1290
Znd. Eng. Chem. Res. 1994,33, 1290-1298
Application of a Modified Generalized Flory Dimer Theory to Normal Alkanes Costas P. Bokis and Marc D. Donohue' Department of Chemical Engineering, Johns Hopkins University, Baltimore, Maryland 21218-2694 Carol K. Hall Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695-7905 The applicability of chain equations of state to real systems is discussed in this paper. For this purpose, we have compared four theories: the perturbed-hard-chain theory (PHCT) of Prausnitz and co-workers, the generalized Flory (GF) and generalized Flory dimer (GFD) theories of Hall and co-workers, and the statistical associating fluid theory (SAFT) of Radosz, Gubbins, and co-workers. In our comparison, the perturbation expansion in the attractive term was truncated after the firstorder term for all theories. Comparison of these theories with Monte Carlo simulation data for hard chains and square-well chains showed that the GFD theory, which explicitly takes into account the effect of the formation of chains in both the repulsive and the attractive part of the equation, is in best agreement with the data. The GFD theory was further improved by using the simulation data directly to reevaluate the shape parameters c and q, which were found to be density dependent. The new simplified GFD theory gives a significantly better correlation of the properties of normal alkanes than the other four theories.
Intraduction There are a number of different equations of state that can be used to predict the properties of chain molecules found in natural gas and petroleum processing. The best known are the Peng-Robinson (PR; Peng and Robinson, 1976)and the Redlich-Kwong-Soave (RKS;Soave, 19721, but there are several others that are of comparable accuracy. These equations give reasonable predictions of both pure-component and mixture properties for lower molecular weight species, but they develop large errors when the systems contain molecules that differ greatly in size or polarity. For such systems, a significant improvement in the accuracy of both predictions and correlations can be obtained by using a 'chain" equation of state, such as the perturbed-hard-chain theory (PHCT;Donohue and Prausnitz, 1978). This is illustrated, for example, by the work of Peters et al. (19921,who found the simplified perturbed-hard-chain theory (SPHCT, a derivative of PHCT; Kim et al., 1986)to be superior to the PR equation of state in predicting the phase behavior of propanetetratriacontane mixtures in the near-critical region. They concluded that 'This study also supports earlier findings of Peters et al. (1988) and Gasem and Robinson (1990) that, with increasing carbon number, the SPHCT equation of stateshould be preferred over the cubics currently in use." In this paper we discuss several of the common chain equations of state and compare them to molecular simulation data. We also propose a modification of the generalized Flory dimer (GFD) equation of state that improves ita agreementwith the simulation data and then use this equation to correlate experimental data on real fluids.
Review of Equations of State for Chain Molecules Perturbed-Hard-Chain Theory. The perturbedhard-chain theory (PHCT), originally proposed by Prausnitz and co-workers (Beret and Prausnitz, 1975;Donohue
* Author to whom correspondence should be addressed.
and Prausnitz, 19781,represented an important advance in applied thermodynamics because it was the first significant attempt to incorporate the vast body of knowledge developed in statistical thermodynamicsduring the 1960s and 1970s into a practical correlation for compounds of industrial interest. PHCT had the advantage that it was valid over a wide range of densities, could be applied to both simple (argon-like) and complex (polymer) molecules, and could predict pure-fluid and mixture equilibria with a minimum number of adjustable parameters. PHCT was derived using the perturbation theory of statistical thermodynamics. In perturbation theory, one expands the properties of a system of spherical molecules around a reference system with known properties, in this case the hard-sphere fluid. The resulting equation of state is a sum of terms; the first is the equation of state for the hard core system and the others are corrections or perturbations which take into account the effects of molecular attractions. The primarycontributionof PHCT was to show how to apply perturbation theory to a fluid containing highly asymmetric molecules which could be modeled as chains of hard spheres surrounded by squarewell attractions. In their derivation, Prausnitz and coworkers used Prigogine's shape parameter c (Prigogine et al., 1953, Prigogine, 1957), which is a measure of the density-dependent degrees of freedom in the partition function. Mathematically, the PHCT leads to an equation of state that is of the form
where 2 is the total compressibility factor of the chain molecule, z;Bp and Z!tt are the repulsive and attractive contributions to the compressibility factor for spherical molecules (monomers),respectively, and c is Prigogine's shape parameter, which is a characteristic of each chain molecule. In PHCT, zf"P was calculated from the Carnahan-Starling (1972)equation for hard spheres, and qttwas calculated using a fourth-order perturbation
0888-5885/94/2633-1290$04.50/00 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1291 expansion due to Alder et al. (19721, i.e.,
where An, are universal constants, and the reduced temperature and volume are defined by
where Zy and 2 : are the hard-chain compressibility factors for monomers and dimers, respectively; qttand .Zft are the attractive contributions to the compressibility factors for monomers and dimers, respectively. Honnell and Hall (1989) used the Carnahan-Starling (1972) equation for hard spheres and the Tildesley-Streett (1980) equation for hard dimers. Yn is a function of excluded volumes, defined by
(3)
Here, u* is a characteristic volume per mole, u is the molar volume, c is the depth of the square-well potential, q is a measure of the surface area per molecule,k is Boltzmann's constant, and T is the temperature. Although eq 1 seems to imply that both the repulsive and attractive contributions to the compressibilityfactor are multiplied by the same factor, c, this is not the case. Because of the way the reduced temperature is defined in PHCT (eq 31, the first-order term in the perturbation expansion of the attractive term is actually multiplied by the parameter q (whichis generallydifferent from c); thus, eq 1may be rewritten as
higher order terms (4) In comparing PHCT with simulation data, the parameters c and q are treated as adjustable parameters. GeneralizedFlory Theories. More recently,Hall and co-workers have developed theories for chain molecules which rigorously take into account the chainlike structure and flexibility of these molecules. They began with a molecular picture in which chain molecules are modeled as a collection of freely jointed, tangent hard spheres. First, Dickman and Hall (1986) developed the generalized Flory (GF) theory by extending Flory's lattice model to continuous-spacefluids. They used the concept of the "excluded volumen to take into account the effect of size in the equation of state. Later, Yethiraj and Hall (1991a) showed that both the repulsive and the attractive terms of the equation of state have the same dependence on the excluded volume. The GF equation of state has the form
where a is the ratio of the excluded volume of an n-mer molecule to that of a monomer molecule (a= ue(n)/ue(l)). In the GF theory, the parameter a multiplies both the repulsive and the attractive compressibility factors for monomersto yield the compressibilityfactor for the chain; furthermore, a is calculated from molecular geometry and is not an adjustable parameter, as are c and q in PHCT. Honnell and Hall (1989)proposed a more sophisticated theory for chain molecules, the generalized Flory dimer (GFD) theory. The GFD theory explicitly takes into account the connectivity of a chain molecule. Also, accordingto Yethiraj and Hall (1991a),both the repulsive and attractive contributions to the equation of state have the same dependence on molecular size. The compressibility factor for a chain molecule in GFD is calculated in terms of those for a monomer and a dimer molecule, and has the form
(7) where ue(l), ~e(2),and ue(n) are the excluded volumes of monomers, dimers, and n-mers, respectively. After algebraic rearrangement, eq 6 can be written in the form
[
z=1+
T P -r e p
l+(Y,+l)
2
TP
]T'+
TP
where and qp are the repulsive contributions to the compressibility factors for monomers and dimers, respectively. Equation 8 is of the same form as PHCT and GF; the difference is that, in GFD, the quantities inside the square brackets are density dependent, in contrast to the constants c and q in PHCT and a in GF. Statistical AssociatingFluid Theory. The statistical associatingfluidtheory ( S m )was proposed by Chapman et al. (1990) and Huang and Radosz (1990,1991). SAFT is an engineering application of Wertheim's first-order thermodynamic perturbation theory of polymerization (Wertheim, 1987). For nonassociating chain molecules, the compressibility factor from SAFT has the form
z = 1+ m T p + (1- m)Pk + mqtt
(9)
where m is the number of segments in the chain, and (1
- m ) Z c k is the compressibility factor increment due to the formation of the chains. The attractive contribution to the compressibilityfactor, which takes into account the dispersion forces, is a power series similar to that in the PHCT, with the coefficientsrefitted to experimental data on argon by Chen and Kreglewski (1977). Equation 9 can be rearranged in the form
higher order terms (10) where Du are the coefficientsof the first-order term in the perturbation expansion. The reduced volume (B) in SAFT is defined in the same way as in PHCT (eq 3); however, the reduced temperature is defined as p = kT/c and, therefore, the attractive term of the monomer is multiplied by the number of segments m to yield the attractive term of the chain. The monomer repulsive term is multiplied by the quantity inside the square brackets,which is density dependent, as is the case with GFD.
Comparison with Simulations In this section we examine the accuracy of these four theories by comparingthe resulting compressibilityfactors with those obtained from simulation data on hard-chain
1292 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 Table 1. Size Parameters in Various Theories n
C
9
4 8 16
2.881 4.812 8.323
3.959 6.422 9.447
m 4 8 16
a
Y"
3.004 5.637 10.90
1.915 5.745 13.41
and square-well chain molecules. To ensure that we are comparing equivalent expressions, we included only the first-order term in the perturbation expansion for the attractive term for SAFT and PHCT. This has little effect on the calculations, since the first-order term dominates the behavior of the attractive term at moderate and high densities where simulation data are available. The monomer and dimer attractive contributions to the compressibility factor that are needed in the GF and GFD theories are calculated using mean-fieldexpressions(Bokis et al., 1992),which are given in the Appendix. Yethiraj and Hall (1991a) used integral equations to evaluate the monomer and dimer attractive contributions. However, their equation of state is not pressure explicit, and therefore, numerical iterations are required to calculate the pressure at each point of interest. The mean-field expressions that we use here are equivalent to the firstorder perturbation theory used in SAFT and PHCT, and therefore,a meaningfulcomparisonof the different models is obtained. Three of these theories (GFD, GF, and SAFT) can be compared directly to the simulation data. However, this is not the case with PHCT; the shape parameters c and q must be fitted to the data. To achieve this, first we fit the hard-chain part of PHCT (eq 1) to Monte Carlo simulation data for freely-jointed, tangent hard 4-mers, 8-mers, and 16-mers (Dickman and Hall, 19881, in order to obtain the parameter c associated with the repulsive term. Then, we fit the whole eq 1 to Monte Carlo simulation data for square-well 4-mers, 8-mers, and 16mers (Yethirajand Hall,1991b),and determined the shape parameter q. Values of these parameters are presented in Table 1. Figure 1 shows a comparison of the four equations of state with the simulation data for hard 8-mers and 16mers (Dickman and Hall, 1988;Denlinger and Hall, 1990). The GF theory overpredicts the compressibilityfactors in all the cases considered here. The PHCT, in which the parameter c was fitted to the very same data, although better than GF, is not as accurate as one would like. It overestimates the data at low densities, and it underestimates the data at high densities. However, the GFD and SAFT models, which explicitly take into account the formation of the chains, are in reasonable agreement with the simulation data over the entire density range. Figures 2 and 3 show the compressibility factors for square-well 4-mers and 8-mers, respectively, calculated with the GF, GFD, SAFT, and PHCT models at two values of the reduced temperature. Comparison is made with the Monte Carlo simulation data of Yethiraj and Hall (1991b). The GF theory overpredicts the data over the entire density range, as was the case for hard chains. SAFT, although quite accurate for hard chains, as Figure 1 illustrated, is not very accurate for the square-well molecules. This is because it does not take into account the effect of the formation of the chains in the attractive term. The GFD theory,which explicitlytakes into account the fact that chain molecules consist of connected segmenta, has the best performance. PHCT also appears to be in good agreement with the simulation data. This suggests that a cancellation of errors occurs between the repulsive and attractive terms in PHCT, sincecalculations
.
N
14
PHCT -.-.SAFT
-
7
0
18
N 12
01 0
1
1
I
0.1
0.2
0.3
0.4
ll Figure 1. Comparison of various equations of state for hard-chain &men and 16-men. Monte Carlo simulation data of Dickman and Hall (1988) and molecular dynamics simulation data of Denlinger and Hall (1990) also are shown.
12
N
E
4
I
10
7
.
Mcdrta
GFD OF ....... PHCT SAFT
N
4
1 -21 0
'
' 0.1
'
'
'
0.2
' 0.3
'
' 0.4
0.5
ll Figure 2. Comparison of various theories for squarewell 4-men at two values of the reduced temperature. Simulationdata of Yethiraj and Hall (1991b) ale0 are shown.
for the repulsive term with PHCT were not in good agreement with simulations (Figure 1). The comparison of calculationsusing these four theories to simulation data shows that the GFD theory, which rigorously takes into account the formation of the chains
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1293
0
lo
f
a
i-.;/.j 0 ,
GFD OF ..- .- - - PHCT SAFT
N ' 6
10 4
4
-2
P
20
6
i4
16
I 3 3
N 10
.......
0
2
SI 0
'
'
1
0.1
I
"
"
0.3
0.4
0.2
I? 0.6
11 Figure 3. Comparison of various theories for square-well &mers at two values of the reduced temperature. Simulationdata of Yethiraj and Hall (1991b) also are shown.
in both the repulsive and attractive terms, can predict the properties of chain molecules with reasonable accuracy. Development of the New Expression Although GFD is reasonably accurate in comparison with simulations, it is relatively complicated since it combines equations for both monomers and dimers. Our goal in this paper is to develop a new equation of state for chain molecules, based on information obtained from the GFD theory and from simulations, that will be simpler than GFD and yet very accurate in comparison to both simulations and real data. In the GFD equation, eq 8,the quantity inside the square brackets that multiplies qp is density dependent. Bokis and Donohue (1992) showed that this quantity is a linear function of the reduced density for each hard n-mer. In addition, the slope and intercept of these straight lines increase, as the chain length increases. Therefore, Bokis and Donohue (1992) developed a simple expression for the compressibilityfactor of hard-chain molecules,which mimics the behavior of the GFD theory. This expression has the form
zhc= 1+ c(q)ZfeP
(11)
where c(q) is a linear function of the density and the chain length. In Figure 4 we illustrate the density dependence of the shape parameter for the repulsive term in the GF, GFD, PHCT, and SAFT theories, by plotting the ratio of the repulsive contribution to the compressibility factor for a chain moleculeto that for a monomer molecule versus the reduced density, q. Calculations are presented for hard 4-mer, &mer, and 16-mer molecules, and Monte Carlo simulation data (Dickman and Hall, 1988) also are included. For GF and PHCT, this ratio is identical to the shape parameters a and c, respectively, which are constant
3J
ii3 a
.F
2.6
-
2 0
,
0
0.1
0.2
11
0.3
0.4
0.5
Figure 4. Ratio of the repulsive Compressibility factors of chain moleculesto those of sphericalmoleculesv e r w the reduced density, for hard 4-mers, &men, and 16-men. Calculations are made with the PHCT, GF, GFD, and SAFT equations. Monte Carlosimulation data of Dickman and Hall (1988) (filled symbole) and molecular dynamicssimulation data of Gao and Weiner (1989) (open symbole) also are shown.
for each n-mer. Therefore, a horizontal straight line is obtained for each n-mer with these two models. In GFD and SAFT, the ratio Z:p/qp is equal to the quantities in eqs 8 and inside the square brackets that multiply 10, respectively. Figure 4 shows that the GFD and SAFT theories predict a linear (or approximately linear) ratio of Z:p/~p.This ratio also is approximately linear in the Monte Carlo simulation data; however, it can be seen in Figure 4 that the slopes and intercepts are different from those predicted by either GFD or SAFT. In particular, the slopes from the simulation data are larger than those predicted by GFD or SAFT, and the intercepts are smaller than those predicted by these two theories. This suggests using the simulation data for hard chains directly to evaluate the coefficients of c(q) in eq 11; this will result in an equation more accurate than either SAFT or GFD in comparison to simulations, especially at low densities. This equation has the form
cp
zhc= 1 + (c, + c b q ) q p
(12)
where ca and Cb are constants for each n-mer. It has been found that these constants vary linearly with the chain length, according to
+ 0.25n
(13a)
= 1.171(n - 1)
(13b)
c, = 0.75 cb
1-1
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
24
repulsive part of the equation). In Figure 5 simulation data of Yethiraj and Hall (1991b) also are included. Yethiraj and Hall calculated the compressibility factors for square-well 4-mers,8-mers, and 16-mers. The attractive contribution was obtained by subtracting the hardchain contribution (calculated from eq 12) from the total compressibilityfactors. One sees that the simulationdata varies also confirm the fact that the ratio qeiqtt approximately linearly with density; however, the slopes and intercepts are different from those predicted by GFD. The simulationdata yield larger slopesand lower intercepts than the GFD theory. Therefore, the disagreement between GFD and simulations is larger at low densities. Since we used first-order perturbation theory in our calculations, temperature cancels out when the ratio qtt/gtt is calculated with all four theories considered here. This is not the case with the simulation data. Nevertheless, this ratio calculated from the simulations appears to be independent of the temperature at moderate to high densities, as Figure 5 shows. By correlating the simualation data directly, the following simple expression for the attractive term is obtained
-
.... ..
- - - . SAFT
16 12 3-
-.-.-........................... ... I
-- -
n
2" = (qa + Q b M y
(14)
where qa and Qb are constants for each chain and vary approximately linearly with n according to
1l i " " 0.1 0
qa = 0.856 Qb =
I
0.2
rl
0.3
+ 0.144n
(15a)
SAFT
-
0.4
0.5
Figure 5. Ratio of the repulsive compreseibility factors of chain moleculesto those of sphericalmoleculesversus the reduced density, for hard 4-mers, &men, and 16-mers. Calculations are made with the PHCT, GF, GFD, and SAFT equations. Monte Carlo simulation dataofYethirajandHall(1991b)aleoareehown(x,kT/e= 1.5;filled symbols, kT/c = 2.0; open symbols, kT/c = 3.0.
Equation 12 is expected to describe accurately the volumetricbehavior of chain molecules. However,it does not reduce to the correct second virial coefficient limit as the density approaches zero. In a recent paper, Bokis et al. (1993) used molecular dynamics results of Gao and Weiner (1989)to show that the linear shape parameter in the repulsive term is a good approximation at moderate and high densities, but that it is not very accurate at low densities. They developed an expression that is very accurate over the entire density range: from liquid-like densities to the second virial coefficient limit. However, the goal of this work is to develop an equation of state that is simple in form and yet describes the properties of chain molecules with reasonable accuracy;therefore, the second virial limit is not taken into account here. Figure 5 shows a plot of the ratio of the attractive contribution to the compressibility factor for n-mers to that for monomers versus the reduced density, for 4-mers, 8-mers, and 16-mers. This ratio is plotted for the GF, GFD, SAFT, and PHCT theories. For GF, PHCT and SAFT, this ratio is the value of the parameters a,q, and m,respectively; therefore, horizontal straight lines are obtained with these theories for each n-mer, as shown in Figure 5. In contrast, the GFD theory predicts this ratio to be density dependent. Although some curvature is observed at high densities, this density dependence can be correlated quite well with a linear function of density. In addition, the slope and intercept of these lines increase, as the chain length increases (as was the case for the
2.54(n - 1)
Wb)
Putting the attractive and repulsive pieces together, we find that the equation of state for the total compressibility factor of the square-well fluid is given by 1+ (Ca
Cb?)Fp
+ (Qa
4bS)g'
(16)
The second virial coefficient limit of the attractive term
was not investigated,sincewe were trying to obtain a simple expression, similar to that for the repulsive part of the equation. Therefore, eq 14 (and as a result eq 16)does not reduce to the correct second virial coefficient for chain molecules as the density approaches zero. Equation 16 is expected to provide an accurate description of the PVT properties for chainlike molecules, but it will not be accurate when low-density calculations are performed (such as calculations of dew points). Nevertheless, eq 16 will have a better performance at low densities than the mean-field version of PHCT, GF, SAFT, and GFD.
Comparison with Experimental Data In this section we examine the applicability of the theories discussed previously to real systems. For this purpose, we used PHCT, SAFT, GF, GFD, and the modified GFD (eq 16) to correlate vapor pressure and liquid density experimentaldata on severalnormal alkanes from methane to eicosane. For each substance, 10 vapor pressure data points were used that cover the entire range from the triple point to the critical, together with about 12high-pressure liquid volume data points. Table 2 gives details about the temperature and pressure range of the data used for the regression. For each theory, three purecomponent parameters were estimated by the data reduction: a characteristic energy parameter, a characteristic volume parameter, and a size or shape parameter. Table 3 shows the pure-component parameters for each theory. As noted above, only the first-order term in the
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1295 Table 2. Temperature Range and Pressure Range of the Experimental Data Used for Regression vapor press. liquid vol compound T range (K)NO rep Trange (K)P range (bar) Na rep 7-300 11 1 114-171 methane 91-188 10 1 13 2 160-600 72-746 ethane 130-305 10 2 12 3,4 253-533 1-203 butane 173-423 10 4 258-393 12-1000 13 4 octane 293-669 10 4 138-7846 12 3,4 273-510 decane 344-563 10 3 1-lo00 16 4 dodecane 373-658 10 3,4 303-393 305-383 1 9 3 nonadecane 473-613 10 3 1-lo00 18 4 eicosane 483-775 10 3,4 373-523
ck e!
P
,"
._e
ii
w
*
a N: number of data points used for regression. (1) Goodwin, R. D.;Haynes,W. D. "TheThermophysicalPropertiea of Methane from 90to MX)K at Pressurea to 700Bar";NBS TechnicalNote 653, Boulder, CO, 1974. (2) Goodwin, R. D.; Roger, H. M.;Straty, G. C. "The Thermophyaical Properties of Ethane from 90 to 600K at Pressures to 700 Bar"; NBS Technical Note 684, Boulder, CO, 1976. (3) Vargaftik, N. B. Tables on the Thermophysical Properties of Liquids and Gases;John Wiley & Sone: New York, 1975. (4) "SelectadValues of Properties of Hydrocarbons and related Compounds";Research Project 44 of the American Petroleum Institute and the Thermodynamic Research Center, Texas A&M University: College Station, Tx,1989.
- - -r- l PHCVGF O C T A N E ....
Table 3. Parameters for Each Theory That Are Evaluated from Pure-ComDonent Data Reduction theory PHCT
SAFT GF GFD eQ 16
energy cqlck ulk elk elk elk
vol
/
~
size (shape)
U*
C
u*
m
u*
a
V*
Y,
u*
n
perturbation expansion in the attractive term was considered in all calculations. Detailed results are presented in Table 4. Calculations with the Peng-Robinson (PR) equation of state are also included for comparison. We should note that no parameters were fitted to experimental data for the PR equation, (Le., literature values of the critical constants and the acentric factors were used). Therefore, the comparison of PR with the other models may not be entirely fair; nevertheless, the PR is included in order to illustrate the relative trends in the errors produced by PR and the theories discussed in this paper, rather than the absolutevalues of these errors. From the tabulated results, one sees that the PR equation of state gives large errors in the liquid volume calculations. The vapor pressure predictions are reasonably good for small molecules, but errors become large for compounds larger than decane. All the other theories give better results than the PR equation for large molecules. The PHCT and GF theories give identical results, since they are identical in form (although parameters are defined differently in each theory). PHCT (or GF) is not in good agreement with the experimental data compared to the newer models. The quality of the fits improves significantly when we use
/
0.4
0.5
0.6
0.7
0.8
0.9
1
TflC
Figure 6. Vapor pressure calculationsfor octane and eicoaane with PR, PHCT, GFD, SAFT,and eq 16.
SAFT,which takes into accountthe formationof the chains in the repulsive term. Further improvement is obtained with the GFD theory, which explicitly accounts for chain connectivity in both the repulsive and the attractive terms. The expression that we propose in this study gives results similar to those obtained by the GFD equation; however, it has the advantage that it is simpler than the GFD equation and therefore more suitable for engineering calculations. Figure 6 shows percent error in the vapor pressures of octane and eicosane for each theory. The PR equation of state gives large errors as the carbon number increases, especially at low temperatures (nearthe triple point). Large errors also are obtained with PHCT (or GF), particularly as the critical point is approached. GFD and eq 16 give a very good correlation of the vapor pressure data, and errors are close to experimental uncertainty. SAFT also gives a good description of the vapor pressure behavior; however, the critical point calculation with SAFT is not as accurate as with GFD or eq 16. Figure 7 shows percent error in the liquid volumes of octane and eicosane for each theory. The accuracy of the various equations of state is similar to that found in the vapor pressure calculations (Figure 6). Errors with the PR equation are very large; for eicosane they are larger
Table 4. Reoultr from Fitting Exwmimental Data for Normal Alkanes compound methene ethane butane octane decane dodecane nonadecane eicosane average
PR 0.891 2.630 2.194 1.522 1.488 3.237 13.08 9.530 4.297
av absolute % error in vapor press. PHCT SAFT GFD 2.080 3.613 1.892 3.528 1.148 1.021 1.682 4.330 8.154 1.587 2.014 4.461 2.439 0.854 1.114 3.156 1.442 1.805 0.633 0.164 0.139 3.437 1.498 1.647 3.008 2.236 1.396
ea16 0.649 2.249 2.248 1.540 1.071 1.667 0.870 1.292 1.448
PR 9.712 7.073 2.225 2.865 11.13 10.49 20.89 20.32 10.59
av absolute % error in liquid vol PHCT SAFT GFD 3.725 1.750 1.203 4.558 1.752 1.405 4.907 4.627 3.380 3.280 0.852 0.472 7.362 1.588 1.199 3.828 1.170 0.545 1.793 0.173 0.266 3.434 0.895 0.843 4.111 1.601 1.165
ea16 1.435 1.463 2.833 0.947 1.957 0.858 0.628 1.131 1.407
1296 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
1l-
12
-~ --
GFD Eq. (16)
EICOSANE T=25OoC
I
y
9 ~
..
,
- -
/
~~
6
3
OCTANE T=30°C
PHCT/GF -.SAFT
-
GFD Eq. (16)
-.
N
.......
-
0
Eq. (16)
0
0.01)
0.16
0.24
0.32
II *-
.-
-
Figure 8. Compressibility factors of an equimolar mixture of hard spheres and hard 8-mers. Calculations are made with PHCT, GF, GFD,SAFT, and eq 16. Monte Carlo simulation data of Honnell and Hall (1991) also are shown.
/
~
14 4
6
5
7
8
WP) (bar) Figure 7. Liquid volume calculations for octane and eicoeane with PR, PHCT, GFD,SAFT, and eq 16. The PR equation of state gave errors larger than -15% for eicosane.
than -15 % ,so they were not included in Figure 7. PHCT also gives large errors. Again, the best performance is obtained with the GFD equation and the model proposed in this study (eq 16).
Extension to Mixtures In this section we extend this comparison of equations of state to mixtures of hard chains and mixtures of squarewell chains. In doing this we followthe method of Honnell and Hall (1991), who replaced the actual mixture by hypothetical one-component fluid exhibiting the same thermodynamic behavior. This idea is similar to conformal-solution theory, in the sense that all site-site interactions take the same functional form. Therefore, the mixture is modeled by a hypothetical one-component fluid at the same volume fraction, 9, and whose molecules have a chain length equal to the molefraction average of the chain lengths of the components in the mixture, Le., ii
= Cxini
(17)
where fi is the average chain length, X i is the mole fraction of component i, and ni is the chain length of component i. This is the average length that we will use in SAFT and in the equation developed in this work (eq 16). In addition, the excluded volume and the PHCT parameters c and q of a chain molecule scale linearly with chain length, so that the mixture value for these parameters also is given by the mole-fraction average of the corresponding parameters of the components of the mixture. As a basis of comparison for mixtures of hard chains, we used the Monte Carlo simulation data of Honnell and Hall (1991). In their simulations, monomers were modeled by hard spheres of diameter u and each chain was represented by a “pearl necklace”of freelyjointed, tangent, hard spheres of the same diameter. Figure 8 shows
N
0
0.1
0.16
0.24
0.32
tl Figure 9. Compressibility factors of an equimolar mixture of hard 4-mers and hard &mers. Calculations are made with PHCT, GF, GFD,SAFT, and eq 16. Monte Carlo simulation data of Honnell and Hall (1991) also are shown.
calculations for an equimolar mixture of hard spheres and 8-mers,using GF, GFD, SAFT,PHCT, and eq 16. Similar behavior to the one-component case is observed. SAFT, GFD, and eq 16 are in very good agreement with the simulation data. In contrast, GF overestimates the mixture compressibilityfactors in the entire density range; PHCT, although the c parameter for 8-mers was fitted to simulation data for hard 8-mers, is not in very good agreement with the mixture simulation data either. Similar results were obtained for the case of an equimolar mixture of hard 4-mers and hard 8-mers,as shown in Figure 9 (simulation data from Honnell and Hall (1991)). GFD and SAFT are in good agreement with the data, whereas GF and PHCT overestimate the simulation data. Equation 16 is in the best agreement with the data over the entire density range. We also performed calculations for mixtures of squarewell chains. In particular, we compared all the theories considered in this work to Monte Carlo simulation data for an equal site fraction mixture of square-well spheres
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1297 I,
f
N
01 0
0.1
I
1
I
0.2
0.3
0.4
Figure 10. Compressibilityfactors of an equal site fraction mixture of square-well spheres and square-well8-mers. Calculationsare made with PHCT, GF, GFD, SAFT, and eq 16. Monte Carlo simulation data of Wichert and Hall (1993) also are shown.
and square-well8-mers at a reduced temperature of kTle = 2.0. (simulation data by Wichert and Hall (1993)).The results are illustrated in Figure 10. One sees again that the GF theory overestimates the mixture compressibility factors everywhere. GFD is in better agreement with the simulations than SAFT; this is to be expected, since GFD accounts for the chain formation in both the repulsive and the attractive terms, whereas SAFT accounts for the chain formation only in the repulsive term. Equation 16 also is in very good agreement with the data. Good accuracy also is obtained from the PHCT equation; however, parameters c and q for 8-mers had been fitted to simulation data for pure 8-mers, as was mentioned before. In addition, the mole fraction of 8-mers in this mixture is relatively small (1/9), which gives rise to the good agreement of PHCT with the Monte Carlo data.
Conclusions In this paper we have presented a review of several theories for chainlike molecules. We examined the accuracy of their repulsive and attractive contributions separately, by comparingthem with molecular simulation data for hard chains and square-well chains. The generalizedFlory dimer theory (Honnelland Hall, 1989),which explicitly accounts for chain connectivity in both the repulsive and the attractive parts of the equation of state, gives the best agreement with the simulations. We showed that the GFD theory can be simplified, if the rather complicated density dependence which results from the dimer expressions is included into densitydependent shape parameters c and q in the repulsive and attractive term, respectively. These shape parameters vary approximately linearly with density and molecular size. The coefficientsfor these shape parameters were evaluated using the simulation data, rather than the GFD equation. In order to examine the applicability of all the theories considered in this paper, we fit pure-component experimental data for several normal alkanes. First-order perturbation theory was used in all theories to account for attractive forces. The PHCT and GF equations of state, which do not account for the effect of the formation of chains, are not in good agreement with the experimental data. In contrast, SAFT and GFD, which explicitly
account for chain connectivity,describe the vapor pressure and volumetric behavior of normal alkanes up to eicosane with good accuracy. The GFD theory is more accurate than SAFT at the pure-component critical point. The model that we propose in this paper (eq 16) is of the same order of accuracy with GFD; however, it has the advantage that it is simpler than GFD and, therefore, more appealing from an engineering point of view. We believe that the accuracy of eq 16 can be improved when simulation data for more chain molecules are included in the evaluation of the density-dependent shape parameters, c(7)and q(7). Using the idea of conformal-solution theory, we have extended this new equation of state to mixtures. A comparison of all equationsconsideredin this study against Monte Carlo simulation data for mixtures for hard chains and mixtures of square-wellchains showed that GFD and its simplified form developed here (eq 16) are in the best agreement with the simulation data.
Acknowledgment This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S.Department of Energy under Contract Nos.DE-FG02-87ER13777and DE-FG0591ER1481. Additional support was provided by the Gas Research Institute under Contract No. 5089-260-1888.
Nomenclature A,,,,, = universal constants in PHCT c = shape parameter in PHCT c,, Cb = constants in eq 12 Dv = universal constants in SAFT k = Boltzmann’s constant n, m = number of segments q = shape parameter in PHCT qp, qb = constants in eq 14 T = temperature
?’
= reduced temperature
u = molar volume u* = characteristic volume u, = excluded volume
ir = reduced volume x = mole fraction Y, = size parameter in GFD
2 = compressibility factor Greek Letters a = shape parameter in GF e
= square-well depth
= reduced density
Subscripts 1 = monomers 2 = dimers i = component i n = n-mers Superscripts att = attractive hc = hard chain rep = repulsive
Appendix In order to implementthegeneralizedFlory dimer theory for square-well chains, one needs expressions for the monomer and dimer attractive contributions to the equation of state. Bokis et al. (1992) developed such
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expressions using mean-field theory:
+ +
