New Variant of the Universal Constants in the Perturbed Chain

Article pubs.acs.org/IECR

New Variant of the Universal Constants in the Perturbed Chain-Statistical Associating Fluid Theory Equation of State Xiaodong Liang* and Georgios M. Kontogeorgis Center for Energy Resources Engineering (CERE), Department of Chemical and Biochemical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark S Supporting Information *

ABSTRACT: The Perturbed Chain-Statistical Associating Fluid Theory Equation of State (PC-SAFT EOS) has been successfully applied to model phase behavior of various types of systems, while it is also well-known that the PC-SAFT EOS has difficulties in describing some second-order derivative properties. In this work, the temperature and volume dependencies of the PC-SAFT EOS have been analyzed based on the total reduced residual Helmholtz free energy from the well-established reference equations. The ranges of parameters and temperature, in which the original PC-SAFT EOS give zero or more than three volume roots, have been analyzed. Then, a practical procedure has been proposed to refit the universal constants of the PC-SAFT EOS with the purpose of fixing the numerical pitfalls in the real application ranges and reusing the original parameters. It is shown that the new universal constants have practically resolved the mostly criticized numerical pitfall, that is, the presence of more than three volume roots at real application conditions. Finally, the possibility of using the original PC-SAFT EOS parameters with the new universal constants has been investigated for the phase equilibria of the systems containing hydrocarbons, chemicals, water, or polymers.

1. INTRODUCTION

could practically avoid the numerical pitfalls of the additional fictitious critical point of the pure components. On the other hand, it has also been found that these new universal constants perform worse for the density of long chain fluids6,15 and isobaric heat capacity. Polishuk16 showed, in a later article, that these new universal constants give poor prediction for virial coefficients, even negative third virial coefficients at high temperatures. Moreover, it is not convenient to develop general characterization procedures for modeling ill-defined systems, for example, petroleum fluids, since the speed of sound data used in the parameter estimation.6 Equally importantly, unrealistic phase behavior can be predicted by the original PC-SAFT EOS. All these inspired us to revisit the fundamentals of this model. The purposes of this work are (1) to analyze the temperature and volume dependencies of the PC-SAFT EOS; (2) to discuss the reasonable parameter ranges of the original PC-SAFT EOS; (3) to propose a new variant of universal constants with the aim to fix the numerical pitfalls in the real application ranges and to reuse the original pure component parameters; (4) to compare the universal constant sets and to investigate the possibility of using the original PC-SAFT parameters with the new universal constants for various systems.

As a popular and promising model, the Perturbed ChainStatistical Associating Fluid Theory Equation of State (PC-SAFT EOS) has been applied into many different fields, including chemicals, polymers, biochemicals, pharmaceuticals, and so on,1−5 and it has also demonstrated powerful capability on modeling the complex phase behavior of petroleum fluids and water containing systems.6 However, it is also well-known that the PC-SAFT EOS has difficulties in describing the second-order derivative properties.7,8 Moreover, it has been discussed for some numerical pitfalls over the recent years,9−13 for which the presence of more than three volume roots at some conditions and/or the discontinuity of the volume roots as the temperature and/or pressure change, are believed to be the reason. As denoted by its name, the dispersion term of the PC-SAFT EOS is a perturbation for hard chains instead of hard spheres, but in fact, it is represented by the interaction energy fitted to the real fluids in the form of sixth degree polynomials of reduced density. There are, in total, 42 empirical coefficients in these sixth degree polynomials, which are the so-called universal constants. In one of our previous works,14 the universal constants are readjusted because the original ones could not provide satisfactory correlation and/or prediction for the speed of sound in most systems, even with the pure component parameters fitted to the speed of sound data. The universal constants were obtained by including the vapor pressure, liquid density, and speed of sound data of saturated methane to decane in the regression. On one hand, it has been found that these new universal constants improve the description of the speed of sound in most systems from both qualitative and quantitative points of view, while keeping satisfactory description of phase equilibrium.6,15 Very recently, Polishuk et al.13 have found that these new universal constants © XXXX American Chemical Society

2. ANALYSIS OF THE ORIGINAL PC-SAFT EOS 2.1. Temperature and Volume Dependencies. The PCSAFT EOS can be expressed as a r = a hc + adisp = (a hs + achain) + adisp

(1)

Received: October 6, 2014 Revised: December 12, 2014 Accepted: December 15, 2014

A

DOI: 10.1021/ie503925h Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research ⎧ ⎫ ∂ 2η ∂ 2F ∂F ⎨ ⎬ = − + ( V / R ) QVV 2 QV ∂η ∂(V /R )2 ∂η2 ⎩ ⎭

The detailed expressions of these terms can be found in the work of Gross and Sadowski1 or in the recent monograph of Kontogeorgis and Folas.3 If it is assumed, for a given temperature and volume (density), that the reduced residual Helmholtz free energy can be calculated from a well-established model, for instance from NIST reference equations,17 and the hard-sphere chain contribution is calculated from PC-SAFT, the reduced dispersion term can then be calculated: a

disp

r

=a −a

hc

⎛ ∂η ⎞−2 ⎟ ×⎜ ⎝ ∂V /R ⎠

If F is further formulated as

F = FASS1 + FCFBSS2

(2)

FA = ( −12I1),

First, the following variables are introduced for easy mathematical manipulations: Q = adisp = a r(NIST) − a hc(PC‐SAFT)

QV =

FC = (mC),

(3)

FA + FB

∂ 2adisp QVV = ∂(V /R )2 (5)

∂ 2FA ∂η2

⎧ ⎛ ε ⎞ F = ⎨( −12I1) ⎜m2 σ 3⎟ + (mC) ( −6I2) ⎝ kT ⎠ ⎩ ⎪

+

⎪

⎪

⎪

1 F V /R

(16)

(17)

2 2 ⎛ −1 ∂ FC 2 ⎛ ∂FC ⎞ ⎜ +⎜ + F FB ⎜ ⎟ B 3 (FC)2 ∂η2 ⎝ (FC) ⎝ ∂η ⎠

∂ 2FB 1 ⎞ SS2 − 2 ∂FC ∂FB 1 ∂ 2F ⎟ + = SS1 ∂η2 ∂η2 FC ⎟⎠ SS1 (FC)2 ∂η ∂η

(18)

(7)

1 F, SS1

∂ 2Q ∂(V /R )2 2 ⎞ ∂ 2η ∂F 1 ⎛ ∂ 2F ⎛ ∂η ⎞ ⎜⎜ 2 ⎜ ⎟ + − 2QV ⎟⎟ = 2 ∂η ∂(V /R ) (V /R ) ⎝ ∂η ⎝ ∂V /R ⎠ ⎠

QVV =

(9)

Then, (10)

⎛ ∂η ⎞−1 ∂F = {(V /R )QV + Q } ⎜ ⎟ ∂η ⎝ ∂(V /R ) ⎠

(11)

1 ∂F , SS1 ∂η

1 ∂ 2F SS1 ∂η2

(20)

In this way, it is possible to have a deterministic procedure to obtain the universal constants, as long as the framework and the assumed pure component parameters could fulfill the requirements that the three properties in eq 20 are all linear functions of 1/T. The procedure is documented in detail as follows: (1) Setup NIST and PC-SAFT for the specified compound (2) Specify reduced density η and temperature T (3) Setup PC-SAFT for temperature dependent parameters/ variables, and calculate V from η and T by V = (π/6) (md3/η) (4) Call NIST to calculate the required properties (ar, Pr, dP/dV)) (5) Call PC-SAFT to calculate the contributions of the hardchain term (6) Calculate the dispersion term from NIST and hard-chain term from PC-SAFT (eqs 3-5) (7) Calculate F, ∂F/∂η, ∂2F/∂η2 from eqs 10-12 (8) Calculate linear coefficients of (1/SS1)F, (1/SS1) (∂F/∂η), (1/SS1) (∂2F/∂η2) as a function of 1/T for the given η (9) If we assume (1/SS1)F = A + B/T, we could get FA = A; FB = FC (B/(ε/k))

(8)

F = (V / R )Q

(19)

So the following three variables should be all linear functions of 1/T.

∂Q ∂(V /R ) ∂η 1 ⎛ ∂F F ⎞ = − ⎜ ⎟ V /R ⎝ ∂η ∂(V /R ) V /R ⎠ ⎞ ∂η 1 ⎛ ∂F − Q⎟ ⎜ (V /R ) ⎝ ∂η ∂(V /R ) ⎠

⎡ ⎛ ε ⎞2 ⎤ SS2 = ⎢m2⎜ ⎟ σ 3⎥ ⎣ ⎝ kT ⎠ ⎦

1 ⎛ SS2 ⎞ F ⎟= ⎜ FC ⎝ SS1 ⎠ SS1

SS2 ε = SS1 kT

QV =

=

(14)

For a pure fluid, (6)

where the constant factor ((π/6) (NA/R)) is ignored here, so Q=

FB = ( −6I2),

⎛ −1 ∂F ∂FB 1 ⎞ SS2 ∂FA 1 ∂F C ⎟ F +⎜ + = B 2 SS1 ∂η ∂η ∂η FC ⎠ SS1 ⎝ (FC) ∂η

Furthermore, a new function F is defined as

⎡ ⎛ ε ⎞ 2 ⎤⎫ ⎢m 2 ⎜ ⎟ σ 3 ⎥⎬ ⎣ ⎝ kT ⎠ ⎦⎭

⎛ ε ⎞ SS1 = ⎜m2 σ 3⎟ ⎝ kT ⎠

Then, (4)

dP dV (NIST) ∂ 2a hc (PC‐SAFT) − T /R ∂(V /R )2

(13)

(15)

P r(NIST) ∂adisp ∂a hc =− − (PC‐SAFT) ∂(V /R ) T ∂(V /R )

=−

(12)

B


Article

Industrial & Engineering Chemistry Research (10) If we assume (1/SS1)(∂F/∂η) = C + D/T, we could get (∂FA/∂η) = C; (∂FB/∂η) = FC((D/ε/k) + (1/(FC)2) (∂FC/∂η) FB)) (11) If we assume (1/SS1) (∂2F/∂η2) = G + H/T, we could get (∂2FA/∂η2) = G; (∂2FB/∂η2) = FC((H/ε/k) − (2/(FC)3) (∂FC/∂η)2 FB + (1/(FC)2) (∂2FC/∂η2) FB + (2/(FC)2) (∂FC/∂η) (∂FB/∂η)) (12) When all the information is available, the sixth polynomial coefficients could be fitted to get the universal constants The examples of the converted dispersion term (eq 16) of methane (C1) and propane (C3) are demonstrated for the reduced density dependence and temperature dependence, respectively, in Figures 1 and 2. It can be seen that the reduced

Figure 2. Temperature dependence of the converted dispersion term (eq 16) for methane and propane. The reduced residual Helmholtz free energy is calculated by NIST.17

Figure 1. Reduced density dependence of the converted dispersion term (eq 16) for methane and propane. The reduced residual Helmholtz free energy is calculated by NIST.17

density dependence, sixth degree polynomials, is sufficient, but the temperature dependence is only valid either at low density region or in a narrow range of temperature. This explains why PC-SAFT has difficulties in describing the highly temperature dependent properties, for example, residual isochoric heat capacity, which solely depends on the derivatives of residual Helmholtz free energy with respect to temperature. 2.2. Isothermal Curves. It has been criticized that the original PC-SAFT EOS can give more than three volume roots at some temperature and pressure conditions. Privat et al.11 stated that it is a problem to have more than three volume roots, since the classical volume-root solvers only search for three roots. It could be argued that this issue can be solved by developing sophisticated volume-root solvers, for example, setting limit

Figure 3. Isothermal curves of n-decane (C10) at 135 K and n-eicosane (C20) at 150 K from the original PC-SAFT EOS. The curve of C10 was first reported by Privat et al.11

values or checking the derivatives. The isothermal curves of n-decane (C10) at 135 K and n-eicosane (C20) at 150 K from the original PC-SAFT EOS are presented in Figure 3. The one of C10 was first reported by Privat et al.,11 in which the detailed information on its curve shape in the low reduced density region can be found. It can be seen that it is possible to have two stable liquid volume roots in some pressure ranges, and there is only one single volume root locating in the very heavy reduced density region. More severely, the volume roots are not continuous as the C


Article

Industrial & Engineering Chemistry Research pressure increases, so the volume-root solver will probably not be an appropriate solution anymore. In order to have a more complete picture of the isothermal curves, the parameters and temperature are screened to test how many volume roots the original PC-SAFT EOS will have at most by using the following procedure. (1) Produce the parameters from {m = [1:1:200], σ = 3.8 Å, ε/k = [100:10:1000] K} (2) Produce the temperature from {T = [100:20:2000] K} (3) Check the number of volume roots at P = 0 bar, NVR(P = 0), and P(η → 0) (4) If [NVR(P = 0)] is zero, check the number of volume roots of dP/dV = 0, NVR(dP/dV = 0) (5) If [NVR(P = 0)] is two, check the number of volume roots of dP/dV = 0, NVR(dP/dV = 0) for P > 0 (6) LOOP step (1) to (5) until testing all the parameters According to the work of Polishuk et al.,13 the parameter σ is arbitrarily set to 3.8 Å, which is a typical value for hydrocarbons. When the isothermal curve has zero or two volume roots at P = 0 bar, it does not mean that it is free from the numerical pitfalls simply because the working pressure is not zero bar. This is why it is checked in steps 4 and 5 for the number of volume roots of dP/dV = 0. Figure 4 presents the examples of the parameter combinations with which the original PC-SAFT EOS has one or three volume

Figure 5. Isothermal curves from the original PC-SAFT EOS with different parameter combinations for the situations, (a) four volume roots at P = 0 bar, and (b) more than two volume roots satisfying dP/dV = 0 when two volume roots at P = 0 bar.

application ranges. It is possible to use this procedure to define the application ranges of the original PC-SAFT EOS. For example, the ranges of segment number and dispersion energy parameters, with which the phenomena of Figure 5 will occur, are presented in Figure 6 for the given segment size (3.8 Å) and temperature (300 K). Figure 6a indicates that it is not recommended to use the dispersion energy higher than 460 K if the segment number is larger than 10 at temperature 300 K.

Figure 4. Isothermal curves from the original PC-SAFT EOS with different parameter combinations for the situations of one root (red dash-dot line) and three roots (blue dash line) at P = 0 bar with P (η → 0) > 0 bar.

3. NEW UNIVERSAL CONSTANTS On one hand, as discussed by Polishuk et al.,13 it is a feasible approach to fix the numerical pitfalls by changing the universal constants. On the other hand, the PC-SAFT EOS has already been successfully applied into modeling various types of systems.1−5 It will fundamentally extend the application ranges of the PC-SAFT EOS if a new set of universal constants can avoid the numerical pitfalls and reuse the original parameters as much as possible. 3.1. Practical Way for Regression. In the previous work,14 the universal constants are regressed with a stepwise procedure by assuming that methane has a segment number parameter m = 1. It is further assumed that propane has a segment number parameter m = 2 to develop the following practical procedure: (1) Estimate the pure component parameters for C1 to C10 with the original universal constants (2) Regress the coefficients {a0i} and {b0i} from the properties of C1 with m = 1

roots at P = 0 bar when P(η → 0) > 0 bar. It can be seen that it is probably that no single solution will be found in most of the interested pressure ranges, while it could be argued that the dispersion energy (ε/k) has been too large and temperature has been too low for real applications, but this examination gives the limitations of the feasible ranges of the parameters, which might be very useful for automatic parameter tuning in large scale simulations. Figure 5 presents the examples of the parameter combinations with which the original PC-SAFT EOS has four volume roots at P = 0 bar, and more than two volume roots satisfying dP/dV = 0 when the NVR(P = 0) is two, respectively. It can be seen that the functionalities of pressure versus reduced density from these two situations are quite similar to those shown in Figure 3 for C10 and C20, that is, two stable liquid volume roots existing at some ranges of pressure and the only valid liquid volume root locating in very high reduced density region for higher pressures. It can be also seen that some combinations have already located in the real D


Article

Industrial & Engineering Chemistry Research

Figure 6. Ranges of m and ε/k for the original PC-SAFT EOS to have the same type of P−η curve as those presented Figure 5 with fixed σ = 3.8 Å at 300 K. (a) Four volume roots at P = 0 bar, and (b) more than two volume roots satisfying dP/dV = 0 when two volume roots at P = 0 bar.

Figure 7. Comparisons of the dispersion term I1 from the two universal constant sets from (a) this work and (b) Gross and Sadowski.1

where Ω is vapor pressure and density (both saturated and isothermal−isobaric conditions), and second-order derivative property dP/dV. The vapor pressure and saturated liquid density are the only properties used for pure component parameter estimation, while the second-order derivative property dP/dV is included for the regression of the universal constants, with a weight factor wk = 0.01. The reason is 3-fold: first, the objective function with only vapor pressure and density has a very flat respond surface; second, the current PC-SAFT framework is not enough for simultaneously accurate description of the first- and second-order derivative properties over a wide range of temperature with the associated pure component parameters, as demonstrated above; third, the possibility of reusing the original PC-SAFT parameters can be kept with a small weight factor. The data are the same ones from NIST17 as those used in the previous work,14 and the details of the temperature and pressure ranges can also be found in the PhD thesis of Liang.6

(3) Regress the coefficients {a1i} and {b1i} from the properties of C3 with m = 2 (4) Regress the coefficients {a2i} and {b2i} from the properties of butane (C4) to C10 (5) Estimate the pure component parameters for C1 to C10 with the new universal constants In the previous work,14 the universal constant regression and pure component parameter estimation are iterated, but the steps (2) to (5) are not repeated in this work, because we aim to keep the possibility of reusing the original PC-SAFT EOS parameters as much as possible. The following objective function is employed fmin =

∑ wk

1 Nk

Nk

⎛ Ωexp − Ωcalc ⎞2 i i ⎟⎟ exp Ω ⎝ ⎠ i

∑ ⎜⎜ i=1

(21)

Table 1. New Universal Constants first dispersion term (I1)

second dispersion term (I2)

i

a0i

a1i

a2i

b0i

b1i

b2i

0 1 2 3 4 5 6

0.961597 0.414449 0.689253 −7.43899 31.8755 −54.8833 27.3613

−0.333416 0.358440 −0.219088 0.285586 5.88256 −22.4931 21.2109

−0.0632483 0.287134 −0.105309 7.16607 −37.5008 68.8002 −40.3177

0.548398 2.07176 −1.84013 −29.9683 160.445 −206.106 51.6201

−0.277226 0.0621105 4.29640 −39.9453 214.911 −232.563 60.9734

−0.150684 −0.507877 1.24227 −48.7052 63.1155 205.380 −262.437

E


Article

Industrial & Engineering Chemistry Research 3.2. I1/I2 versus η. The obtained new universal constants are listed in Table 1. From the viewpoint of individual coefficients, it can be seen that these new universal constants are quite different from the original ones of Gross and Sadowski1 and the ones developed with speed of sound data in the regression.14 The later one will not be further discussed in this work, and its applications can be found in the works of Liang et al.6,14,15 In the following discussions, the PC-SAFT models with the original universal constants (from Gross and Sadowski1) and the new ones (from this work) will be denoted as PC-SAFT (GS) and PC-SAFT (LK), respectively. The curves of the dispersion terms I1 and I2 versus reduced density (packing fraction η) from the two universal constant sets are compared for m = [1, 1.5, 2, 3, 10, 10 000] in Figures 7 and 8.

are refitted to the same vapor pressure and saturated liquid density data with the two universal constant sets. The fitted model parameters and corresponding deviations are listed in the Supporting Information Tables S1 and S2, respectively. It is known from Table S2 that the PC-SAFT (GS) gives deviations 0.65% and 0.35%, respectively, for the vapor pressure and density of n-alkanes from C1 to C20, and the PC-SAFT (LK) gives deviations 0.30% and 0.27% for the corresponding properties. This illustrates that both universal constant sets provide quite satifactory correlating capability for vapor pressure and saturated liquid density. The differences of the pure component parameters from PC-SAFT with the two universal constant sets are compared in Figure 9a, which is defined as %RD of parameters =

Figure 8. Comparisons of the dispersion term I2 from the two universal constant sets from (a) this work and (b) Gross and Sadowski.1

LK − GS × 100% GS

(22)

Figure 9. Comparisons of the PC-SAFT parameters and the corresponding %AADs for vapor pressure and liquid density of n-alkanes (from C1 to C20) from the two universal constant sets. The experimental data are taken from NIST17 for C1 to C10 and from DIPPR18 for C11 to C20.

It can be seen that, in general, the two universal constant sets produce similar I1 and I2 curves over a wide range of reduced density, that is, η = 0−1.0, from the qualitative point of view, except for m = 1, and they give similar values in the typical range of liquid volume roots, that is, η = 0.25−0.40, except for I2 of m = 1. However, the PC-SAFT (GS) gives, respectively, larger and smaller values for I1 and I2 than those from PC-SAFT (LK) in the high reduced density region, that is, η = 0.6−1.0, or in other words, the curvatures of I1 and I2 from these two universal constant sets occur in different reduced density regions, that is, the PC-SAFT (LK) shifts the larger stationary point to a higher reduced density region. 3.3. Parameters and Physical Properties. In order to have a fair comparison, the pure component parameters of n-alkanes

GS and LK are used instead of PC-SAFT (GS) and PC-SAFT (LK) just for simplicity, which will also be used in the following figures. It can be seen that the PC-SAFT (LK) gives larger parameter m and smaller parameters σ and ε/k for all the investigated n-alkanes, except C1, and the differences themselves have small variations for n-alkanes longer than C10. The differences are smaller than 1%. The differences of the deviations for vapor pressure and saturated liquid density from PC-SAFT with the two universal constant sets are compared in Figure 9b, which is defined as F


Article

Industrial & Engineering Chemistry Research %AAD differences = LK − GS

(23)

It is shown in Figure 9b that the PC-SAFT (LK) presents a slightly better correlation for vapor pressure than the PC-SAFT (GS), whereas the two sets perform quite similarly for density. The second-order derivative properties, speed of sound (u), dP/dV, residual isochoric and isobaric heat capacities (CV(r) and CP(r)), and Joule−Thomson (JT) coefficients are calculated from PC-SAFT with the two universal constant sets, and compared with the data from NIST.17 The detailed deviations for these properties are reported in the Supporting Information Tables S3 and S4. The deviations are compared for speed of sound and dP/dV in Figure 10a and for residual and total

Figure 11. (a) Comparisons of the %AADs for Joule−Thomson (JT) coefficients of C1 to C10 over wide ranges of temperature and pressure using eq 23, and (b) Joule−Thomson coefficients of saturated nC8 from the two universal constant sets. GS and LK denote the universal constants from Gross and Sadowski1 and this work, respectively. The experimental data are taken from NIST.17

from which the two universal constant sets present quite similar results. As discussed above, it is possible for different universal constant sets to have equally good correlations for vapor pressure and density, and it is also possible to obtain different universal constant sets by taking different property combinations into account, which will give different balances on describing the properties depending on the designed objective function. To the best of our experience, it seems not possible to find a unique set of universal constants, within the current framework, which is able to give absolutely better description for all the properties. It could be anticipated that it is a fundamental deficiency that the current PC-SAFT framework is not able to simultaneously describe all of the first- and second-order derivative properties, based on this work and the previous one.14 3.4. Application Ranges. The same procedure proposed in Section 2.2 is also conducted for the new universal constants. The results show that the behavior of having one or three volume roots at P = 0 bar when P (η → 0) > 0 bar have not been observed, as those presented in Figure 4. It has also been found that it is possible to have at most two volume roots satisfying dP/dV = 0 when there are zero or two volume roots at P = 0 bar, which are typical cubic-like P−η curves, so there will be no numerical pitfalls for these two situations as well. In the entire investigated ranges of parameters and temperature, all possible combinations, with which the new universal constants

Figure 10. Comparisons of the %AADs from the two universal constant sets using eq 23. (a) Speed of sound (u) and dP/dV, and (b) residual (r) and total heat capacities of C1 to C10, over wide ranges of temperature and pressure. The experimental data are taken from NIST.17

isochoric and isobaric heat capacities in Figure 10b by using eq 23 as well. It can be seen that the PC-SAFT (LK) gives better description of speed of sound and dP/dV, except for C1, and the PC-SAFT (GS) gives better prediction of residual isochoric heat capacity for short chain molecules. They present quite similar performance for residual isobaric heat capacities as well as the total heat capacities, both isochoric and isobaric ones. The deviations of Joule−Thomson coefficients are compared in Figure 11a. It can be seen, from Figure 11a and Supporting Information Tables S3 and S4, that the PC-SAF (LK) predicts quite large deviations for Joule−Thomson coefficients of saturated nC8. This is because one “experimental” datum is very close to 0.0. The whole Joule−Thomson coefficient curves of saturated nC8 from these two models are plotted in Figure 11b, G


Article


Figure 12. Ranges of m and ε/k for the PC-SAFT EOS with the new universal constants to have four volume roots at P = 0 bar, which presents the potential risk to have more than three volume roots and discontinuity of the volume roots versus temperature or pressure change when the parameters locate in these ranges.

Figure 14. Phase equilibria of (a) water + n-hexane, data from Tsonopoulos et al.23 and water parameters from Gross and Sadowski;2 (b) water + cyclo-hexane, data from Tsonopoulos et al.24 and water parameters from Liang et al.22 The kij values are shown in the parentheses. GS and LK denote the universal constants from Gross and Sadowski1 and this work, respectively.

However, it has to be pointed out that it is possible for the PCSAFT (LK) to give at most six volume roots at P = 0 bar when the working temperature is lower than 55 K, which could be argued that it is out of the real application ranges for most interested processes. This can be improved by tuning the universal constants to obtain the I1 and I2 curves as those from our previous work14 or Polishuk,16 but the ultimate solution is believed to use a fewer degree polynomials, for example, fourth degree polynomials, or other types of functions to represent the dispersion interactions. In this case, however, the pure component parameters are probably needed to be refitted. 3.5. A Real Example. It is common practice to use large parameter values for dispersion energy and/or segment number when modeling asphaltene.19−21 For instance, Hustad et al.19 used the parameters {m = 11, σ = 4.53 Å, ε/k = 500 K} for asphaltene. The isothermal curve of this asphaltene at 280 K is shown in Figure 13a, and the one from the PC-SAFT (GS) has the typical behavior of four volume roots at P = 0 bar. It can be seen that the volume root will locate in the very heavy reduced density area (η > 0.74) when the pressure exceeds 600 bar, and the discontinuity is seen for the volume roots as the pressure increases. In order to study the relationship of isothermal curve and phase equilibrium, the isobaric−isothermal flash calculations have been conducted for the original reservoir fluid, that is, without nitrogen injection, from 200 to 1000 bar at 280 K. All the

Figure 13. (a) Isothermal curves of asphaltene at 280 K with the parameters from Hustad et al.;19 (b) the molar numbers of asphaltene in its rich phase versus pressure. GS and LK denote the universal constants from Gross and Sadowski1 and this work, respectively.

give four volume roots at P = 0 bar, are shown in Figure 12. It can be seen that it is not possible to have more than three roots, or two stable liquid volume roots, in the entire pressure range if the dispersion energy is not larger than 780 K, or if the working temperature is not lower than 140 K. In another word, the PCSAFT EOS with the new universal constants is practically free of numerical pitfalls of the presence of more than three volume roots. H


Article


Figure 15. Phase equilibria of (a) water with MEG, data from ChiavoneFilho et al.25 Water parameters from Grenner et al.26 and MEG parameters from Tsivintzelis et al.;27 (b) water + 1-butanol, data are from Boublik28 and Sørensen et al.29 The model parameters are from Gross and Sadowski.2 The kij values are shown in the parentheses. GS and LK denote the universal constants from Gross and Sadowski1 and this work, respectively.

Figure 16. LLE of water + methanol + n-heptane, (a) the solubility of n-heptane in aqueous phase; (b) the solubility of methanol in n-heptane rich phase. The data are from Letcher et al.30 The model parameters of water and methanol are from Liang et al.22 GS and LK denote the universal constants from Gross and Sadowski1 and this work, respectively.

It can be seen that the phase diagrams from the two universal constant sets are quite similar to each other. 4.2. 1-Alcohol + n-Alkane Mixtures. The VLE of the binary mixtures of ethanol + n-hexane and 1-octanol + n-dodecane have been modeled by PC-SAFT with both universal constant sets. The results are presented in Supporting Information Figure S2. The kij is fitted to the data for the original parameters with the PC-SAFT (GS). Again it can be readily seen that the phase diagrams are almost identical. The possibility of using the original parameters with the new universal constant sets is also checked for the LLE of methanol + n-alkanes (n-hexane, n-octane, or n-decane). The results with the same model parameters and kij are presented in Supporting Information Figure S3a. The model parameters of methanol are from Liang et al.,14 with speed of sound data in the parameter estimation. The kij is fitted to the data for the original parameters with the PC-SAFT (GS). It can be seen that the phase diagrams are similar, but the differences from the two universal constant sets are not as small as what have been seen in the aforementioned VLE modeling results. It is demonstrated in Supporting Information Figure S3b that a slightly different kij could give almost identical phase diagrams. 4.3. Water Containing Systems. From the phase equilibrium modeling point of view, the water containing systems are good candidates to investigate the possibility of using the original PC-SAFT parameters with the new universal constants, since

parameters, including the binary interaction parameters, are taken from Hustad et al.19 The molar numbers of the asphaltene in its rich phase are presented in Figure 13b. It can be seen that the two models give similar prediction below 600 bar, and the PC-SAFT (GS) predicts nonphysical phase split results when the pressure exceeds 600 bar; that is, one pure asphaltene phase, whereas the PC-SAFT (LK) gives a smooth prediction. The discontinuity of the volume roots versus pressure is believed to the reason for the discontinuous phase equilibrium results along the pressure curve.

4. POSSIBILITY TO REUSE THE ORIGINAL PC-SAFT PARAMETERS As discussed above, the same pure component parameters from the original universal constants are used to fit the new universal constants, and the differences of the final pure component parameters of n-alkanes from these two universal constant sets are less than 1%. In this section, we will investigate the possibility of reusing the parameters from the original PC-SAFT EOS with the new universal constants. 4.1. Binary Hydrocarbon Systems. The VLE of the binary mixtures of ethane + n-decane and propane + benzene have been modeled by PC-SAFT with both universal constant sets using the parameters from Gross and Sadowski.1 The results are presented in Supporting Information Figure S1. The kij is 0.0 in both cases. I


Article


Supporting Information Figure S5. It can be seen that the PC-SAFT (LK) predicts considerably different cloud points by using the kij from the PC-SAFT (GS), while it could give similar prediction with a slightly different kij as shown in Supporting Information Figure S5b. 4.5. Natural Gas Systems. Since it is based on mean-field theory, the PC-SAFT EOS is believed to have difficulties in predicting the critical points when the model parameters are fitted to the vapor pressure and liquid density only. The critical points of 50 multicomponent systems are predicted from PC-SAFT with both universal constant sets using the model parameters from the PC-SAFT (GS), which are compared in Figure 17. It can be seen that the two models predict quite similar critical points. The phase envelope of two natural gas systems are investigated with both universal constant sets by using the original model parameters1 in a predictive way; that is, no binary interaction parameter is used. The results are shown in Supporting Information Figure S6, and it can be seen that the shapes of the phase envelopes are quite similar.

5. CONCLUSIONS Based on the total residual Helmholtz free energy calculated from well-established equations, such as NIST reference models in this work, the temperature and volume dependences of the PC-SAFT EOS have been analyzed. The results reveal that the sixth degree polynomials are sufficient for representing the volume dependence of the model, but the temperature dependence is only valid at low density region or in a narrow range of temperature. Moreover, this will lead to a deterministic procedure for obtaining the universal constants of the dispersion term if the framework and assumed pure component parameters could correctly represent the temperature dependence. The behavior of isothermal curves, that is, pressure versus reduced density at a given temperature, is investigated by screening the parameters and temperature over wide ranges. The results show that it is possible for the original PC-SAFT EOS to have more than three volume roots, or two stable liquid volume roots, and to have discontinuous volume roots as the pressure increases, which will lead to unrealistic prediction of phase behavior. This investigation gives the limitation of the feasible parameter ranges, in which the PC-SAFT EOS will be free of numerical pitfalls. Based on the stepwise procedure developed in the previous work, a practical way is proposed to obtain the 42 universal constants by further assuming that propane has a fixed segment number m = 2. The same pure component parameters from the original PC-SAFT EOS are used. The new universal constants are compared with the original ones on the properties of n-alkanes from methane to decane. The results show that the new universal constants perform better on speed of sound and dP/dV, while the original ones have better description for the residual isochoric heat capacity of short chain n-alkanes and Joule− Thomson coefficients of long chain n-alkanes, and they show similar performance on vapor pressure, density, and isobaric heat capacities. Different universal constant sets could give equally good correlations for vapor pressure and density, but it does not seem possible to obtain a unique set having the best description for all properties. More importantly, the PC-SAFT EOS with the new universal constants is free of numerical pitfalls of the presence of more than three volume roots or the discontinuity of volume roots versus pressure in the real application conditions. The investigation on various types of mixtures reveals that it is feasible to use the

Figure 17. Critical temperature and pressure of 50 natural gas systems. The collections of the detailed data, including mixture composition, critical temperature and pressure, and literature can be found in Sørensen.31 GS and LK denote the universal constants from Gross and Sadowski1 and this work, respectively.

water is a challenging molecule for any EOS.22 The PC-SAFT modeling results of the LLE of water + nonaromatic hydrocarbon (n-hexane or cyclo-hecane), VLE of water + monoethylene glycol (MEG), VLE and LLE of water +1-butanol, and LLE of water + methanol + n-heptane are presented in Figures 14−16 with both universal constant sets. All of the model parameters and binary interaction parameters are based on the PC-SAFT (GS). The combining rule CR-13 is used for cross associations. It can be seen that the two models give similar results for all the systems, except for the 1-butanol rich phase in the water +1-butanol binary mixture and the methanol solubility in the hydrocarbon rich phase of the ternary mixture, which can be tuned by the binary interaction parameters as discussed above for the methanol + n-alkane binary mixtures. It could be argued, however, that the deviations from these two universal constant sets are much smaller than the deviations of modeling results from the experimental data. 4.4. Polymer Containing Systems. The possibility is also evaluated for the polymer containing systems. The VLE of two polymer systems have been modeled by PC-SAFT with both universal constant sets. As seen from Figure S4 in the Supporting Information, the two sets give quite similar pressure− composition curves with the same parameters. The two universal constant sets are further compared for their behavior on modeling the LLE of polypropylene (PP) + propane by assuming PP is monodisperse. The results are presented in J


Article

Industrial & Engineering Chemistry Research original PC-SAFT model parameters with the new universal constants. It is even possible to use the original binary interaction parameter for the VLE modeling, while it might be necessary to tune the binary interaction parameter for some LLE systems. However, it is believed that better description of phase behavior and physical properties could be obtained if the pure component parameters and binary interaction parameter are refitted to the new universal constants.

■

■

REFERENCES

(1) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An equation of state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (2) Gross, J.; Sadowski, G. Application of the perturbed-chain SAFT equation of state to associating systems. Ind. Eng. Chem. Res. 2002, 41, 5510. (3) Kontogeorgis, G. M.; Folas, G. K. Thermodynamic Models for Industrial ApplicationsFrom Classical and Advanced Mixing Rules to Association Theories; John Wiley and Sons: New York, 2010. (4) de Hemptinne, J. C.; Ledanois, J. M.; Mougin, P.; Barreau, A. Select Thermodynamic Models for Process Simulation; Editions Technip: Paris, 2012. (5) von Solms, Nicolas; Kouskoumvekaki, I. A.; Michelsen, M. L.; Kontogeorgis, G. M. Capabilities, limitations, and challenges of a simplified PC-SAFT equation of state. Fluid Phase Equilib. 2006, 241, 344. (6) Liang, X. D. Thermodynamic modeling of complex systems. PhD Thesis, Technical University of Denmark, Kgs. Lyngby, Denmark, 2014. (7) de Villiers, A. J.; Schwarz, C. E.; Burger, A. J.; Kontogeorgis, G. M. Evaluation of the PC-SAFT, SAFT, and CPA equations of state in predicting derivative properties of selected non-polar and hydrogenbonding compounds. Fluid Phase Equilib. 2013, 338, 1. (8) de Villiers, A. J. Evaluation and improvement of the sPC-SAFT equation of state for complex mixtures. PhD Dissertation, Stellenbosch University, Stellenbosch, South Africa, 2011. (9) Yelash, L.; Müller, M.; Paul, W.; Binder, K. A global investigation of phase equilibria using the perturbed-chain statistical-associating-fluidtheory approach. J. Chem. Phys. 2005, 123, 14908. (10) Yelash, L.; Müller, M.; Paul, W.; Binder, K. Artificial multiple criticality and phase equilibria: An investigation of the PC-SAFT approach. Phys. Chem. Chem. Phys. 2005, 7, 3728. (11) Privat, R.; Gani, R.; Jaubert, J. N. Are safe results obtained when the PCSAFT equation of state is applied to ordinary pure chemicals? Fluid Phase Equilib. 2010, 295, 76. (12) Privat, R.; Conte, E.; Jaubert, J.-N.; Gani, R. Are safe results obtained when SAFT equations are applied to ordinary chemicals? part 2: study of solid-liquid equilibria in binary systems. Fluid Phase Equilib. 2012, 318, 61. (13) Polishuk, I.; Privat, P.; Jaubert, J. N. Novel methodology for analysis and evaluation of SAFT-type equations of state. Ind. Eng. Chem. Res. 2013, 52, 13875. (14) Liang, X.; Maribo-Mogensen, B.; Thomsen, K.; Yan, W.; Kontogeorgis, G. M. Approach to improve speed of sound calculation within PC-SAFT framework. Ind. Eng. Chem. Res. 2012, 51, 14903. (15) Liang, X.; Thomsen, K.; Yan, W.; Kontogeorgis, G. M. Prediction of the vapor−liquid equilibria and speed of sound in binary systems of 1alkanols and n-alkanes with the simplified PC-SAFT equation of state. Fluid Phase Equilib. 2013, 360, 222. (16) Polishuk, I. Standardized critical point-based numerical solution of statistical association fluid theory parameters: The perturbed chainstatistical association fluid theory equation of state revised. Ind. Eng. Chem. Res. 2014, 53, 14127. (17) REFPROP, Reference Fluid Thermodynamic and Transport Properties, NIST Standard Reference Database 23, Version 9.0; NIST: Gaithersburg, MD, 2010. (18) DIPPR Database, Design Institute for Physical Property Data; DIADEM, Information and Data Evaluation Manager, version 6.0.1; DIPPR: New York, 2012. (19) Hustad, O. S.; Jia, N. J.; Pedersen, K. S.; Memon, A.; Leekumjorn, S. High pressure data and modeling results for phase behavior and asphaltene onsets of GoM Oil mixed with nitrogen. SPE Annual

ASSOCIATED CONTENT

* Supporting Information S

The refitted pure component parameters of n-alkanes (Table S1), %AADs for vapor pressure and saturated liquid density (Table S2), %AADs for second-order derivative physical properties along the saturated lines (Table S3), %AADs for density and second-order derivative physical properties over wide isobaric−isothermal conditions (Table S4). Phase diagrams of two binary hydrocarbon mixtures (Figure S1), phase diagrams of two binary alcohol + hydrocarbon mixtures (Figure S2), LLE of methanol + hydrocarbons (Figure S3), phase diagrams of two binary polymer containing mixtures (Figure S4), LLE of polypropylene + propane (Figure S5) and phase envelopes of two natural gas systems (Figure S6). This material is available free of charge via the Internet at http://pubs.acs.org.

■

η = reduced density (packing fraction) u = speed of sound CV (r) = residual isochoric heat capacity CP (r) = residual isobaric heat capacity JT = Joule−Thomson coefficients

AUTHOR INFORMATION

Corresponding Author

*Tel.: 045-45252891. Fax: 045-45882258. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge Prof. Michael L. Michelsen for instructive suggestions and comments. The authors also thank Dr. Ilya Polishuk and Dr. Romain Privat for many useful discussions. The project is funded by the Danish National Advanced Technology Foundation (DNATF) and the Department of Chemical and Biochemical Engineering, Technical University of Denmark, to whom the authors express their gratitude.

■

LIST OF SYMBOLS EOS = equation(s) of state SAFT = statistical associating fluid theory PC-SAFT = perturbed-chain statistical associating fluid theory ar = molar residual Helmholtz free energy hs = hard sphere (segment) term of reduced residual Helmholtz free energy hc = chain term of reduced residual Helmholtz free energy disp = dispersion term of reduced residual Helmholtz free energy I1, I2 = the two dispersion terms C = an abbreviation for the hard-chain compressibility expression m = PC-SAFT parameter segment number σ = PC-SAFT parameter segment size ε = PC-SAFT parameter dispersion energy T = temperature P = pressure V = volume ρ = molar density R = gas constant k = Boltzmann constant K


Article

Industrial & Engineering Chemistry Research Technical Conference and Exhibition, New Orleans, LA, Sept. 30−Oct. 2, 2013; SPE 166097. (20) Punnapala, S.; Vargas, F. M. Revisiting the PC-SAFT characterization procedure for an improved asphaltene previpitation prediction. Fuel 2013, 108, 417. (21) Panuganti, S. R.; Tavakkoli, M.; Vargas, F. M.; Gonzalez, D. L.; Chapman, W. G. SAFT model for upstream asphaltene applications. Fluid Phase Equilib. 2013, 359, 2. (22) Liang, X. D.; Tsivintzelis, I.; Kontogeorgis, G. M. Modeling water containing systems with the simplified PC-SAFT and CPA equations of state. Ind. Eng. Chem. Res. 2014, 53, 14493. (23) Tsonopoulos, C.; Wilson, G. M. High-temperature mutual solubilities of hydrocarbons and water. Part I. Benzene, cyclohexane, and n-hexane. AIChE J. 1983, 29, 990. (24) Tsonopoulos, C.; Wilson, G. M. High-temperature mutual solubilities of hydrocarbons and water. Part II. Ethylbenzene, ethylcyclohexane, and n-octane. AIChE J. 1985, 31, 376. (25) Chiavone-Filho, O.; Proust, P.; Rasmussen, P. Vapor−liquid equilibria for glycol ether + water systems. J. Chem. Eng. Data 1993, 38, 128. (26) Grenner, A.; Schmelzer, J.; von Solms, N.; Kontogeorgis, G. M. Comparison of two association models (Elliott−Suresh−Donohue and simplified PC-SAFT) for complex phase equilibria of hydrocarbon− water and amine-containing mixtures. Ind. Eng. Chem. Res. 2006, 45, 8170. (27) Tsivintzelis, I.; Grenner, A.; Economou, I. G.; Kontogeorgis, G. M. Evaluation of the nonrandom hydrogen bonding (NRHB) theory and the simplified perturbed-chain-statistical associating fluid theory (sPC-SAFT). 2. Liquid−liquid equilibria and prediction of monomer fraction in hydrogen bonding systems. Ind. Eng. Chem. Res. 2008, 47, 5651. (28) Boublik, T. Gleichgewicht flüssigkeit-dampf XX. Bestimmung des gleichgewichts flüssigkeit-dampf in systemen, deren komponenten in flüssiger phase beschränkt mischbar sind. Collect. Czech. Chem. Commun. 1960, 25, 285. (29) Sørensen, J. M.; Arlt, W. Liquid−Liquid Equilibrium Data Collection (Binary Systems); DECHEMA Chemistry Data Series; DECHEMA: Frankfurt/Main, Germany, 1995; Vol. V, Part 1. (30) Letcher, T. M.; Wootton, S.; Shuttleworth, B.; Heyward, C. Phase equilibria for (n-heptane + water + an alcohol) at 298.2 K. J. Chem. Thermodyn. 1986, 18, 1037. (31) Sørensen, C. H. Thermodynamic modeling of hydrocarbon mixtures with the PC-SAFT Model. Msc Thesis, Technical University of Denmark, Kgs. Lyngby, Denmark, 2008.

L


New Variant of the Universal Constants in the Perturbed Chain

Recommend Documents