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Ind. Eng. Chem. Res. 1999, 38, 1718-1722
Effect of the Power Series Dispersion Term on the Pressure-Volume Behavior of Statistical Associating Fluid Theory Naveen Koak and Theo W. de Loos* Laboratory of Applied Thermodynamics and Phase Equilibria, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
Robert A. Heidemann Department of Chemical and Petroleum Engineering, The University of Calgary, 2500 University Dr. N.W., Calgary, Alberta T2N 1N4, Canada
It is shown that the statistical associating fluid theory (SAFT) equation of state with the power series dispersion term can have multiple volume roots at a specified pressure and temperature. Examples considered are based on published correlations for the nonassociating pure components methane and poly(ethylene-propylene) 790 (PEP790). Conditions have been found where two of the five volume roots (one mechanically stable and one unstable) occur at reduced densities greater than theoretical close packing and the equation of state has a large negative pressure in the nominal close-packed state. The SAFT fluid can have the least chemical potential at the smallest volume root, which appears not to have been considered in the correlation of the methane and PEP790 behavior. Introduction The statistical associating fluid theory (SAFT) equation of state is based on Wertheim’s theory of associating fluids1-3 and is applicable to small, large, and associating molecules. Because of its diverse applications, SAFT has become a very popular model for describing the phase behavior of complex fluids and their mixtures. The most used version of SAFT is the one developed by Huang and Radosz.4,5 This version of SAFT employs a mean-field dispersion term that is a power series based on the molecular dynamics calculations for square-well fluids of Alder et al.6 which were fitted and adjusted for argon behavior by Chen and Kreglewski.7 This work focuses on the effect of the Chen-Kreglewski dispersion term on the pressure-volume behavior of the nonassociating pure components methane and poly(ethylenepropylene) (PEP) 790. SAFT Compressibility Factor Equation (No Association) The compressibility factor (Z) equation for SAFT4 consists of two parts: (i) the repulsive term and (ii) the attractive term.
Z ) ZRep + ZAtt with,
[
]
(1)
[
]
5 η - η2 4η - 2η2 2 + (1 - m) ZRep ) 1 + m 1 (1 - η)3 (1 - η) 1 - η 2 (2)
(
)
* Corresponding author. E-mail:
[email protected]. Telephone: +31 15 2788478. Fax: +31 15 2788047.
The repulsive contribution to the compressibility factor was derived by Chapman and co-workers8,9 from Wertheim’s statistical theory of fluids with highly directional attractive forces.1-3 The Carnahan and Starling10 compressibility factor and the implied radial distribution function were used by Chapman and coworkers as the basis for the model. Molecules are assumed to contain m spheres which are combined in chains because of an infinite association energy. Chain formation is accounted for as a perturbation to the hardsphere compressibility factor. The variable η is a reduced density defined for pure fluids by
η ) m(NAVπd3/6)/v ) mvo/v
(3)
In this expression, d is the diameter of the spheres and v is the molar volume. Note that eq 2 is defined to reduced densities ranging to η ) 1. However, close packing of spheres occurs at a reduced density τ, where τ ) x2π/6 ) 0.740 48. Carnahan and Starling10 derived their compressibility expression, in the first instance, by approximating the virial expansion for hard spheres, and therefore the expression is not strictly applicable at the higher densities. The Alder and Wainwright11 (also see McQuarrie12) molecular dynamics calculations for an assembly of hard spheres, which have been used to validate the Carnahan-Starling equation, are interpreted as showing two separate lines of Z vs η with a transition from liquidlike behavior to solidlike behavior at a reduced density of around η ) 0.45. The CarnahanStarling closed form represents an extrapolation of fluid behavior when the reduced density exceeds the transition value. The attractive (or dispersion) part of the SAFT compressibilty factor for pure fluids is
10.1021/ie9804069 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/26/1999
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1719 4
9
[ ][ ] u
∑ ∑jDij kT i)1 j)1
ZAtt ) m
i
η τ
Table 1. Pure-Component Parameters
j
(4)
This power series expansion in terms of absolute temperature and reduced density is based on a proposal by Chen and Kreglewski.7 Alder et al.6 had used eq 4 to correlate molecular dynamics calculations for a squarewell fluid. Chen and Kreglewski refit the coefficients in the Alder et al. polynomial to match certain data for argon. The SAFT equation of Huang and Radosz4,5 uses the Chen and Kreglewski polynomial coefficients and a temperature-dependent hard-sphere diameter, as was also proposed by Chen and Kreglewski. The volume of a mole of segments is calculated from
vo ) NAvπd3/6 ) voo[1 - 0.12 exp(-3uo/kT)] (5) where voo and uo/k are temperature-independent parameters. Also following Chen and Kreglewski, the energy in the dispersion term of eq 4 is made a temperature function through
u/k ) (uo/k)[1 + (e/k)/T]
(6)
This temperature dependence introduces e/k as an additional constant parameter for each substance. Alder et al. and Chen and Kreglewski noted that the dispersion term showed poor convergence properties at high densities and the data available to produce the correlations extend to a limited range of the dimensionless density η. The resulting equation, however, can be extrapolated to values of η greater than the theoretical close-packed limit. Results Methane. The parameters for methane were obtained from Huang and Radosz4 and are contained in Table 1. Parts a-f of Figure 1 show the pressure-volume behavior for methane at a few temperatures ranging from well over the critical temperature to well below the critical temperature from the model. It can be seen that as the temperature is lowered the P-v isotherms at low volumes start to have multiple volume roots. The temperature of 190 K is quite close to the model critical temperature if the smallest mechanically stable root is neglected. Table 2 contains the values of the vapor pressures, the three mechanically stable volume roots, and the corresponding chemical potentials for four temperatures below the model critical temperature of methane (if the smallest mechanically stable root is neglected in vapor pressure calculations). It can be seen from Table 2 that the smallest mechanically stable root is always found to have the smallest chemical potential. The chemical potential µ was calculated as given by the following equation:
µres µ ) + ln(RT/vPo) RT RT
(7)
µres is the residual chemical potential from the equation of state. Po is the reference pressure for the ideal gas state and is taken to be unity in the same units as pressure P. The ideal gas reference state chemical potential which is a function of temperature only is not included in the calculation of the chemical potential. The
methane PEP790
m
uo/k (K)
voo (cm3/mol)
e/k
molar mass
1 25.5
190.29 404.8
21.576 22.5
1 10
16.043 790
Table 2. Methane Saturation (sat) Pressures, Mechanically Stable Volume Roots, and Chemical Potentials Tsat (K)
Psat (MPa)
vsat vsat vsmall (cm3/mol) (cm3/mol) (cm3/mol) µsat/RT µsmall/RT
190 180 160 150
4.49442 3.25718 1.57758 1.03070
137.7941 260.5238 636.0290 990.4140
74.1499 58.1856 47.8992 44.9901
17.7535 17.6280 17.4265 17.3397
3.3986 3.1506 2.5451 2.1691
-12.8401 -26.7673 -68.5289 -99.2587
Table 3. PEP790 Volume Roots and Chemical Potentials at T ) 400 K and P ) 0.1 MPa vsp (cm3/g)
µ/RT
0.5938 1.2643
-805.40096 -69.29217
limiting volume (i.e., the volume which corresponds to η ) 1) depends on temperature according to eqs 3 and 5. The value is τvo, which equals 15.69 cm3/mol for methane at 190 K. The reduced density at the smallest volume root is η ) 0.884 and well above the nominal close-packed density. Nonetheless, this would be the most stable volume root at T ) 190 K and P ) 4.494 42 MPa if SAFT is to be treated in the same way that other equations of state are handled in classical thermodynamic modeling. Also, for T ) 190 K, at the reduced density of 0.740 48 (close-packed reduced density) the pressure is found to be below -2000 MPa, which is physically unreasonable. PEP790. The behavior of SAFT for poly(ethylenepropylene) 790 (PEP790) is typical of what can be seen with a number of other polymers. The parameters for PEP790 were obtained from the paper of Chen et al.13 and are contained in Table 1. With these parameters, SAFT has a high critical temperature well above 1700 K. Parts a-d of Figure 2 show the P-v behavior of this polymer at a few temperatures. All temperatures are below the model critical temperature for the polymer. Part of the isotherm at 501.2 K (T/TC < 0.3) is shown in Figure 2c. Here it can be seen that a second volume interval where the fluid would be mechanically unstable has developed in the dense fluid and there is, consequently, a pressure region where five volume roots can be found. It is questionable whether the attractive term of Chen and Kreglewski7 can be used at this low reduced temperature. At 400 K (T/TC < 0.3) and a pressure 0.1 MPa, the two smaller mechanically stable volume roots along with the corresponding chemical potentials are given in Table 3. Note that the phase with the lower chemical potential is the one with the smaller volume, i.e., the phase with vsp ) 0.5938 cm3/g. At this temperature, the limiting volume (corresponding to infinite pressure) is 0.5285 cm3/g and the theoretical close-packed volume is 0.7138 cm3/g. At T ) 400 K the pressure corresponding to the nominal close-packed reduced density of 0.740 48 is below -6000 MPa and again is a physically unreasonable result. It appears that Chen et al.13 used the larger of the two mechanically stable volume roots when correlating parameters for PEP790.
1720 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999
Figure 1. SAFT pressure-volume isotherms for methane: T ) (a) 250, (b) 240, (c) 230, (d) 220, (e) 190, and (f) 150 K.
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1721
Figure 2. SAFT pressure-volume isotherms for PEP790: T ) (a) 1700, (b) 1600, (c) 501.2, and (d) 400 K.
Conclusions The two examples above, for one pure substance and for a polymer, seem to be typical of the behavior of SAFT when eq 2 for hard-sphere chains is combined with the Chen-Kreglewski dispersion term. Our calculations indicate that, for any value of m, there are multiple volume roots at low enough reduced temperatures. However, for some other compounds examined, the P-v isotherms in the vicinity of the model critical temperature do not show multiple volume roots as was found for methane. In our opinion the ambiguity in the choice of volume roots where there are multiple roots needs to be resolved. This kind of behavior would present obstacles to including this version of SAFT in process simulators. The theoretical implications of our findings may be more significant. In the specific cases examined, the middle of three mechanically stable volume roots does not have the lowest chemical potential and, if the requirements of classical thermodynamics are strictly adhered to, this root would play no role in the phase equilibrium behavior dictated by the model. On the
other hand, the smallest of the stable volume roots, particularly at moderate temperatures, is typically below the nominal close-packing volume, which implies that use of this root is an extrapolation of the theories underlying SAFT into regions where they could not apply. We also note that the equation of state can produce extreme negative pressures at densities corresponding to nominal close packing. This behavior appears to us to be inconsistent with the intention of the original theories. Acknowledgment The authors are grateful to Dr. A. Galindo for helpful discussions. Nomenclature η ) τFmvo ) reduced density µ ) chemical potential τ ) close-packed reduced density ) 0.740 48 F ) molar density d ) temperature-dependent diameter of a segment
1722 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 e/k ) pure component constant (K) k ) Boltzmann constant m ) number of segments NAV ) Avogadro’s number P ) pressure R ) universal gas constant T ) temperature TC ) critical temperature u/k ) temperature-dependent energy dispersion term, eq 6 uo ) temperature-independent segment-segment energy v ) molar volume vo ) volume occupied by a mole of segments, eq 5 voo ) temperature-independent constant vsp ) specific volume Z ) compressibility factor
Literature Cited (1) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J. Stat. Phys. 1984, 35, 19. (2) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. II. Thermodynamic Perturbation Theory and Integral Equations. J. Stat. Phys. 1984, 35, 35. (3) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. III. Multiple Attraction Sites. J. Stat. Phys. 1984, 35, 459. (4) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284.
(5) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules: Extension to Fluid Mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994. (6) Alder, B. J.; Young, D. A.; Mark, M. A. Studies in Molecular Physics. X: Corrections to the Augmented van der Waals Theory for Square-Well Fluids. J. Chem. Phys. 1972, 56, 3013. (7) Chen, S. S.; Kreglewski, A. Applications of the Augmented van der Waals Theory of Fluids. I. Pure Fluids. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 1048. (8) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT Equation of State Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31. (9) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (10) Carnahan, N. F.; Starling, K. E. Thermodynamic Properties of a Rigid-Sphere Fluid. J. Chem. Phys. 1969, 51, 635. (11) Alder, B. J.; Wainwright, T. E. Studies in Molecular Dynamics. II. Behavior of a Small Number of Elastic Spheres. J. Chem. Phys. 1960, 33, 1439. (12) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976; p 256. (13) Chen, S.-J.; Chiew, Y. C.; Gardecki, J. A.; Nilsen, S.; Radosz, M. P-V-T Properties of Alternating Poly(EthylenePropylene) Liquids. J. Polym. Sci., Part B: Polym. Phys. 1994, 32, 1791.
Received for review June 23, 1998 Revised manuscript received January 5, 1999 Accepted January 18, 1999 IE9804069
