor–liquid equilibrium data for the system N2 + n-octane. T (K)

p (MPa)

xN

yN

KN

KnC

344.5

0.02a 2.39 3.82 5.67 7.34 10.04 14.31 18.50 20.82 24.31 27.74 31.28 34.79 37.81 41.86 45.91 50.14

0.0000 0.0309 0.0491 0.0715 0.0911 0.1216 0.1668 0.2062 0.2275 0.2580 0.2875 0.3136 0.3398 0.3629 0.3914 0.4176 0.4448

0.0000 0.9883 0.9928 0.9942 0.9947 0.9946 0.9941 0.9932 0.9925 0.9914 0.9903 0.9890 0.9875 0.9861 0.9841 0.9818 0.9787

31.9838 20.2200 13.9049 10.9188 8.1793 5.9598 4.8167 4.3626 3.8426 3.4445 3.1537 2.9061 2.7173 2.5143 2.3511 2.2003

0.0121 0.0076 0.0062 0.0058 0.0061 0.0071 0.0086 0.0097 0.0116 0.0136 0.0160 0.0189 0.0218 0.0261 0.0313 0.0384

0.19a 2.42 3.38 4.04 4.93 7.63 10.13 15.25 19.92 25.38 29.81 35.27 40.17 45.00 49.70

0.0000 0.0369 0.0518 0.0620 0.0765 0.1175 0.1530 0.2261 0.2884 0.3592 0.4125 0.4791 0.5405 0.6080 0.6991

0.0000 0.8948 0.9196 0.9301 0.9392 0.9519 0.9559 0.9586 0.9569 0.9517 0.9451 0.9359 0.9229 0.9018 0.8565

24.2493 17.7529 15.0016 12.2771 8.1013 6.2477 4.2397 3.3180 2.6495 2.2912 1.9535 1.7075 1.4832 1.2251

0.1092 0.0848 0.0745 0.0658 0.0545 0.0521 0.0535 0.0606 0.0754 0.0934 0.1231 0.1678 0.2505 0.4769

0.56a 2.54 4.02 5.62 7.83 10.64 12.88 15.27 17.70 19.99 23.12 27.01 29.92 32.13

0.0000 0.0394 0.0693 0.1011 0.1429 0.1949 0.2354 0.2773 0.3202 0.3617 0.4180 0.4922 0.5591 0.6603

0.0000 0.7227 0.8062 0.8457 0.8703 0.8853 0.8902 0.8920 0.8899 0.8865 0.8776 0.8582 0.8311 0.7618

18.3426 11.6335 8.3650 6.0903 4.5423 3.7816 3.2167 2.7792 2.4509 2.0995 1.7436 1.4865 1.1537

0.2887 0.2082 0.1717 0.1513 0.1425 0.1436 0.1494 0.1620 0.1778 0.2103 0.2792 0.3831 0.7012

1.02a 2.05 3.26 4.23 5.40 6.93 7.90 9.06 10.28 11.53 12.85 13.99 15.44 16.94 18.08 20.56 21.21

0.0000 0.0256 0.0552 0.0782 0.1059 0.1412 0.1632 0.1910 0.2191 0.2487 0.2802 0.3072 0.3426 0.3813 0.4141 0.5083 0.5534

0.0000 0.4201 0.5885 0.6606 0.7082 0.7444 0.7587 0.7715 0.7796 0.7847 0.7875 0.7871 0.7846 0.7751 0.7653 0.7210 0.6720

16.4102 10.6612 8.4476 6.6874 5.2720 4.6489 4.0393 3.5582 3.1552 2.8105 2.5622 2.2901 2.0328 1.8481 1.4185 1.2143

0.5951 0.4355 0.3682 0.3264 0.2976 0.2884 0.2824 0.2822 0.2866 0.2952 0.3073 0.3277 0.3635 0.4006 0.5674 0.7344

1.75a 2.29 2.92 3.39 3.72 4.13 4.57 5.14 5.65 6.94 7.65 8.33 9.08 9.83 10.57

0.0000 0.0204 0.0409 0.0571 0.0680 0.0809 0.0967 0.1146 0.1326 0.1767 0.2021 0.2269 0.2572 0.2970 0.3508

0.0000 0.1849 0.2888 0.3392 0.3737 0.4046 0.4337 0.4642 0.4858 0.5230 0.5311 0.5380 0.5374 0.5263 0.4907

9.0637 7.0611 5.9405 5.4956 5.0012 4.4850 4.0506 3.6637 2.9598 2.6279 2.3711 2.0894 1.7721 1.3988

0.8321 0.7415 0.7008 0.6720 0.6478 0.6269 0.6052 0.5928 0.5794 0.5877 0.5976 0.6228 0.6738 0.7845

424.0

473.5

508.1

543.5

a

2

2

Calculated vapor pressure of pure n-octane [14].

2

8

Fig. 1. Experimental pressure-composition phase diagram for the N2 + n-octane system. Full symbols: data reported in this work; open symbols: data from Llave and Chung [13]. Solid lines used to better identify the experimental data.

we have the errors given by

⎡

zi =

1 n2T

⎣(nT − ni )ni + ni

N 

⎤ nj ⎦

(2)

j= / i

where ni and nj are the mean quadratic derivations in mole number for components i and j, resulting from a data fitting on the results of the chromatographic detector calibration. Table 2 also presents the vapor pressures of n-octane at each temperature level. These values were calculated with the Wagner equation in the “3, 6” form using the parameters reported by Reid et al. [14] as follows: ln

p s

pc



= (1 − )−1 −7.91211 + 1.380076 1.5 − 3.80435 3 − 4.50132 6



 =1−

T Tc

(3)

where ps is the vapor pressure in MPa, pc the critical pressure in MPa, T the temperature in K, and Tc the critical temperature in K. The calculated K values, defined as the equilibrium ratio between the vapor and liquid for each component at a given temperature and pressure, are also given in Table 2. As can be seen in this table, N2 have K values higher than one and it concentrate in the vapor phase whereas n-octane have K values lower than the unity which concentrate in the liquid phase. Fig. 1 shows the results of the vapor–liquid equilibrium data obtained on a pressure-composition diagram for the various experimental temperatures, including the results of Llave and Chung [13]. An examination of this figure shows that our experimental results at 344.5 K are in good agreement with those determined by these authors at 344.3 K. In this figure, it can also be observed that the solubility of N2 in the n-octane-rich liquid phase increases as pressure increases for all isotherms investigated; however, the solubility of N2 in the n-octane-rich liquid phase decreases with decreasing temperature. Isotherms obtained above the critical temperature of N2 (Tc = 126.2 K) end up in the mixture critical point, which was approached by adding carefully small quantities of N2 to avoid

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G. Eliosa-Jiménez et al. / Fluid Phase Equilibria 282 (2009) 3–10

Table 3 Estimated mixture critical data for the N2 + n-octane system. T (K)

p (MPa)

xN2

T (K)

p (MPa)

xN2

424.0 473.5 508.1

50.98 32.33 21.43

0.7908 0.7065 0.6085

543.5 568.83a

10.81 2.487a

0.4031 0.0000

a

Critical point of pure n-octane [22].

upsetting the phase equilibrium. After every step of adding N2 , the cell content was stirred about 4 h before withdrawing the samples of the equilibrium phases. It is worth mentioning that the measurements at (424.0, 473.5, 508.1, and 543.5) K and, respectively, (49.70, 32.13, 21.21, and 10.57) MPa were associated to an opalescence phenomenon, indicating the proximity of a critical state for this mixture; however, it was no possible to attain this terminal state because the apparatus is not suitable to measure this point directly. When the pressure was increased just above the critical point, only one phase was observed. All isotherms studied in this work lie between the critical temperatures for the two components of this system. Since the equilibrium ratios for the two components converge to unity at the critical point of the mixture then it is possible to obtain the critical pressure corresponding to each experimental temperature from the K values versus pressure diagram. Here, we estimated the mixture critical point for each isotherm by adjusting a series of pressure-composition data to Legendre polynomials. Fredenslund [15] showed that these polynomials are able to correlate this type of data within the experimental uncertainty. Once having correlated these data, the pressure-composition phase diagram is calculated in order to locate the maximum pressure. For a system with two components, this maximum corresponds to the critical pressure of the mixture at constant temperature. The composition associated to this maximum pressure corresponds to the critical composition of the mixture. The estimated numerical values of the critical points for the N2 + n-octane system are reported in Table 3. Fig. 2 shows the isotherms plotted in a log K value versus log pressure diagram. In this figure, there are two branches for each isotherm, one for each component. It can be seen that N2 tends to concentrate in the vapor phase whereas the n-octaneB does it

Fig. 3. Pressure–temperature phase diagram for the N2 + n-octane system.

in the liquid phase. The two branches will converge at the mixture critical point where the K values are equal to the unity. The degree of smoothness of the curve for each isotherm reflects the internal consistency of the data; however, it would be convenient to subject these data to a thermodynamic consistency test involving comparison between experimental and calculated vapor phase mole fractions by using, for instance, the procedure given by Christiansen and Fredenslund [16] or the one given by Won and Prausnitz [17]. The estimated mixture critical data reported in Table 3 were plotted in a pressure–temperature diagram shown in Fig. 3. This figure shows that, starting at the critical point of pure n-octane, the mixture critical line runs to lower temperatures and higher pressures exhibiting a positive curvature. The data displayed in this figure allow establishing that the system N2 + n-octane is a type III system according to the classification of van Konynenburg and Scott [4]. Notwithstanding, to substantiate this claim it is necessary to know how the other critical line that departs from the critical point of pure N2 and that this system exhibits a three-phase line liquid–liquid–vapor. In this case, we have no experimental evidence of these phase equilibrium phenomena for this system because they occur at low temperatures and the apparatus used in this work is limited to be used at moderate and high temperatures. However, Eakin et al. [18] and Schindler et al. [19] measured the three-phase line liquid–liquid–vapor up to the upper critical end point for the systems N2 + ethane and N2 + propane, respectively. On this basis, we believe that the system N2 + n-octane will also exhibit the same phase behavior as that displayed by the systems N2 + ethane and N2 + propane. 4. Data representation

Fig. 2. Effect of pressure on vapor–liquid equilibrium ratios for the N2 + n-octane system.

It is well known that mixtures containing components that markedly differ in their critical temperatures (e.g., N2 + hydrocarbon mixtures) behave highly asymmetric so that the calculation of phase equilibria using an equation with pure-component information only is generally unsatisfactory. Consequently, to increase the usefulness of the combining rules in the equations of state for predicting the phase behavior of the

G. Eliosa-Jiménez et al. / Fluid Phase Equilibria 282 (2009) 3–10

N2 + n-octane system, we have estimated the interaction parameter for the PR [9] and PC-SAFT [10] equations of state by fitting the experimental vapor–liquid equilibrium data presented in Table 2 for this system.

7

The hard-chain reference contribution is given by ¯ a˜ hs − a˜ hc = m

N 

xi (mi − 1) ln giihs (ii )

(14)

i=1

¯ is the mean segment number in the mixture: where m

4.1. The PR equation of state The explicit form of the PR equation of state [9] can be written as

¯ = m

N 

xi mi

(15)

i=1

p=

RT a(T ) − v − b v(v + b) + b(v − b)

(4)

The Helmholtz energy of the hard-sphere fluid is given on a persegment basis as

where constants a and b for pure-components are related to a = 0.45724

RTc ˛(T ) pc

(5)

b = 0.07780

RTc pc

(6)



1/2

˛(Tr ) = 1 + (0.37464 + 1.5422ω − 0.26992ω2 )(1 − Tr

gijhs = (7)

N 

3 1 2 2 3 + + (1 − 3 ) 3 (1 − 3 )2

1 + (1 − 3 )





di dj di + dj

 k  xi mi di , 6

3 2



− 0



ln(1 − 3 )

(16)



3 2 (1 − 3 )

2

+

di dj di + dj

2

2 2 2 (1 − 3 )3

(17)

k = 0, 1, 2, 3

(18)

i=1

(8)

The temperature-dependent segment diameter di of component i is given by



xi bi

(9)

i=1

and aij is defined as



ai aj ,

kij = kji ; kii = 0

(10)

where kij is an interaction parameter characterizing the binary formed by components i and j. Eq. (4) can be written in terms of the compressibility factor Z, as Z 3 − (1 − B)Z 2 + (A − 3B2 − 2B)Z − (AB − B2 − B3 ) = 0

(11)

A = ap/(RT)2

and B = bp/(RT). where The expression for the fugacity coefficient of component i is given by b ln ϕi = i (Z − 1) − ln(Z − B) b A − √ 2 2B

2 3

N

xi xj aij

aij = (1 − kij )



with k defined as k =

i=1 j=1

b=



2

)

For mixtures, constants a and b are given by a=

1 0

and the radial distribution function of the hard-sphere fluid is

and ˛(Tr ) is expressed in terms of the acentric factor ω as

N N  

a˜ hs =

 N 2

xa j=1 j ij a



di = i 1 − 0.12 exp −3

εi kT



(19)

where k is the Boltzmann constant and T is the absolute temperature. The dispersion contribution to the Helmholtz energy is given by ¯ m2 ε 3 a˜ disp = −2  I1 ( , m)



¯ 1+Z −  m

hc

∂Z hc + ∂

−1 ¯ m2 ε2  3 I2 ( , m)

(20)

where Zhc is the compressibility factor of the hard-chain reference contribution, and m2 ε 3 =

N N  

xi xj mi mj

ε ij

kT

ij 3

(21)

i=1 j=1

 b − i b

 ln

 √   Z + 1+ 2 B  √  (12) Z + 1−

m2 ε2  3 =

N N  

xi xj mi mj

 ε 2 ij

kT

ij 3

(22)

i=1 j=1

2 B

The parameters for a pair of unlike segments are obtained by using conventional combining rules: 4.2. The PC-SAFT equation of state

ij =

In the PC-SAFT equation of state [10], the molecules are conceived to be chains composed of spherical segments in which the pair potential for the segment of a chain is given by a modified square-well potential [20]. Non-associating molecules are characterized by three pure component parameters: the temperature-independent segment diameter , the depth of the potential ε, and the number of segments per chain m. The PC-SAFT equation of state written in terms of the Helmholtz energy for an N-component mixture of non-associating chains consists of a hard-chain reference contribution and a perturbation contribution to account for the attractive interactions. In terms of reduced quantities, this equation can be expressed as res

a˜

hc

= a˜

disp

+ a˜

(13)

εij =

1 ( + j ) 2 i



εi εj (1 − kij )

(23) (24)

where kij is a binary interaction parameter, which is introduced to correct the segment–segment interactions of unlike chains. ¯ and I2 ( , m) ¯ in Eq. (20) are calculated by The terms I1 ( , m) simple power series in density: ¯ = I1 ( , m)

6 

¯ i ai (m)

(25)

¯ i bi (m)

(26)

i=0

¯ = I2 ( , m)

6  i=0

8

G. Eliosa-Jiménez et al. / Fluid Phase Equilibria 282 (2009) 3–10

where the coefficients ai and bi depend on the chain length as given in Gross and Sadowski [10]. The density to a given system pressure psys is determined iteratively by adjusting the reduced density until pcalc = psys . For a converged value of , the number density of molecules , given in Å−3 , is calculated from: =

 N  6



−1 xi mi di

3

(27)

Table 4 Estimated binary interaction parameters for the N2 + n-octane system. PR equation of state kij

p

PC-SAFT equation of state y

kij

p

Bubble-point pressure method 0.1396 11.2

x

4.0

0.1233

7.6

Flash method 0.1040

3.3

0.1109

3.1

x

y 1.2

2.1

1.4

i=1

Using Avogadro’s number and appropriate conversion factors,  produces the molar density in different units such as kmol mol−3 . Equations for the compressibility factor are derived from the relation:



Z =1+

∂˜ares ∂



= 1 + Z hc + Z disp

(28)

T,xi

The pressure p can be calculated in units of Pa = N m−2 by applying the relation:



p = Z kT 1010 Å /m

3

(29)

The expression for the fugacity coefficient of component i is given by



ln ϕi = where



∂(n˜ares ) ∂ni



∂(n˜ares ) ∂ni

+ (Z − 1) − ln Z

(30)

 = a˜ res +

,T,nj = / i

−

∂˜ares ∂xi

  N  xk

k=1

 ,T,xj = / i





(31)

The binary interaction parameter kij , defined in Eqs. (10) and (24) for the PR and PC-SAFT equations of state, respectively, was estimated by minimizing the sum of squared relative deviations of bubble point pressures and the sum of squared deviations in mole fraction of phase equilibrium compositions. The calculation of the phase equilibria was carried out by minimization of the Gibbs energy using stability tests (based on the tangent plane criterion) to find the most stable state of the system, according to the solution approach presented by Justo-García et al. [21]. Physical properties of N2 and n-octane (i.e., critical temperature Tc , critical pressure pc , and acentric factor ω) for the PR equation of state were taken from Ambrose [22], while the three pure-component parameters (i.e., temperature-independent segment diameter , depth of the potential ε, and number of segments per chain m) of these compounds for the PC-SAFT equation of state were taken from Gross and Sadowski [10]. The simplex optimization procedure of Nelder and Mead [23] with convergence accelerated by the Wegstein algorithm [24] was used in the computations by searching the minimum of the following objective functions:

exp

i=1

pi

exp

− yicalc )

2

(33)

i=1

for the flash calculation method. exp exp exp In these equations, pi − pcalc , xi − xicalc and yi − yicalc are i the residuals between the experimental and calculated values of, respectively, bubble-point pressures, liquid mole fractions, and vapor mole fractions for an experiment i, and M is the total number of experiments. Once minimization of objective functions S1 and S2 was performed, the agreement between calculated and experimental values was established through the standard percent relative deviation in pressure,  p , and standard percent deviation in mole fraction for the liquid,  x , and vapor,  y , phases of the lightest component:



M



exp

pi

M

2 1/2

− pcalc i

(34)

exp

pi

1/2 (35)

i=1

,T,xj = / k



2 − yicalc )

2

− xicalc ) + (yi

2 1  exp x = 100 (xi − xicalc ) M

4.3. Estimation of interaction parameters and results

S1 =

exp

(xi



∂˜ares ∂xk

exp + (yi

M 

i=1

In Eq. (31), partial derivatives with respect to mole fractions are N x = 1. calculated regardless of the summation restriction i=1 i

 2 M exp  pi − pcalc i

S2 =

1 P = 100 M

,T,nj = / i



for the bubble-point pressure method, and

(32)



2 1  exp y = 100 (yi − yicalc ) M M

1/2 (36)

i=1

where  p ,  x , and  y were obtained with the optimal values of the binary interaction parameters, and they are given in Table 4. This table shows the correlative capabilities of the PR and PCSAFT equations by using the van der Waals one-fluid mixing rules and a temperature-independent binary interaction parameter for the N2 + n-octane system. Overall, it can be said that the quality for correlating the experimental vapor–liquid equilibrium data of this binary system with the PC-SAFT equation is superior to that obtained with the PR equation. The results of the correlation with the PR equation of state to the N2 + n-octane system are shown in Fig. 4. As can be seen in this figure, the PR equation fits the data well at low and moderate pressures but it is less satisfactory when pressure and/or temperature increase. The fact that predictions at the different temperatures are not precise indicates that a temperature-dependent interaction parameter is necessary to adequately model the N2 + n-octane system when calculations are made over a wide range of temperatures, as done by Privat et al. [2,3] for the PPR78 model (a group contribution method for estimating binary interaction parameters of the PR equation). Nonetheless, this is outside the scope of this work and no attempt was made to either determine this temperature dependence or apply other complex mixing rules or combining rules. Fig. 5 shows the results of the correlation with the PC-SAFT equation of state to the N2 + n-octane system. In this figure, it can be seen that the overall agreement between experimental and calculated values is satisfactory, in particular at the highest temperatures. The superior quality of the PC-SAFT equation for predicting the phase

G. Eliosa-Jiménez et al. / Fluid Phase Equilibria 282 (2009) 3–10

9

low level of the random errors in the measurements. Moreover, the phase measurements on the system N2 + n-octane showed that it belongs to the type III class of systems according to the classification of van Konynenburg and Scott. The PR and PC-SAFT equations of state with van der Waals onefluid mixing rules were used to represent the measured data of this binary system by adjusting a single temperature-independent interaction parameter for each equation. Results of the representation showed that the PR equation fits the data well at low and moderate pressures but it is less satisfactory when pressure and/or temperature increase, while PC-SAFT equation fits better the data on the whole temperature and pressure range studied. List of symbols A Helmholtz energy (J) a attractive term in PR EoS a˜ reduced Helmholtz energy, a˜ = A/NkT b co-volume in PR EoS d temperature dependent segment diameter (Å) hs g˛ˇ site-site radial distribution function of hard-sphere fluid Fig. 4. Experimental and calculated pressure-composition phase diagram for the N2 + n-octane system. Solid lines: PR EoS with kij = 0.1396 fitted to the vapor–liquid equilibrium data of this work.

I1 , I2 , Ki k kij m ¯ m ni nT M p R S1 S2 T

v xi yi Z

Fig. 5. Experimental and calculated pressure-composition phase diagram for the N2 + n-octane system. Solid lines: PC-SAFT EoS with kij = 0.1233 fitted to the vapor–liquid equilibrium data of this work.

behavior of asymmetric mixtures is due to that this model is based on a perturbation theory for chain molecules that can be applied to mixtures of small spherical molecules such as gases, non-spherical solvents, and chainlike polymers by using conventional one-fluid mixing rules. 5. Conclusions New experimental vapor–liquid data have been obtained for the N2 + n-octane system at temperatures from (344.5 to 543.5) K and at pressures up to 50 MPa by using an experimental static-analytical apparatus with pneumatic capillary samplers. The experimental measurements reported in this work are in good agreement with those determined by other authors. The smoothness of the equilibrium ratio–pressure curve for each isotherm showed indicates a

abbreviations, defined in Eqs. (25) and (26) equilibrium ratio (=yi /xi ) of component i Boltzmann constant (J K−1 ) binary interaction parameter number of segments per chain mean segment number in the system, defined in Eq. (15) mole number of component i total mole number molecular weight pressure (MPa) gas constant objective function, defined in Eq. (32) objective function, defined in Eq. (33) temperature (K) molar volume liquid mole fraction of component i vapor mole fraction of component i compressibility factor

Greek letters ˛ alpha function in PR EoS fugacity coefficient of component i ϕi packing fraction, = 3  total number density of molecules (Å−3 )  segment diameter (Å) standard percent relative deviation in pressure, defined in p Eq. (34) x standard percent deviation in liquid mole fraction, defined in Eq. (35) standard percent deviation in vapor mole fraction, defined y in Eq. (36) abbreviation (n = 0, . . ., 3), defined in Eq. (18) (Ån−3 ) n ω acentric factor in PR EoS Superscripts calc calculated property disp contribution due to dispersive attraction exp experimental property hc residual contribution of hard-chain system hs residual contribution of hard-sphere system res residual property Subscripts c critical property i, j components i, j

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Acknowledgments This work was supported by the Molecular Engineering Research Program of the Mexican Petroleum Institute under projects D.00406 and I.00432. Eliosa-Jiménez gratefully acknowledges the National Council for Science and Technology of Mexico (CONACYT) for the financial support received through a PhD fellowship (192773). References [1] F. García-Sánchez, G. Eliosa-Jiménez, G. Silva-Oliver, R. Vázquez-Román, Vapor–liquid equilibria of nitrogen–hydrocarbon systems using the PC-SAFT equation of state, Fluid Phase Equilib. 217 (2004) 241–253. [2] R. Privat, J.N. Jaubert, F. Mutelet, Addition of the nitrogen group to the PPR78 model (predictive 1978 Peng Robinson EOS with temperature-dependent kij calculated through a group contribution method), Ind. Eng. Chem. Res. 47 (2008) 2033–2048. [3] R. Privat, J.N. Jaubert, F. Mutelet, Use of the PPR78 model to predict new equilibrium data of binary systems involving hydrocarbons and nitrogen. Comparison with other GCEOS, Ind. Eng. Chem. Res. 47 (2008) 7483–7489. [4] P.H. van Konynenburg, R.L. Scott, Critical lines and phase equilibria in binary van der Waals mixtures, Phil. Trans. R. Soc. Lond., Ser. A 298 (1980) 495–540. ˜ [5] G. Silva-Oliver, G. Eliosa-Jiménez, F. García-Sánchez, J.R. Avendano-Gómez, High-pressure vapor–liquid equilibria in the nitrogen–n-pentane system, Fluid Phase Equilib. 250 (2006) 37–48. [6] G. Eliosa-Jiménez, G. Silva-Oliver, F. García-Sánchez, A. de Ita de la Torre, Highpressure vapor–liquid equilibria in the nitrogen + n-hexane system, J. Chem. Eng. Data 52 (2007) 395–404. [7] F. García-Sánchez, G. Eliosa-Jiménez, G. Silva-Oliver, A. Godínez-Silva, Highpressure (vapor–liquid) equilibria in the (nitrogen + n-heptane) system, J. Chem. Thermodyn. 39 (2007) 893–905. ˜ [8] G. Silva-Oliver, G. Eliosa-Jiménez, F. García-Sánchez, J.R. Avendano-Gómez, High-pressure vapor–liquid equilibria in the nitrogen–n-nonane system, J. Supercrit. Fluids 42 (2007) 36–47.

[9] D.-Y. Peng, D.B. Robinson, A new two constant equation of state, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. [10] J. Gross, G. Sadowski, Perturbed-chain SAFT: an equation of state based on perturbation theory for chain molecules, Ind. Eng. Chem. Res. 40 (2001) 1244–1260. [11] P. Guilbot, A. Valtz, H. Legendre, D. Richon, Rapid on-line sampler-injector: a reliable tool for HT-HP sampling and on-line GC analysis, Analusis 28 (2000) 426–431. [12] R. Battino, M. Banzhof, M. Bogan, E. Wilhelm, Apparatus for rapid degassing of liquids. Part III, Anal. Chem. 43 (1971) 806–807. [13] F.M. Llave, T.H. Chung, Vapor–liquid equilibria of nitrogen–hydrocarbon systems, J. Chem. Eng. Data 33 (1988) 123–128. [14] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York, 1987. [15] A. Fredenslund, Thermodynamic consistency of high pressure vapor–liquid equilibrium data, in: Proceedings of International Symposium on Vapor–Liquid Equilibria in Multicomponent Systems, Warszawa-Jablonna, Poland, November 2–6, 1975. [16] L.J. Christiansen, A. Fredenslund, Thermodynamic consistency using orthogonal collocation or computation of equilibrium vapor compositions at high pressures, AIChE J. 21 (1975) 49–57. [17] K.W. Won, J.M. Prausnitz, High-pressure vapor–liquid equilibria. Calculation of partial pressures from total pressure data. Thermodynamic consistency, Ind. Eng. Chem. Fundam. 12 (1973) 459–463. [18] B.E. Eakin, R.T. Ellington, D.C. Gami, Physical–chemical properties of ethane– nitrogen mixtures, Inst. Gas Technol. Res. Bull. 26 (1955), IGTR, Chicago. [19] D.L. Schindler, G.W. Swift, F. Kurata, More low temperature V–L design data, Hydrocarbon Process. 45 (1966) 205–210. [20] S.S. Chen, A. Kreglewski, Applications of the augmented van der Waals theory of fluids. I. Pure fluids, Ber. Bunsen-Ges. Phys. Chem. 81 (1977) 1048–1052. [21] D.N. Justo-García, F. García-Sánchez, A. Romero-Martinez, Isothermal multiphase flash calculations with the PC-SAFT equation of state, Am. Inst. Phys. Conf. Proc. 979 (2008) 195–214. [22] D. Ambrose, Vapour–Liquid Critical Properties, NPL Report Chem. 107, National Physical Laboratory, Teddington, 1980. [23] J.A. Nelder, R.A. Mead, Simplex method for function minimization, Comput. J. 7 (1965) 308–313. [24] J.H. Wegstein, Accelerating convergence for iterative processes, Commun. Assoc. Comp. Mach. 1 (1958) 9–13.