Molecular Simulation Prediction of Sound Velocity for a Binary Mixture

Ind. Eng. Chem. Res. 2010, 49, 3469–3473

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Molecular Simulation Prediction of Sound Velocity for a Binary Mixture near Miscible Conditions Babak Fazelabdolabadi and Alireza Bahramian* Department of Chemical Engineering, Faculty of Engineering, and Institute of Petroleum Engineering, UniVersity of Tehran, Tehran, 11155-4563, Iran

The speed of sound was computed via Monte Carlo simulation for a binary mixture of methane and n-butane in the subcritical, near-critical, and supercritical regions at the temperature of 311 K. The technique encompasses a sequential implementation of the isobaric-isothermal and canonical ensembles in a simulation box, in which the (residual) thermodynamic derivative properties are evaluated via the fluctuation method [Escobedo, F. A. J. Chem. Phys. 1998, 108, 8761] during the Monte Carlo moves. A united atom Lennard-Jones potential with parameters proposed by Mo¨ller et al. [Mo¨ller, D. et al. Mol. Phys. 1992, 75, 363] was chosen to represent methane. In case of n-butane, we employed an optimized anisotropic united atom intermolecular LennardJones description of Ungerer et al. [Ungerer, P. et al. J. Chem. Phys., 2000, 112, 5499]. In decent agreement with the experiment, we find the technique to provide reasonably quantitative estimates for the speed of sound in the thermodynamic regions studied. 1. Introduction The information of fluid responses to sound waves (i.e., sound velocities) has applications in many areas, including monitoring of the enhanced oil recovery process,1,2 determination of the hydrocarbon saturation in reservoir rocks,3 and modeling.4–7 Sound velocity information is particularly important for within fluid characterization; as such a quantity can be related to the other thermophysical quantities in the fluid via thermodynamic concepts.8 Such knowledge of the fluid characteristics is typically hard to attain in near-critical conditions, in which the precision of the classical fluid models is diminished, owing to thermodynamic anomalies.9 Knowledge of the sound velocity at such critical conditions may be helpful in characterizing the thermophysical behaviors of the system, as well as to serve as an indicator of the state and structure of the fluid in such thermodynamic regions with essentially little experimental data availability. The speed of sound may be related to second-order derivates of thermodynamic potentials (Gibbs energy, Helmholtz energy, enthalpy, and internal energy), whose determination shall be of major difficulty in thermodynamics, owing to deficiency of the available equations of state in accurate prediction of secondorder derivative quantities, albeit their fairly good ability in estimation of properties of the first-order derivative group (i.e., phase equilibria calculations).10 Proximity to the critical state may affect several second-order thermodynamic properties in pure compounds.11–13 Experimental observations, for instance, report a phenomenon of divergence of the isochoric specific heat for argon at the critical point,11 and a reduction in sound velocity in the supercritical steam.12 Nevertheless, such behaviors should be less pronounced in the case of mixtures.14 The present study focuses on the Monte Carlo prediction of the sound velocity in the vicinity of the critical region for a binary mixture representing a real reservoir fluid. While there are plenty of data on the sound velocity of atmospheric binary and ternary mixtures, the work of Plantier et al.14 seems to be the only available data in the literature which provide speed of * To whom correspondence should be addressed. Fax: +98 21 88632976. Tel.: +98 21 61114715. E-mail: [email protected].

sound measurements for fluids representing real reservoir hydrocarbons in near-critical conditions. In a recent article,15 we offered a Monte Carlo computational route for prediction of sound velocity in some pure compounds. The evaluation of sound velocity by our approach shall entail simultaneous determination of other thermodynamic secondorder derivative quantities (i.e., thermal expansivity, isothermal compressibility, heat capacity) for the conditions involved.15 Quantitative prediction of such second-order derivative properties may be accomplished by applying the statistical fluctuation theory16 to analyze the results obtained within a single simulation run, which would essentially keep the required computational time within reasonable length. This theory states that the derivative of a property X with respect to an intensive variable Y is related to the fluctuations of 〈XZ〉 - 〈X〉〈Z〉 in the considered statistical ensemble, in which Z is the conjugate extensive variable of Y.17 Direct computational estimates may therefore be established via this approach for the isothermal compressibility and thermal expansivity, which are derivative properties with respect to pressure and temperature, respectively. The numerical evaluation of heat capacity was attempted via a more elegant approach. Indeed, the kinetic part of energy, which contributes to the value of heat capacities, cannot be computed via the two simulation ensembles considered herein. For this reason, the recommended approach17 suggests the heat capacity being evaluated as the sum of a residual heat capacity and an ideal heat capacity, which are obtained from the Monte Carlo simulation and the experimental correlations, respectively.15 In the section to follow, we will mention some basic equations to be used in the present analysis. A third section is devoted to an explanation of our simulation method, in which we shall mention the intermolecular as well as intramolecular parameters employed for the current molecular simulations. A fourth section is dedicated to mention our Monte Carlo simulation results, ensued by our concluding remarks. 2. Theoretical Background The basic relationship to compute the speed of sound (c) in thermodynamics relates this value in a fluid to the isentropic compressibility.18 Nevertheless, an essentially equivalent equal-

10.1021/ie901422a  2010 American Chemical Society Published on Web 03/04/2010

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ity may be established to express this quantity in terms of the isothermal compressibility as follows: c2(T, P) )

Cp(T, P) υ(T, P) CV(T, 〈V〉) MβT(T, P)

(1)

Table 1. Parameters of the CH4, CH3, and CH2 Groups group b 4 c 3 c 2

CH CH CH a

with M being the molar mass, υ being the molar volume, and βT being the isothermal compressibility of the fluid. Also, CP and CV are the molar heat capacities at constant pressure and constant volume, respectively. The above equality provides generally identical results as for the sound velocity compared to the experimental values18 and, thus, will serve as the basis for our computational analysis in the present work. As a consequence, our methodology should entail simultaneous determination of the heat capacities as well as the isothermal compressibility, which are obtainable by analyzing the fluctuations in the simulation results in their corresponding statistical ensembles.15 The fluctuation formula used to compute the ensemble average of a property (X) under isothermal-isochoric ensemble may be established analytically.19 For a binary mixture, composed of N1 (rigid) molecules of type 1 and N2 molecules of type 2, this can be written as follows: 1 ∆

〈X〉 )

N1+N2

∫ N !NXV!Λ 1

2

3N1 3N2 1 Λ2

ˆ ) dV dζn exp(-βH

(2)

with ∆)

N1+N2

∫ N !NV!Λ 1

2

3N1 3N2 1 Λ2

ˆ ) dV dζn exp(-βH

(3)

Here, Λi is the de Broglie wavelength of the molecule i, ζ is the dimensionless degree of freedom, n is the number of degrees ˆ ) Uext + Uint of freedom, V is the volume of the system, and H ext + PV is the configurational enthalpy, with U and Uint being the intermolecular and intramolecular potential energies, respectively. Since the derivatives of the de Broglie wavelength with respect to temperature cancel out,18 the following equality may be established for the isothermal-isobaric ensemble:

(

∂〈X〉NPT ∂β

)

P

ˆ 〉NPT - 〈XH ˆ 〉NPT ) 〈X〉NPT〈H

(4)

here β ) 1/kT, with k being the Boltzmann constant and T being the temperature of the system. Here, the subscript to the brackets refers to the simulation ensemble under which the average properties are recorded. The thermal expansivity (RP) and the isothermal compressibility βT may hence be computed via17 RP )

1 〈V〉NPTkT2

βT )

ˆ 〉NPT - 〈V〉NPT〈H ˆ 〉NPT) (〈VH

1 (〈V2〉NPT - 〈V〉NPT2) 〈V〉NPTkT

(5)

(6)

As discussed in our earlier paper,15 a more elegant approach should be devised for computation of the heat capacity in the present analysis, as the kinetic energy contribution to this quantity may be neglected through the two simulation ensembles considered herein.15 For this sake, the residual heat capacity term will be complemented with an ideal heat capacity term,

σ (Å)

ε/k (K)

δa (Å)

molecular weight (g/mol)

3.7327 3.6072 3.4612

149.92 120.15 86.29

0 0.21584 0.38405

16.04 15.03 14.03

The carbon-to-force center distance. b Reference 24. c Reference 25.

the latter of which will be responsible for inclusion of the kinetic energy contribution to the heat capacity quantity. On the basis of the fluctuation analysis, the expression for the residual isobaric heat capacity is established as15 Cres P )

1 ˆ 〉NPT - 〈Uext〉NPT〈H ˆ 〉NPT) + P〈V〉NPTRP - Nk (〈UextH kT2 (7)

A similar approach yields the following expression for the residual isochoric heat capacity: Cres V )

1 ˆ2 ˆ 〉NVT2) (〈U 〉NVT - 〈U kT2

(8)

ˆ ) Uext + Uint is the total system energy during where U canonical Monte Carlo simulation. As discussed earlier,15 the isochoric heat capacity (CV) can alternatively be established via the Histogram-reweighing grand canonical Monte Carlo technique,20 or other such recent techniques as the transition matrix Monte Carlo,21 and the Wang-Landau sampling.21 Nevertheless, the bulk of computations required to reach this specific quantity via such (postprocessing) techniques may be enormous compared to the canonical ensemble; hence, we choose the canonical ensemble for the isochoric heat capacity computation in our present analysis. Complemented with an ideal heat capacity contribution at the same temperature in the corresponding ensemble,22 the molar values for the heat capacities can be retrieved by multiplying the corresponding computed values by the ratio of the Avogadro’s number to the number of particles involved in the simulation. Substitution in eq 1 results in a fluctuation formula utilized for evaluation of sound velocity.15 A pseudoensemble may therefore be devised, in which a sequential implementation of the isobaric-isothermal and canonical Monte Carlo simulations are attempted, the latter of which will borrow its value for the system volume from the ensemble average of the latter ensemble. 3. Simulation Methods 3.1. Molecular Models. The intermolecular energy between two interacting sites on different molecules was described by the Lennard-Jones potential as

[( ) ( ) ]

ULJ(rij) ) 4εij

σij rij

12

-

σij rij

6

(9)

where rij denotes the distance between interacting site, σij and εij are the exclusion diameter and the energetic parameter, respectively. The Lorentz-Berthelot combining rule23 was used to estimate the crossed interaction parameters between different CHi groups. The united atom potential (UA) representation of Mo¨ller et al.24 was used for the case of methane, with parameters listed in Table 1. An anisotropic united atom (AUA) potential representation with optimized parameters of Ungerer et al.25 was utilized for the case of n-butane, along with the intramolecular bending potential (Ubend) as well as the torsion potential (Utors) energies evaluated through the following relationships

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010 Table 2. Parameters for the Anisotropic United Atom Model for n-Butanea

a

torsion

constants (K J/mol)

a0 a1 a2 a3 a4 a5 a6 a7 a8

8.32556 17.70555 -2.51974 -30.03364 18.51362 18.34541 -37.32590 -14.43552 23.42457

Reference 25 (Kbend ) 622.75 K J/mol, θ0 ) 114°).

with the parameters listed in Table 2.25–27 The carbon-carbon separation distance in n-butane was considered at a value of 1.535 Å. 1 Ubend(θ) ) Kbend(cos(θ) - cos(θ0))2 2

(10)

8

∑ a (cos(φ))

Utors(φ) )

i

(11)

i

i)0

3.2. Monte Carlo Algorithms. Similar to our previous analysis of the subject,15 we employed a combination of the configurational-bias Monte Carlo technique27–31 within a sequence of isobaric-isothermal and canonical ensembles in order to predict the speed of sound in a mixture. Within the pseudoensemble considered, an isobaric-isothermal simulation was initially performed, which was ensued by a canonical Monte Carlo implementation. The volume ensemble average of the former simulation was considered as the volume at which the latter simulation was attempted. The probability for generating conformations during the configurational-bias moves as well as the acceptance probability for regrowth were evaluated according to the probability devised elsewhere.15 A number of six trial positions were tested for the force center for each segment regrowth. We attempted various moves with a probability fraction of 0.005 for volume changes (if applicable), 0.5 for regrowth, and the complement being assigned to translation and rotation (if applicable). The maximum amplitude of volume change (if

applicable) and particle displacement were continuously monitored and adjusted in order to yield a 50% acceptance probability.15 Data collection was performed after an equilibration period of typically 104 iterations per molecule. A sample size of 10-100 was collected at presumably uncorrelated intervals (typically 104 iterations per molecule). A larger sample size of the order of 103 was essentially recorded in vicinity of the critical region. The Monte Carlo runs were performed on different nodes of a supercomputer, each node equipped with a Dual Core Xeon 3.2 GHz processor. The estimated single-processor CPU time to run 108 Monte Carlo moves was approximated at 5 h, for a particle pool of 1000 molecules. A number of 1000 molecules (N ) 1000) were considered in the computations. Given the adequate number of Monte Carlo moves devised for each particle on an average, reliable results should be expected for a system with such relatively large simulation sample size. A molecular number lower than the one considered (e.g., N ) 100) will indeed make an incorrect representation of the binary mixture being simulated. For instance, the simulation results obtained for a sample size of 100 molecules at x1 ) 0.158 will in fact pertain to a binary mixture with x1 ) 0.15. Also, consideration of any sample size larger than the one considered will be at the expense of increased computational time. 4. Results and Discussions We carried out Monte Carlo simulations for a binary mixture of methane (1) + n-butane (2) in three different compositions pertaining to a subcritical (x1 ) 0.158), critical (x1 ) 0.724), and supercritical (x1 ) 0.894) fluid15 at 311 K (Table 3). The ideal heat capacities in either case were calculated using the correlations prescribed elsewhere.22 The mixture of methane + n-butane at the composition of x1 ) 0.158 and temperature of 311 K resides in the subcritical region.14 Figures 1A and 2A show our simulation results for the isothermal compressibility and the sound velocity for this mixture, respectively. The order of magnitude of computed values for the isothermal compressibility indicates that the mixture resides inside the liquid reign; this is essentially consistent with the experimental observations.14 The statistical uncertainties are illustrated in the figures as twice the standard deviations computed. As evident from the

Table 3. Monte Carlo Results for Molar Volumes for the Mixture of Methane (1) + n-Butane (2) x1 ) 0.158

a

x1 ) 0.724 -1 a

x1 ) 0.894

P (MPa)

ν (m mol )

P (MPa)

ν (m mol )

P (MPa)

ν (m3 mol-1)a

3.930 3.999 4.137 4.826 5.516 6.205 6.895 10.34 13.79

1.0835 × 10-4 1.0764 × 10-4 1.0746 × 10-4 1.0674 × 10-4 1.0639 × 10-4 1.0616 × 10-4 1.0603 × 10-4 1.0344 × 10-4 1.0155 × 10-4

13.128 13.148 13.231 13.307 13.472 13.845 14.479 15.168 15.858 16.547 17.237

1.14840 × 10-4 1.0715 × 10-4 1.0499 × 10-4 1.0395 × 10-4 1.01912 × 10-4 1.0063 × 10-4 9.73747 × 10-5 9.57973 × 10-5 9.48164 × 10-5 9.28131 × 10-5 8.95358 × 10-5

2.068 2.758 3.447 4.137 4.826 5.516 6.205 6.895 7.584 8.274 8.693 9.653 10.34 11.03 11.72 12.41 13.10 13.79 14.47 15.16

1.1989 × 10-3 8.76720 × 10-4 6.90418 × 10-4 5.6173 × 10-4 4.7057 × 10-4 4.04213 × 10-4 3.5789 × 10-4 3.20923 × 10-4 2.83811 × 10-4 2.52824 × 10-4 2.30711 × 10-4 2.0609 × 10-4 1.93813 × 10-4 1.7958 × 10-4 1.6766 × 10-4 1.5645 × 10-4 1.4583 × 10-4 1.4152 × 10-4 1.3529 × 10-4 1.2835 × 10-4

3

The subscripts give the statistical accuracies of the last decimal(s).

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3

-1 a

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Figure 1. Monte Carlo simulation results for isothermal compressibility for a mixture of methane + n-butane (squares) with x1 ) 0.158 (A), 0.724 (B), and 0.894 (C) at 311 K.

figure, the Monte Carlo simulation results for sound velocity have been in decent agreement with the experiment, with maximum deviation at about 3.6%. We have enlisted our simulation results for the molar volumes for the three compositions analyzed in Table 1. Figure 2A reveals also the predicted values of sound velocity data by the Peng-Robinson (PR) equation of state. As it is evident from the figure, both the Monte Carlo simulation technique and the PR equation reasonably predict the trend of increasing the sonic velocity data with increasing pressure, as a typical behavior of liquid states. However, the predicted values by PR equation largely deviate from the measured data, showing the minimum absolute deviation of 7.6% compared to the maximum deviation of 3.6% obtained by the Monte Carlo simulation. The inaccuracy of the PR equation for prediction of sound velocity deteriorates into more deviation from the measured values when applied for near critical mixtures. The estimated experimental uncertainty was reported at about (0.03% by Plantier et al.14 For the next composition considered (x1 ) 0.724), the mixture remains in the near-critical region.15 Due to the thermodynamic anomalies typically encountered in such regions,10 we obtained our Monte Carlo ensemble averages from an essentially larger sample size, as mentioned previously. Likewise, the predicted sound velocity results for this composition have been in good agreement with the experiment, with maximum deviation

Figure 2. Monte Carlo simulation results for sound velocity for a mixture of methane + n-butane (squares) with x1 ) 0.158 (A), 0.724 (B), and 0.894 (C) at 311 K, compared with the experimental values of Plantier et al.14 (diamonds) and the Peng-Robinson equation of state (triangles).

estimated at about 5.0% (Figure 2B). The isothermal compressibility results indicate an interesting intermediate value (i.e., order of 10-8 Pa-1) between that of the liquid state (typical order of 10-9 Pa-1) and that of the gas state (typical order of 10-6-10-7 Pa-1) (Figure 1B). The results of PR equation of state, however, show a large deviation from the measured data, resulting in a prediction with minimum deviation estimated at about 54.2%, though the predicted trend is still the case. The thermodynamic conditions for the last mixture composition analyzed (x1 ) 0.894) will be in the supercritical domain,14 in that, at constant temperature no condensation would happen by changing the pressure. The predicted sound velocity results in this case, however, have been less satisfactory compared with the former two cases (Figure 2C). For this supercritical mixture, the sound velocity results have exhibited a maximum deviation of roughly 9.7% in the pressures range between 4.137 and 7.584 MPa, and a maximum deviation of 5.48% in the other regions. The predicted values by PR equation of state less deviate from the measured data (minimum error ) 17.5%), in comparison with the previous case, but wrongly predict an increasing trend for the speed of sound with increasing the pressure. The experimental data, as reasonably predicted by the Monte Carlo technique, passes through a minimum by changing the pressure. The Monte Carlo simulation results indicate the isobaric heat capacity passing through a local minimum in the supercritical

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010

region, where a minimum in the sound velocity occurs. Although, this value is not a global minimum of the isobaric heat capacity, the ratio established according to eq 1 will hold its global minimum for sound velocity at such state within the range of supercritical conditions investigated. As mentioned earlier, the work of Plantier et al.14 has served as the mere experimental basis for the current computational analysis of sound velocity. Hence, the present analysis may have provided additional information to the literature on the thermodynamic behaviors of such near-critical mixtures which represent real reservoir hydrocarbons. 5. Conclusions The Monte Carlo simulation methodology16 was employed to compute the speed of sound in a binary mixture representing real reservoir fluid in the vicinity of the critical region. In excellent agreement with the experiment, we find the technique to be capable of providing sound velocity values, which may be used for fluid characterization in such thermodynamic conditions with essentially little data availability. Represented by the PR equation, it is also shown that cubic equations of state are incapable in accurate prediction of sound velocity (or other second derivative properties), especially at near critical conditions. Literature Cited (1) Greaves, R. J. Three-Dimensional Seismic Monitoring of an Enhanced Oil Recovery Process. Geophysics 1987, 52 (9), 1175. (2) Wang, Z. Wasve Velocities in Hydrocarbons and Hydrocarbon Saturated Rocks - with Applications in EOR Monitoring. Ph.D. Dissertation, Stanford University, Stanford, CA, 1988. (3) Wang, Z.; Nur, A. Ultrasonic Velocities in Pure Hydrocarbons and Mixtures. J. Acoust. Soc. Am. 1991, 89 (6), 2725. (4) Estrada-Alexanders, A. F.; Justo, D. New Method for Deriving Accurate Thermodynamic Properties from Speed of Sound. J. Chem. Thermodyn. 2004, 36 (5), 419. (5) Estrada-Alexanders, A. F.; Trustler, J. P. M.; Zarari, M. Determination of Thermodynamic Properties from the Speed of Sound. Int. J. Thermophys. 1995, 16 (3), 663. (6) Gillis, K. A.; Moldover, M. R. Practical Determination of Gas Densities from the Speed of Sound Using Square-well Potentials. Int. J. Thermophys. 1996, 17 (6), 1305. (7) Scalabrin, G.; Marchi, P.; Benedetto, G.; Gavios, R. M.; Spagnolo, R. Determination of a Vapour Phase Helmholtz Equation for 1,1,1trifluoroethane (HFC-143a) from Speed of Sound Measurements. J. Chem. Thermodyn. 2002, 34 (10), 1601. (8) Sun, T. F.; Seldam, T.; Kortbeek, P. J.; Trappeniers, N. J.; Biswas, S. N. Acoustic and Thermodynamic Properties of Ethanol from 273.15 to 333.15K and up to 280 MPa. Phys. Chem. Liq. 1988, 18 (2), 107. (9) Levelt Sengers, J. M. H.; Morrison, G.; Chang, R. F. Critical Behavior in Fluids and Fluid Mixtures. Fluid Phase Equilib. 1983, 14, 19. (10) Gregorowicz, J.; O’Connel, J. P.; Peters, C. J. Some Characteristics of Pure Fluid Properties that Challenge Equation-of-state Models. Fluid Phase Equilib. 1996, 116, 94.

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ReceiVed for reView August 10, 2009 ReVised manuscript receiVed February 21, 2010 Accepted February 22, 2010 IE901422A