Thermodynamic and transport properties of nitrogen and butane

MOLECULAR PHYSICS, 2000, VOL.98, No. 1, 43-55

Thermodynamic and transport properties of nitrogen and butane mixtures

Downloaded by [VUL Vanderbilt University] at 20:59 15 December 2012

J. L. RIVERA’, J. ALEJANDRE’’3*, S. K. NATH4 and J. J. DE PABL02*3 I Departmento de Quimica, Universidad Autonoma Metropolitana-Iztapalapa, Apdo, Postal 55-534; 09340 Mlxico D.F., Mtxico Department of Chemical Engineering, University of Wisconsin-Madison, Madison, WI 53706-1691, USA 3 . Simulacion Molecular Instituto Mexican0 del Petroleo, Eje central Lazaro Cardenas 152, Apdo. Postal 14-805, 07730 Mtxico D.F., Mlxico 4 Molecular Simulations Inc., 9685 Scranton Road, San Diego, CA 92121, USA (Received 16 March 1999; revised version accepted I7 August 1999)

A force field has been developed to describe the phase behaviour, interfacial, and transport properties of nitrogen and hydrocarbon mixtures under conditions relevant to those found in the high pressure extraction of oil from underground reservoirs. A Gibbs ensemble Monte Carlo method is used to parametrize intermolecular potentials for the pure components by matching experimental and simulated liquid and vapour coexisting densities. Also the surface tension, diffusion coefficient and shear viscosity of nitrogen and its mixtures with butane have been determined. The latter properties were obtained by canonical molecular dynamics simulations. The diffusion coefficient and shear viscosity were calculated by a Green-Kubo method. Results for pure nitrogen are given for temperatures ranging from 7 0 K to llOK. For mixtures of nitrogen with butane, results are presented at 339.4 K and 380.2 K. Good agreement is found between the results of simulations and available experimental data.

1. Introduction Injection of nitrogen into oil reservoirs [ 1, 21 has been considered as a way to maintain high pressure in oil reservoirs and permit extraction of oil. Nitrogen-hydrocarbon mixtures are becoming more and more relevant as primary oil resources become depleted. An increasing number of wells require that pressure be preserved to sustain current levels of production; this can be achieved by natural gas or nitrogen injection. As nitrogen is fed into a reservoir, a number of different scenarios can arise. Depending on pressure, temperature, and composition, several liquid and vapour phases can coexist. From an oil recovery point of view, some of these phases are more attractive than others (e.g., they are richer in hydrocarbons); it is therefore important to know the phase boundaries, compositions and densities of nitrogen-hydrocarbon mixtures so that a well can be operated under conditions which facilitate efficient recovery. Unfortunately, experimental thermodynamic and transport property data for such mixtures are scarce and, in the absence of adjustable parameters, equation-of-state predictions are only qualitatively correct. A predictive, quantitative method for the predic-

* Author

for correspondence. e-mail: [email protected]

tion of the thermodynamic and transport properties, including phase behaviour, of nitrogen-hydrocarbon mixtures would be of significant value to the oil industry. Conducting controlled experiments under conditions relevant to oil extraction is time consuming and expensive. While it is important that such experiments be carried out, it would be useful to complement them with a predictive theoretical tool. This is a field where computer simulations could have a significant impact; therefore it is timely to investigate whether or not computer experiments can be used to describe quantitatively the behaviour of nitrogen-hydrocarbon mixtures under extreme thermodynamic conditions. Some thermodynamic and transport properties, including vapour-liquid (VLE) and liquid-liquidvapour (LLVE) equilibria, involving nitrogen and small hydrocarbons, have been measured experimentally at high pressures [3-71. These data have been used to evaluate several engineering equations of state and their corresponding mixing rules. It has been argued that simultaneous determination of both compositions and densities of coexisting phases poses a more stringent test for such models and mixing rules than analysis of compositions alone.

Molecular Physics ISSN 00268976 print/ISSN 1362-3028 online 0 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals/tf/00268976.html

J. L. Rivera et al.

44

Table 1 . Potential parameters used in equations (1H3).

N-N = 1.089: A C--C = 1.54 A

Bond distances

ks = 65 000 K rad-2 and 0, = 114.0

Bond angles Torsion potential


United atom CH3-CH3 CH24H2 N-N

Co = 1116K C3 = -368 K

C , = 1462K C, == 3156K

o/A 3.901 3.93 3.31

(&/4/K 104 45.8 36.0

Recent developments in the field of molecular simulation have facilitated significantly the calculation of thermodynamic properties of realistic systems. The Gibbs ensemble Monte Carlo (GEMC) and its extensions [8, 91, canonical molecular dynamics (CMD) [lo] and Gibbs-Duhem [I I] methods are only a few of the techniques that allow direct simulation of coexistence diagrams. Furthermore, recent force fields have been shown to be capable of accurate predictions of phase equilibria for a variety of hydrocarbon mixtures [12]. It is therefore reasonable to expect that simulations could provide an accurate description of many of the mixtures encountered in the pressurization of reservoirs with nitrogen. We address this issue by presenting a systematic investigation of the phase behaviour and transport properties that can be expected from a molecular simulation approach. 2. Potential models Throughout this work, a united-atom representation is adopted to describe nitrogen and butane. For butane, bond angle restrictions are introduced by a quadratic potential energy function of the form

where 6 is the bond angle, subscript 0 is used to denote its equilibrium value, and ke is a spring constant. For butane, the torsion around a carbon+arbon bond is governed by a torsional potential energy of the form

c ci 5

U ( 4 )=

cosi 4,

i=O

with constants Ci taken from the work of Ryckaert'and Bellemans [ 131. Interaction sites located on different molecules interact through a Lennard-Jones potential energy function. The parameters for alkane interactions were taken from the literature [12], and those for nitrogen were derived in this work by taking as an initial guess those used by

C2 = -1578K Cs = -3788K

Cheung and Powles [14]. Interactions between atoms from different molecules are thus given by a spherical truncated potential of the form

where [T is the diameter of the united atom, E is a measure of the strength of the interaction, r is the distance between any two atoms, and R, is the cutoff radius. Unlike interactions were handled by means of conventional Lorentz-Berthelot mixing rules. The parameters used in equations (1H3) are given in table 1. At this point we emphasize that, depending on the property and simulation method of interest, slightly different attractive-tail functional forms and cutoffs were occasionally employed for Lennard-Jones interactions. It is also important to note that, for simplicity, in canonical molecular dynamics (CMD) simulations bond distances were kept constant at their corresponding equilibrium values. 3. Simulation methods Depending on the property of interest, a number of different simulation methodologies were employed in this work. Coexistence densities and compositions were determined by means of hybrid Gibbs ensemble simulations (GEMC) [IS, 161, and with CMD. We have shown that CMD and canonical Monte Carlo (CMC) are equivalent to GEMC when the same potential model is used [171. Surface tensions were determined by molecular dynamics simulations of interfaces. Transport properties such as viscosities or diffusion coefficients were obtained from conventional, canonical molecular dynamics simulations and Green-Kubo methods [18]. Details concerning the hybrid GEMC method can be found in the literature [15, 161; only a few essentials are provided here. Molecules are displaced using the

45



velocity version of the Verlet molecular dynamics algorithm. Such displacements are accepted according to modified Metropolis criteria. In order to conduct simulations at constant pressure and temperature, the volume of coexisting simulation boxes and their particle numbers are allowed to fluctuate. The potential was cut off at lOA and standard tail corrections were implemented [ 181. Molecular dynamics simulations of interfaces and bulk phases were carried out by means of a conventional Verlet position algorithm [18]. Note, however, that all of our MD simulations are conducted using a cut and shifted potential. Consequently, the results of such calculations are not entirely equivalent to those of GEMC simulations using a truncated potential with complete long range corrections. However, to facilitate comparison between the results of these two methods, we have attempted to shift the potential at relatively long values of distances (e.g., 40). Liquid-vapour interfaces were simulated using CMD by constructing a liquid slab in the centre of a parallelepiped of dimensions L, = L, = 28.54 A and L, = 85.62A, for pure nitrogen, and L, = L, = 32.42A and L, = 209.27A for mixtures of nitrogen and butane. This is a common practice when setting up an initial configuration for the study of phase equilibria by means of an interfacial method [lo]. To approximate the full potential as close as possible, we truncated inter-

140

actions at R, = 14.2A for pure nitrogen and at R, = 16.2 A for mixtures. Interface simulations were carried out at constant temperature, with a timestep of 4.0fs. For pure nitrogen 500 molecules were used at temperatures ranging from 70 K to 1 10K, while for mixtures, 500 molecules of butane were mixed with changing amounts of nitrogen ranging from 75 to 425 molecules in order to cover a complete range of concentration. To investigate system size and truncation effects on orthobatic densities and surface tension, additional simulations of pure nitrogen were conducted using the GEMC and CMD methods at 70 K and 100 K. Systems with 1000 molecules and R, = 16.5 A were used for the CMD calculations. The GEMC simulations were carried out with 400 molecules and a cutoff distance of 13.2 After 50 000 equilibration steps, average thermodynamic properties were determined from an additional 200000 steps for nitrogen and 400000 steps for mixtures. Transport properties were determined from CMD simulations of bulk liquid and their average values were obtained for 150000 steps for nitrogen and mixtures. The equilibration step involved 50000 steps in both cases. To assess the validity and correctness of our models and codes, for pure nitrogen, simulations of diffusion were conducted with 500 molecules for thermodynamic states for which literature simulation data

A.

I

,

I

,

0.2

0.4

0.6

0.8

120

Y F 100

Figure 1 . Coexistence curve for nitrogen. The line represents experimental data [19]; open circles ( N = 300) and open squares ( N = 400) are results of hybrid GEMC simulations; and black circles ( N = 500) and black squares ( N = 1000) correspond to CMD results.

80

60

0.0

1 .o

46

J. L. Rivera et al.

are available. Equimolar mixtures of nitrogen and butane with a total number of 864 molecules were simulated at 339.4 K for different densities using R, = 15.72 A. The timestep employed for these calculations was 1.O fs. The surface tension was determined from interface simulations using the Irving and Kirkwood definition for the components of the pressure tensor in the molecular version [lo], i.e.,

direction. The inclusion of two surfaces in the simulation is taken into account by the factor The profile of the tangential component of the pressure is given by

i.

where A is the surface area of one interface, p ( z ) is the molecular density profile along the z direction, k is Boltzmann's constant, T is the temperature, and @(x) where PN(z) and PT(z) are profiles of the normal and is the step function. Subindices i and j refer to molecules tangential components of the pressure tensor, respectand a and b to atoms in a molecule. The distance riajb is ively, and L, is the size of the simulation box in the z between atom a in molecule i and atom b in moleculej. The normal component, PN(z),is given by an expression Table 2. Results for pure nitrogen. Liquid and vapour coexof the form of equation (5) with ZijZiajbreplacing isting densities as a function of temperature and calcu(xijxiajb+ yijyiajb/2. lated vapour pressure and surface tension as a function The diffusion coefficient and shear viscosity were of temperature. The densities are given in units of g ~ m - ~ , obtained from Green-Kubo equations [ 181 for the veltemperature in K, pressure in bar, surface tension in ocity autocorrelation function and the off-diagonal comdynecm-l . Simulation results are indicated using CMD or GEMC while experimental results are indicated with ponents of the pressure tensor autocorrelation function, Exp. respectively. The diffusion coefficient D was determined from the Pv Pv PL PL velocity autocorrelation function according to T GEMC EXP ~ 9 1 GEMC Exp [I91


(4)

roo

70" 77.2 90 96.5 100' 106.1 114 120

3.OE-03 4.OE-03 1.6E-02 2.9E-02 3.5E-02 4.7E-02 8.5E-02 1.3E-01

1.8961E-03

Pv

Pv

PL

PL

T

CMD

EXP 1191

CMD

Exp [I91

T

0.807 0.814 0.765 0.707 0.656 0.668 0.530

0.838 54 0.838 54 0.792 37 0.745 01 0.689 23 0.689 23 0.621 44

PV

Y

Y

CMD

CMD

1.5077E-02 3.2089E-02 8.1587E-02 1.2509E-01

~

70 70b 80 90 100 100b 110 T

2.78E-03 1.95E-03 6.85E-03 1.85E-02 4.25E-02 4.03E-02 7.78E-02

0.827 0.802 0.739 0.710 0.677 0.643 0.585 0.536

1.8961E-03 1.8961E-03 6.1356E-03 1.5077E-02 3.2089E-02 3.2089E-02 6.2578E-02

0.838 54 0.74501

D

'N

0.582 0.557 1.749 3.829 9.956 10.228 17.077

0.3847 0.3847 1.3659 3.5969 7.7664 7.7664 14.6285

= 400, R, = 13.2 Ad N = 1000, R, = 16.5 A .

(V(t) . V(0)) dt

where V is the velocity of the centre of mass of a molecule and t is the time.

0.689 23 0.587 98 0.523 39

Table 3. Transport properties for pure nitrogen: diffusion coefficient in cm2s-l) and shear viscosity (in gcm-' s- 1.

C

D/1OK5 M D [I41

D/10-5

PL

106.7 57.8 111.2 98.5

0.8993 0.8993 0.7054 0.7054

2.55 0.98 7.02 6.08

2.77 0.91 7.59 6.60

90

0.8290 0.7896 0.7537 0.7909 0.7498 0.6992 0.7637 0.7202 0.7453 0.7236 0.6406 0.5243

1.5011 1.1958 0.9485 1.1869 1.1520 0.8783 1.1237 0.9386 0.6386 0.9665 0.5955 0.3779

1.6694 1.2893 1.9345 1.2537 1.0173 0.7896 1.0845 0.8636 0.6167 0.8708 0.6008 0.3919

CMD

~

EXP

POI

~

70 70b 80 90 100 1OOb 110

f

= JO

10.9 11.5 8.3 6.2 3.9 4.6 1.6

10.5339 10.533 9 8.3126 6.191 3 4.191 8 4.191 8 2.350 3

100

110

120

47

Thermodynamic and transport properties of nitrogen and butane mixtures 12.5

10.0

E

7.5

,o 0) C

%

0

Y


%

5.0

2.5 Figure 2. Surface tension of nitrogen. The continuous line shows exDerimental data [20], and -the black circles ( N = 500) and black squares ( N = 1000) are CMD results.

i 0.0

I

I

I

I

I

TIK

2.0

I

I

I

I

1.5

0.5 Figure 3. Shear viscosity of pure nitrogen. The continuous and dashed lines are experimental results [21] at T = 90 K and 120 K, respectively. The black circles are CMD results.

0.0 0.5

I

0.6

I

0.7 P /(g/cm3)

I

0.8

0.9

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J. L. Rivera et al. 300

I

I

250 /

/

/'

I

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0

-

0

a

200

z


g 150 a 100

50

0' 0.0

1

I

I

0.1

0.2

0.3

I

I

0.5

0.4

P /(g/cm3)

300

200 Figure 4. (a) Vapour pressure as a function of liquid and vapour densities for nitrogen-butane mixtures; (b) vapour pressure as a function of molar fraction of nitrogen in the vapour and liquid phases. The temperature is 339.4 K. Open squares are for experimental data [3]; open and filled circles are simulation results obtained using hybrid GEMC and CMD methods, respectively; and dashed line is results of Peng-Robinson equation of state [22].

z

$ 100

/

II

I bLm

0'

1

I

I

Mole fraction of nitrogen

I

0.6


49

150

100

/

\ \ \

/

z

s

\

w

50


\a

PI /-

/

/

/

m

/

\ \ \ \

\m \

\w ‘D

0 P /(s/cm3)

150

100

Figure 5. (a) Vapour pressure as a function of liquid and vapour densities for nitrogen-butane mixtures; (b) vapour pressure as a function of molar fraction of nitrogen in the vapour and liquid phases. The temperature is 380.2 K. Open squares are for experimental data [3]; filled circles are simulation results obtained using CMD; and the dashed line is results of a PengRobinson equation of state [22].

z

9

2

50

0 0.0

I

0.2

1

0.4 Mole fraction of nitrogen

I

0.6

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J. L. Rivera et al.

The shear viscosity 77 was evaluated using the stress autocorrelation function with off-diagonal elements a and ,B of the components of the pressure tensor Pap, i.e., 1 " (7) 77 = I/kTJ-@P r n d W < Z d O ) ) dt7


where V is the volume of the system. Components ap of the pressure tensor are given in terms of their centre-ofmass definition.

where a = x, y , z and p = x , y , z , with a # ,B, Via is the velocity of the centre of mass of molecule i in the (Y direction, (rii)ais the distance between molecules i and j in the a direction and (FiUjb)@ is the force that atom a in molecule i exerts on atom b on molecule j in the p direction.

3. Results and discussion 3.1. Pure nitrogen Figure 1 shows results of vapour-liquid simulations of nitrogen using the GEMC and CMD methods. The results are also given in table 2. As pointed out earlier, CMD calculations were conducted with a shifted force field; their results are therefore slightly different from those of GEMC simulations, which were conducted with the full force field (through calculation of cutoff corrections). For the full force field, agreement with experimental VLE data [19] is highly satisfactory. The uncertainties in coexisting densities in both GEMC and CMD calculations are smaller than the symbol size in figure 1. Figure 2 shows CMD results for the surface tension of nitrogen. Long range corrections (LRC) were applied at the end of the simulation according to the procedure described in [lo]. The LRC contributions are less than 8% of the values obtained directly in the simulation. Agreement with experiment is also satisfactory. Our nitrogen force field, which was derived by fitting the experimental coexistence curve, slightly overpredicts the surface tension. Simulation results for vapour pressure and surface tension are given in table 2. Calculation of transport coefficients by means of Green-Kubo methods requires integration of various autocorrelation functions. Such functions can exhibit fairly long tails, and it is therefore important to observe a complete decay during the range of calculations. The results obtained in this work are given in table 3. These calculations were performed using the cut and shifted potential. Correlations do decay to zero within a few picoseconds and our Green-Kubo estimates of viscosity are therefore expected to be reliable. Figure 3 shows

results for the shear viscosity of nitrogen obtained by integrating the stress autocorrelation function. For the two temperatures examined in this work (90 K and 120 K), agreement with experiment is satisfactory. At elevated densities our model appears to underpredict the experimental data slightly [21], but the deviations from experiment are less than 10%. Table 3 also presents results for the diffusion coefficient of liquid nitrogen in the temperature range 57.8 K < T < 11 1.2 K and at two densities, namely p = 0.7054 gcmP3 and 0.8993 g cmP3. These conditions were selected to compare our results to those of earlier simulations by Cheung and Powles [14]. In general, the differences between our results and those of these Table 4. Results for mixtures between nitrogen and butane (units are the same as those specified in tables 2 and 3). Vapour pressure, coexisting densities and nitrogen molar fractions in the liquid and vapour equilibria at 339.4 K calculated using the GEMC technique.

339.4

34.35 68.26 135.96 193.88

0.046 0.087 0.180 0.250

0.510 0.509 0.508 0.470

0.065 0.135 0.250 0.390

0.720 0.810 0.820 0.790

Vapour pressure, coexisting densities and nitrogen molar fractions in the liquid and vapour equilibria at 339.4K and 380.2K. The surface tension is given for some states at T = 339.4K. The results are calculated using the CMD method.

339.4

380.2

28.73 41.96 58.99 83.29 95.69 125.93 139.31

0.0388 0.0551 0.0707 0.1050 0.1174 0.1537 0.1795

0.5088 0.5103 0.5088 0.5128 0.5039 0.5014 0.5010

0.0415 0.0724 0.0884 0.1348 0.1810 0.2141 0.2510

0.6626 0.7215 0.7808 0.8044 0.8202 0.8116 0.7949

27.65 42.94 70.44

0.0527 0.0704 0.1203

0.4546 0.4528 0.4336

0.0140 0.0479 0.1403

0.2171 0.4338 0.5112

6.4 5.4 4.0 3.3 2.3

Transport properties for equimolar mixtures in the liquid phase. Diffusion coefficients and shear viscosities are given as a function of density. T 339.4

pL .

0.63 0.74 0.85 0.96 1.08

D/10-4 1.8170 1.094 1 0.63337 0.362 10 0.235 33

17/10p3 1.2111 2.0335 4.7371 11.778

Thermodynamic and transport properties of nitrogen and butane mixtures 0.6

I

I

51

I

0.4 h

3

.

3


Q

0.2

0.0

I

0

50

L~~

-

I

100 z /Angstroms

1

150

200

100.0

Figure 6. ( a ) Density profiles of nitrogen and butane mixtures at 339.4 K . The bold face curves show results for butane and the thinner lines are for nitrogen. The vapour pressures are 28.7 bar and 95.7 bar. At 9 5 . 7 K the liquid density of butane is the lowest and the liquid density of nitrogen is the highest. (b) Difference of the normal and tangential components of pressure as function of z obtained using CMD for a mixture of nitrogen and butane. The temperature is 339.4K and the vapour pressure is 28.7 bar.

50.0

z

e

h

N

v

L--

a' I h

N

Y

nz

0.0

-50.0

z /Angstroms

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J. L. Rivera et al. 8

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h

E

Y

g4

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t

2

Figure 7. Surface tension as a function of vapour pressure for nitrogen and butane mixtures at 339.4 K.

0

0

authors are about 10%. Unfortunately, we have been unable to find experimental data under these conditions to further assess the accuracy of our predictions.

3.2. Nitrogen-hydrocarbon mixtures Figure 4 (a, b) shows coexistence densities and compositions for nitrogen-butane mixtures at T = 339.4 K. Figure 5 ( a , b) shows similar results at T = 380.2 K. The results obtained using hybrid GEMC and CMD methods are given in table 4. The GEMC calculations were performed using the truncated potential with cutoff corrections. The solid line represents the experimental data of Malewski and Sandler [3]. Both densities and compositions are in good agreement with experiment. It is important to emphasize that no adjustable binary parameters were employed for these predictions. In order to compare simulation predictions with those of commonly used engineering methods, figures 4 ( a ,b ) and 5 ( a , b ) also show the predictions of a Peng-Robinson equation of state [22] for the density and composition of the coexisting phases. The necessary pure component parameters for the equation of state were determined from critical constants for nitrogen and butane. The Peng-Robinson predictions for composition are quantitatively correct. However, as can be seen from these figures, in the absence of adjustable binary parameters, the predictions of molecular simulations for density are

100

50

150

Plbar

superior to those of a prototype engineering equation of state. Figure 6 ( u ) shows density profiles corresponding to two different pressures observed during the course of interface CMD simulations of saturated nitrogenbutane mixtures at 339.4 K (the average vapour pressures are 28.7 bar and 95.7 bar, respectively). The bold face curves show results for the density of butane across the interface at these two pressures, and the thinner lines show the corresponding profiles for nitrogen in the mixtures. These calculations were performed using the cut and shifted potential. At the higher pressure, the concentrations of nitrogen in the liquid and vapour phases are higher; the concentrations of butane are correspondingly smaller. While the total density of the liquid phase remains essentially the same at these two pressures, that of the vapour is significantly higher at the higher pressure. Note, however, that the actual density of butane in the liquid phase is smaller at the higher pressure. The results are also given in table 4. The density profiles across the interface are well defined, particularly those for butane, and permit accurate determination of equilibrium densities. These profiles also serve to illustrate that our systems are sufficiently large for our purposes. It is interesting to point out that the nitrogen density profile in the film undergoes a gradual transition from its vapour phase value to its lower, liquid phase value. This is seen


53

2.0


1.5

0.5 Figure 8. Diffusion coefficients of nitrogen in mixtures with butane. The filled circles are results obtained using CMD, and the line is a guide to the eye.

0.0 0.6

more clearly at the pressure of 95.7 bar. The decay from vapour to liquid values occurs in a penetration zone of about 20A from each side of the film. The ‘bulk’ value of the film is about 70 wide. For this system, the extent of nitrogen condensation on the surface of the film is minimal and almost nonexistent. This is in contrast to other mixtures, such as water-alcohol, where surface condensation has been clearly observed. Figure 6 ( a ) shows results of simulations for the difference between the normal and the tangential components of the pressure tensor for the mixture discussed in figure 6 ( a ) at 28.7 bar. This figures exhibits two pronounced peaks corresponding to the liquid-vapour interfaces of the film; as expected from a stable film, the pressure differences in the interior of the film and in the vapour phase are both equal to zero. We have verified that the tangential and normal pressures are also equal in the bulk vapour and liquid phases. This pressure difference is used to determine the surface tension according to equation (5). Bulk phases do not contribute to the surface tension. Figure 7 and table 4 give the surface tension of saturated nitrogen-butane mixtures at 339.4K as a function of pressure. As expected for this system, the surface tension of the mixture decreases almost linearly with pressure. While experimental data for this particular system are not available, it is interesting to compare our results to those of Thomas et al. [l], who have measured surface tensions

A

for mixtures of nitrogen with various types of commercial oil; the range of surface tensions reported by these authors is comparable with that predicted by our simulations. The velocity autocorrelation function, not shown, decreases monotonically to zero at low densities. At high densities, it undergoes a change of sign and then increases to zero after about 1.5 ps. The corresponding diffusion coefficients for nitrogen in equimolar mixtures with butane are shown in figure 8 as a function of density. As expected, the diffusion coefficient decreases as the density increases. The transport properties for mixtures are given in table 4. The stress autocorrelation functions for mixtures are significantly noisier than those for velocities, but they are sufficiently clear to yield reasonable estimates of the viscosity of nitrogen-butane mixtures. The results for the shear viscosity shown in figure 9 for nitrogenbutane as a function of density indicate that viscosity increases by an order of magnitude when the density is raised from 0.63 g cmP3 to 0.96 g cmP3. 4. Conclusion Our calculations indicate that molecular simulations with simple force-fields, conventional Lorentz-Rerthelot combining rules, and without binary adjustable parameters, provide accurate coexistence density and compositions for the nitrogen-hydrogen mixtures studied in

54

J. L. Rivera et al.

12.5

-i

I


10.0

2.5 Figure 9. Calculated shear viscosity of nitrogenbutane mixtures. The filled circles are results obtained using CMD, and the line is a guide to the eye.

0.0

0.6

this work. It is worth emphasizing that for oil-recovery applications it is important to generate reliable density predictions. Our results and those of others suggest that, in the absence of multiple adjustable parameters, simple engineering equations of state are unable to predict quantitatively the orthobaric densities of the mixtures studied in this work; molecular simulations, however, permit their reliable prediction. In the case of surface tension, the predictions of molecular simulations appear to be reasonable. There is, however, a significant lack of experimental data and it is therefore difficult to establish unambiguously the accuracy of our calculations. Our predictions, however, are consistent with experiments on oil-nitrogen mixtures. As pointed out above, engineering correlations for surface tension rely on extrapolation of pure component data over wide ranges of temperature and pressure. For mixtures, such correlations require use of empirical mixing rules whose validity has not been established; in general, estimates obtained by that route should be regarded with caution. The predictions of simulations are based on well defined models and offer a consistent approach for this type of calculation. The transport properties calculated in this work are also consistent with limited available experimental data. Measurement of these properties at elevated pressures and temperatures is demanding; engineering correla-

0.7

0.8

0.9

1 .o

P /(4/cm3)

tions for their prediction suffer from the limitations discussed above. We therefore expect simulations to provide a useful, consistent alternative method for their prediction.

J.L.R. and J.A. thank CONACyT for financial support and the Mexican Petroleum Institute for computing time. This work was also partially supported through a PECASE award from the National Science Foundation to J.J.dP., who is particularly grateful to Gustavo Chapela for his hospitality during a sabbatical visit to the Mexican Petroleum Institute. References [I] THOMAS,L. K., DIXON,T. N., PIERSON, R. G . , and HERMANSEN, H., 1991, Ekofisk nitrogen injection, SPE Formation Evaluation, June, p. 151. [2] PEMEX. Boletin 300/96; September 1996. [3] MALEWSKI, M. K. F., and SANDLER, S. I., 1989, J. chem. Eng. Data, 34, 424. [4] CHEN,W., LLAVE,F. M., LUKS,K. D., and KOHN,J. P., 1989, J. chem. Eng. Data, 34, 233. [5] SHIBATA, S. K., and SANDLER, S. I., 1989, J. chem. Eng. Data, 34, 291. [6] CHEN,W., LUKS,K. D., and KOHN,J. P., 1989, J. chem. Eng. Data, 34, 312. [7] KILLIE,S., HAFSKJOLD, B., BORGEN,O., RATKJE,S. K., and HOVDE,E., 1991, AIChE J., 37, 142. [8] PANAGIOTOPOULOS, A. Z., 1987, Molec. Phys., 61, 813.



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