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Correlation of synthesis gas solubility in n-paraffin solvents and Fischer—Tropsch waxes
Fluid Phase Equilibria, 46 (1989) 179-195 Elsevier Science Publishers B.V., Amsterdam
179 -
Printed
in The Netherlands
CORRELATION OF SYNTHESIS GAS SOLUBILITY SOLVENTS AND FISCHER-TROPSCH WAXES J.S. CHOU
and K.C. CHAO
School of Chemical Engineering, (Received
IN n-PARAFFIN
Purdue University,
May 27, 1988; accepted
West Lafayette,
in final form November
IN 47907 (U.S.A.)
4, 1988)
ABSTRACT
Chou, J.S. and Chao, K.C., 1989. Correlation of synthesis gas solubility and Fischer-Tropsch waxes. Fluid Phase Equilibria, 46: 179-195.
in n-paraffin
solvents
A correlation of the solubility of Fischer-Tropsch synthesis and product gases in heavy n-paraffin waxes is obtained with the Lacombe-Sanchez equation of state (EOS). The EOS parameters for pure n-paraffins are developed and correlated as functions of carbon number. Interaction parameters with the n-paraffins are determined from solubility data and expressed as functions of carbon number of the n-paraffin solvent for the gases hydrogen, carbon monoxide, methane, carbon dioxide, ethylene and ethane.
INTRODUCTION
The Fischer-Tropsch synthesis is a method of indirect coal liquefaction in which hydrogen and carbon monoxide obtained from coal gasification are reacted to produce hydrocarbons and/or oxygen-containing derivatives. One of the more effective reactor types for Fischer-Tropsch synthesis is the three-phase slurry reactor (Kiilbel and Ralek, 1980). Small particles of the synthesis catalyst are suspended in a heavy wax mixture and the reactants are bubbled through the slurry. Satterfield and Stenger (1985), Albal et al. (1984) and Stern et al. (1983) have concluded that the solubilities of the reactant and product gases in the wax mixture are needed to design these reactors. In this work the Lacombe-Sanchez equation of state is used for the correlation of solubilities of hydrogen, carbon monoxide, methane, carbon dioxide, ethylene and ethane in n-eicosane, n-octacosane, n-hexatriacontane and two Fischer-Tropsch waxes to develop a general description of the solubility of these gases in heavy n-paraffins. 0378-3812/89/$03.50
0 1989 Elsevier
Science Publishers
B.V.
180
THE LACOMBE-SANCHEZ
EQUATION
OF STATE
The Lacombe-Sanchez EOS was chosen to correlate the solubility data because the n-paraffins are chain molecules considered in Guggenheim’s statistical mechanical lattice model from which the EOS is derived. Sanchez and Lacombe (1976) showed their equation to represent polymeric liquid densities effectively. Sanchez and Lacombe (1976) derived an expression for the Gibbs free energy of a pure fluid:
G Nrr *
=.G”= -fi+&+i;
(e-l)ln(l-fi)++
In
where the symbols are as described in the symbols list. The EOS obtained from eqn. (1) is fi* + F + f
[
( 11
In(1- fi) + 1 - + p” = 0
PARAMETERS
(2)
FOR n-PARAFFINS
Only three of the six reducing parameters (either T *, P *, p* or e *, r, are required to characterize a fluid. Each set can be converted to the other by using their definitions and the following relations u*)
RT*p* ----= P*
* *=-- M VP r
(3)
and p*v* -= RT*
1
(4)
Values of r and u* have been determined by Sanchez and Lacombe (1976) for the n-paraffins up to n-C,,. Figure 1 shows the dependence of the molecular core volume ru* on the carbon number N,. A linear dependence is observed and is represented by rv* = lS.OSN, + 18.78
(5)
Extrapolation to heavy n-paraffins by this equation seems reasonable in view of the constant incremental value of rv* for each additional CH, group for the large n-paraffins. Since the reciprocal of ru* is p*/M, eqn. (5) is also a correlation of p* as a function of carbon number. With u* given by eqn. (5), we determine T * and P* for the n-paraffins by fitting saturated vapor pressure values taken from the API 44 tables (Wilhoit and Zwolinski, 1971). Figures 2 and 3 show the dependence of T *
181
400. 3. E >_
300.
E s -
200.
1 100.
I
0. i
5.
0.
I
I
I
I
10.
15.
20.
25.
CARBON NUMBER Fig. 1. Molecular core volume of n-paraffins: (1976); - - -, eqn. (5).
30.
o, parameters
from
Sanchez
and Lacombe
and the product MP* on carbon number. Both these quantities are fitted with the growth equation (Gasem and Robinson, 1985a),
y+_y+;-c*)
exp[ -c4(NC - l)(l
- c2)])‘;“-‘*’
The quantity Y can be identified with either T * or MP *. When Y is T * , the
r
695.
645.
?c i
595.
I
I
I
I
I
I
I
I
40.
80.
80.
4 545.
d
495. 0.
20.
CARBON Fig. 2. T* of n-paraffins;
100
NUMBER
o, parameters
fitted from vapor pressure
data;
-
-
-,
eqn. (13).
182
constants are cr = 417, c2 = -0.405, c3 = 414 and cq = 0.044. When Y is MP*, the constants are c1 = 6.13, c2 = -0.0884, cg = 1.504 and cq = 0.00737. The curves shown in Figs. 2 and 3 represent eqn. (6). Table 1 lists the correlated parameters T *, P * and p* for selected n-paraffins between n-C, and n-C,,. Using these parameters, the Lacombe-Sanchez (LS) equation fits the n-paraffin saturated vapor pressure data reported in the API 44 tables with the average absolute deviations (% AAD) presented in Table 1, where % AAD is defined to be
where n is the number of vapor pressure with an overall average of 1.7% AAD.
TABLE
parameters
and
comparison
of
calculated
vapor
pressures
44 data for n-paraffins
NC of n-paraffin 7
Z-* (K)
P* (atm)
P* (gcme3)
%AAD
501.0
3000.0
0.81
2.8
8
512.0
3000.0
0.82
2.5
9
523.0
3000.0
0.83
1.8
10
533.0
3000.0
0.84
1.5
11
542.0
2990.0
0.85
1.5
12
550.0
2970.0
0.85
1.0
13
558.0
2960.0
0.86
0.33
14
565.0
2950.0
0.86
0.67
17
585.0
2900.0
0.87
1.0
20
601.0
2860.0
0.88
1.6
28
631.0
2730.0
0.90
1.6
36
649.0
2620.0
0.90
1.7
44
660.0
2520.0
0.91
1.3 2.2
50
665.0
2450.0
0.91
55
668.0
2390.0
0.91
1.1
60
670.0
2340.0
0.91
2.5
65
672.0
2290.0
0.92
1.3
70
673.0
2240.0
0.92
1.5
75
674.0
2190.0
0.92
1.7
80
675.0
2140.0
0.92
1.7
85
675.0
2100.0
0.92
2.2
90
676.0
2060.0
0.92
1.6
95
676.0
2020.0
0.92
3.2
Overall
good fits,
1
Lacombe-Sanchez project
data. All cases yielded
% AAD
= 1.7.
with
API
183
250. s 8
200.
A e
150.
% -
100.
s z
50.
'3 0. 0.
20.
40.
CARBON Fig. 3. MP*
of n-paraffins:
60.
80.
100.
NUMBER o, parameters
fitted
from vapor pressure
data;
-
-
-
(13).
Table 2 presents the correlated parameters for the three n-paraffins and two industrial waxes studied along with the light gases to be described in the following section. The industrial waxes, SASOL and Mobil, are primarily mixtures of n-paraffins and have average carbon numbers of 43 and 61, respectively (Chou and Chao, 1989a,b).
TABLE 2 Lacombe-Sanchez Species
parameters
for waxes and gases T* (K)
P* (atm)
P* (gem-3)
600.0 250.0 475.0 550.0 150.0 400.0
2550.0 1400.0 3230.0 3200.0 1000.0 1200.0
1.40 0.68 0.64 0.63 0.045 0.40
Waxes
Carbon dioxide Carbon monoxide Ethane Ethylene Hydrogen Methane
184 PARAMETERS
FOR SIX LIGHT GASES
Although the homologous nature of the n-paraffins enabled the correlation of the characteristic parameters, the six gases studied in this work have quite different physical properties and do not conform to a simple correlation. The characteristic parameters for each gas must be determined on an individual basis. Parameters for the light gases obtained from fitting vapor pressure data yielded poor fits of the gas solubility data. The liquid states of the light gases are at temperatures far below those of the solubility phenomenon of interest. Parameters obtained from fitting their pure liquid states do not describe the The characteristic parameters were finally desolubility phenomenon. termined by directly fitting the solubility data in all solvents. The final parameter values for all six gases are presented in Table 2. GAS-PARAFFIN
INTERACTION
PARAMETERS
The correlation of the gas solubilities in the waxes requires the mixture form of the LS equation of state (Sanchez and Lacombe, 1976). Before the mixture equation is presented, some of the new mixture variables will be introduced. The number of sites a molecule occupied on the lattice has been represented by Y. For the mixture calculations, the number of sites occupied in the pure state will now be designated by r,“, and the molecular core volume by r,“u,%. The molecules of component i in the mixture now occupy r, sites where r, = r,”
(7)
(u*/v*)
u* is now the average close-packed volume of a segment in a mixture can be expressed in terms of a pure-state volume fraction ~$0
v* = c+pv*
which
(8)
(9) The average number r = xxi’;
of sites occupied
by molecules
in the mixture
is 00)
185
The volume fraction for the mixture is then
The expression for the Gibbs free energy of the mixture resembles the expression for the free energy of a fluid in the pure state, with the exception of an added term G”= -fi+PC+F
i
(8-l)ln(l-@)++
lnp+C:
I ( III In 2
i
02)
The reduced quantities take the same form as for pure components, but the characteristic parameters for the mixture are calculated from mixing rules. The characteristic energy c * for the mixture is obtained from the mixing rule E* = c c+&Cl”;
(13)
The characteristic volume u* is given by eqn. (8). The cross terms ei”;.are related to the pure fluid energies by Ei”;= (1 - kij)(Ciej)1’2
(14)
The calculated gas solubility crucially depends on the interaction energy between the gas and the wax. In the correlation of the solubility data the cross interaction coefficient kij is adjusted for the best fitting of the data by the EOS calculation. The equation of state for the mixture obtained from eqn. (12) is fi2+j+li
[
ln(l-p”)+
i
1-i
)I
p =0
(15)
This equation is identical in form with eqn. (2), the equation of state for the pure fluid. The difference lies in the mixture parameters used in the reducing variables. The expression for chemical potential derived from eqn. (12) is
(16)
186
where
A,, = -A,, = +
+ (@I- @&
- * 1
2
c.=T/q* Fi = P/P** At equilibrium, the chemical potential in the vapor and liquid phases for each component are equal lu’l = jJk
(17)
This is the basis for the VLE gas solubility calculations. Using the parameter values in Table 2, optimal kjj values were determined to correlate the solubility data of Huang et al. (1988b,c), Tsai et al. (1987, 1988) and Chou (1988). Tables 3-7 show the optimal kjj values obtained. The ki, values were correlated as a function of carbon number by least-squares fitting.
TABLE 3 Comparison of the calculated gas solubilities in n-eicosane with experimental data Gas
t (“C>
Optimal k,,
%AAD
Correlated ki,
%AAD
HZ
100 200 300
0.37 0.31 0.32
2.6 1.6 1.5
0.37 0.33 0.31
2.6 4.2 1.2
co
100 200 300
0.35 0.35 0.38
3.6 3.8 3.1
0.35 0.35 0.37
3.6 3.8 2.8
CH,
100 200 300
0.16 0.20 0.25
1.0 1.7 0.89
0.16 0.20 0.24
1.0 1.7 1.7
CO,
100 200 300
0.30 0.35 0.41
2.0 1.5 1.1
0.30 0.36 0.41
2.0 2.1 1.1
200 300
0.24 0.25 0.27
0.80 0.90 1.2
0.24 0.25 0.26
0.80 0.90 2.0
100 200 300
0.28 0.29 0.31
0.75 1.5 2.1
0.28 0.29 0.31
0.75 1.5 2.1
C2%
C,H,
100
187 TABLE 4 Comparison of the calculated gas solubilities in n-octacosane with experimental data Gas H2
co
CH,
co*
C2H6
C,K,
t (“Cl
Optimal kij
%AAD
Correlated kij
SAAD
100 200 300
0.47 0.43 0.39
2.1 2.5 1.7
0.48 0.42 0.40
4.6 2.9 2.4
100 200 300
0.42 0.41 0.43
4.6 2.7 2.1
0.42 0.42 0.44
4.6 3.8 2.5
100 200 300
0.17 0.22 0.26
2.1 1.8 1.8
0.18 0.22 0.27
2.4 1.8 3.0
100 200 300
0.33 0.39 0.45
2.9 1.5 2.8
0.33 0.39 0.44
2.9 1.5 2.8
100 200 300
0.28 0.28 0.30
2.2 0.70 2.3
0.28 0.28 0.30
2.2 0.70 2.3
100 200 300
0.31 0.32 0.34
1.8 0.75 1.5
0.31 0.32 0.34
1.8 0.75 1.5
Linear forms were found to fit the data, and the parameters are listed in Table 8. As an illustration, Fig. 4 shows the kij of hydrogen and of ethane interacting with the waxes at 200 o C. Tables 3-7 also list for each kii value the error between the calculated and experimental gas solubilities expressed as percentage average absolute deviation (W AAD), defined as
n
&Ml=+
IXiexp-XtcalcI
n I.I I
’
Xi,exp
’
I x 100 1
where n is the number of data points along the isotherm. In addition, Tables 3-7 present the correlated kijs obtained from the carbon number, and their associated 5%AADs. In a few cases, the error for the correlated k,, value is less than that for the optimal value. This occurs because the optimization was performed on the objective function Q
=
C
[1n(xexp,i/xcalc,i)]2 i
rather than % AAD.
188 TABLE 5 Comparison of the calculated gas solubilities in n-hexatriacontane with experimental data Gas
t (“Cl
Optimal kjj
%AAD
Correlated k,,
%AAD
100 200 300
0.60 0.53 0.50
5.1 4.3 3.5
0.60 0.52 0.49
5.1 4.5 3.3
co
100 200 300
0.50 0.50 0.51
4.1 3.7 2.5
0.50 0.48 0.50
4.1 4.8 2.6
CH,
100 200 300
0.20 0.24 0.30
1.8 1.4 1.9
0.20 0.25 0.30
1.8 2.9 1.9
CO,
100 200 300
0.36 0.42 0.48
0.97 2.3 1.7
0.36 0.42 0.48
0.97 2.3 1.7
C,H,
100 200 300
0.31 0.32 0.33
1.9 2.8 2.2
0.31 0.32 0.34
1.9 2.8 2.6
100 200 300
0.34 0.35 0.37
1.7 1.6 2.2
0.34 0.35 0.37
1.7 1.6 2.2
H2
C2H4
0. 10.
I
I
I
I
I
20.
30.
40.
50.
60.
CARBON Fig. 4. k,,
70.
NUMBER
vs. carbon number for H2 and C,H,
at 200°C:
o, H,;
+, C,H,.
189 TABLE 6 Comparison of the calculated gas solubilities in SASOL wax with experimental data Gas
t (“Cl
Optimal kij
%AAD
Correlated ki j
SAAD
HZ!
200 260
0.61 0.55
4.0 1.5
0.60 -
8.1 _
300
0.55
2.8
0.57
3.8
200 260
0.52 0.52
1.2 1.1
0.54 _
4.7 -
300
0.53
0.50
0.56
5.7
CH,
200 260
0.27 0.30
1.0 0.97
0.27 _
1.0 -
300
0.32
0.84
0.33
3.0
CO,
260
0.48
0.80
-
_
300
0.50
0.67
0.51
2.2
200 260
0.35 0.35
2.1 2.1
0.35 _
2.1 -
300
0.37
2.4
0.37
2.4
200 260
0.37 0.39
2.0 1.9
0.37 _
2.0 -
co
C,H,
C,H,
TABLE 7 Comparison of the calculated gas solubihties in Mobil wax with experimental data Gas H2
co
CH,
co2
C2H6
t (“C>
Optimal ki,
190 TABLE Fitting
8 parameters
for k,, correlations
(k = b + mlv,)
t CC)
Gas
Slope, m
Intercept,
100
H2 co CH, CO2 C2H, C,H,
1.44x10-2 9.38 x 1O-3 2.50x10-3 3.75x10-3 4.38 x lo- 3 3.51x10-3
0.0775 0.161 0.107 0.225 0.154 0.221
200
H2 co CH, CO2 C,H, C,H,
1.18 x10-’ 8.21 x~O-~ 3.22~10-~ 3.82 x lop3 4.44x10-3 3.51 x 1o-3
0.0922 0.185 0.131 0.279 0.159 0.221
300
H2 co CH, CO2 C,H, C2H,
1.10x10-* 7.99x10-3 4.26x1O-3 4.28~10~’ 4.93 x 1o-3 3.75 x 10-3
0.0917 0.211 0.150 0.325 0.163 0.235
b
0.21 xi
0.16
q z
0.15
B !: v3
0.12 0.09 L
E
z 2
0.03 0.06 1 0.0 L0
i 10
20
30
PRESSURE
40
50
60
(ATM)
Fig. 5. Comparison of calculations with solubility data for hydrogen n-C2s ; o, n-C,,; X, n-C,,; a, n-C,,; - - -, n-C,, (predicted).
at 300 o C: + , n-C,,;
*
191 0.28 , ,’
/’ /’
0.24
/’
^M
I
+. 0.20 ;; vE 0.16
ic
2 z 3
0.08 -
E: 0.04 -
0
n
I’
x
_i;ji-l: ,’
/’
/’
x
,’
/’
0.12 -
I’
/’
I’
x
*
30
40
*
/’
/’
/’
10
20
50
60
(ATM) Fig. 6. Comparison of calculations with solubility data for carbon monoxide at 300 o C: n-Go; *, n-C,,; 0, n-C,,; X, n-C,,; A, n-C,,; - - -, n-C,, (predicted).
+ ,
The experimental uncertainty was estimated by determining the 95% confidence limits on the measured solubilities. These confidence limits were compared against the average solubility for each experiment and expressed as percentages. These percentage errors were averaged over each isotherm, and ranged from 0.9% to 2.8%, with an average error over all isotherms of 1.4%. This estimation of the experimental uncertainty is generally smaller than the deviations obtained from the equation of state calculations; therefore it is expected that the errors reported for the calculations arise from the equation rather than from experimental error. Figures 5 and 6 compare the solubilities predicted by the correlation against the experimental solubilities for hydrogen and carbon monoxide in several n-paraffins at 300” C. The correlation performs well at the lower pressures, but gives low solubility estimates at higher pressures. The amount of under-prediction tends to increase for heavier n-paraffins. Predicted solubilities where no experimental data exist for hydrogen in n-C,, and carbon monoxide in n-C,, are shown in the figures. Based on the deviations obtained from the optimal kijs, the LS equation correlates the gas solubiliy data very well. With the exception of one isotherm all isotherms fitted to within 5.3% AAD. Of the 85 isotherms, 82 fitted within an AAD of 5% or less. However, the LS equation does not perform as well when the correlated kijs are used. Although employing the correlated kij values decreases the performance in some cases, the overall fitting quality is still very good: 68 of the 73 isotherms fitted to an AAD within 5%.
192
I
1
I
20.
30.
40.
I
0.
10.
PRESSURE Fig. 7. Comparison dioxide: +, n-C,,; TABLE
50
(ATM)
of calculations with Gasem *, n-C2s; o, n-C,,; x , n-C,.
and Robinson’s
data
for carbon
9
Comparison of performance of the Lacombe-Sanchez equation for n-C,c, n-C,, and n-Cs6 Gas
(1985b)
t (“C)
equation
n-C,,
n-C,0
with
the modified
n-C,,
LS %AAD
RKS % AAD
LS %AAD
RKS %AAD
LS % AAD
RKS %AAD
HZ
100 200 300
2.6 4.2 1.2
3.8 3.0 2.6
4.6 2.9 2.4
5.8 3.7 4.3
5.1 4.5 3.3
4.7 5.2 5.1
co
100 200 300
3.6 3.8 2.8
3.3 4.7 4.2
4.6 3.8 2.5
5.1 4.3 3.8
4.1 4.8 2.6
4.0 4.1 4.1
(334
100 200 300
1.0 1.7 1.7
2.7 2.4 2.8
2.4 0.9 1.6
2.7 3.4 4.5
1.8 2.9 1.9
2.5 2.7 4.0
CO,
100 200 300
2.0 2.1 1.1
3.9 3.1 1.9
2.9 1.5 2.8
3.6 3.4 4.2
1.0 2.3 1.7
4.8 3.5 3.8
C,H,
100 200 300
0.8 1.4 2.0
1.8 0.7 2.0
2.2 0.7 2.3
2.5 0.9 2.5
1.9 2.8 2.6
1.8 2.8 3.3
100
0.8 1.5 2.1
4.9 3.2 3.8
1.8 0.8 1.5
4.1 1.8 2.4
1.7 1.6 2.2
2.3 2.9 4.0
GH4
200 300
RKS
193 COMPARISON
OF RESULTS
WITH
DATA OF GASEM
AND ROBINSON
Gasem and Robinson (1985b) studied the solubility of carbon dioxide in heavy n-paraffins. A comparison of the Lacombe-Sanchez correlation with their data is presented in Fig. 7. For all four n-paraffins the correlation works well at low pressures, but tends to under-predict the solubilities at higher pressures (above 20 atm.).
COMPARISON
WITH
HUANG’S
CORRELATION
Huang et al. (19SSa) have developed a correlation for gas solubilities in heavy n-paraffins based on an extended form of the Redlich-Kwong-Soave (RKS) equation of state and a new mixing rule based on polymer solution theory. Although the modified RKS equation was able to satisfactorily correlate the gas solubility data, the LS equation yielded better results, particularly for the higher molecular weight industrial waxes (see Tables 9 and 10). The better performance of the LS equation may come from the direct development of the LS equation of state from Guggenheim’s long-chain statistical mechanical lattice model, in contrast with the modified RKS equation which accounts for the effects of the long-chain n-paraffins by a modification of the mixing rules for the attractive energy term.
TABLE
10
Comparison of performance of the Lacombe-Sanchez equation for SASOL and Mobil wax Gas
1 (“Cl
SASOL wax
equation
with
the modified
Mobil wax
LS %AAD
RKS %AAD
LS %AAD
RKS %AAD
H2
200 300
8.1 3.8
5.0 4.8
5.0 4.2
5.5 5.5
co
200 300
4.7 5.7
5.5 3.2
3.8 4.0
4.3 4.6
CH,
200 300
1.0 3.0
4.0 1.9
3.2 3.2
6.3 5.1
co2
300
2.2
1.7
2.8
5.4
C,H,
200 300
2.1 2.4
2.9 6.6
5.3 6.3
6.6 7.3
C,H,
200
2.0
2.2
_
RKS
194
In both correlations the interaction coefficient k,, is temperature dependent. Though the dependence is not strong, a slight variation of kjj can cause a significant variation in the calculated solubility. Interpolation between the tabulated temperatures will be necessary, if the temperature of interest is not one of the tabulated. ACKNOWLEDGMENT
Financial support for this study was provided by the Department Energy through Contract DE-AC22-84PC70024.
of
SYMBOLS
G k A4
N P
P*
P Y
R T T*
If u ll*
u”
V V*
Gibbs free energy Boltzmann constant molecular weight number of molecules pressure characteristic pressure ( 6 */u * ) reduced pressure ( P/P * ) number of molecular segments gas constant temperature characteristic temperature (6 */k) reduced temperature (T/T * ) specific volume core volume per molecular segment reduced volume (V/V * = l/p”) volume characteristic volume ( N( YU* ))
Greek letters E* P P*
b w
total interaction energy per molecular segment mass density characteristic density (l/ V * ) reduced density ( p/p* ) molecular size/symmetry/flexibility parameter
REFERENCES Albal, R.S., Shah Y.T., Carr, N.L. and Bell, A.T., 1984. Chem. Eng. Sci., 39: 905. Chou, J.S., 1988. Solubility of Synthesis Gases in a Fischer-Tropsch SASOL Wax and in Three High Paraffins. M.S. Thesis, Purdue University.
195
Chou, J.S. and Chao, K.C., 1989a. Solubihty of ethylene in n-eicosane, n-octacosane, and n-hexatriacontane. J. Chem. Eng. Data, 34: 68. Chou, J.S. and Chao, K.C., 1989b. Solubihty of synthesis gases in a Fischer-Tropsch SASOL wax. J. Chem. Eng. Data, in press. Gasem, K.A.M. and Robinson Jr., R.L., 1985a. Prediction of phase behavior for CO, + heavy normal paraffins using generalized parameter Soave and Peng-Robinson equations of state. Paper presented at AIChE Meeting, Houston, Tex., March 24-28. Gasem, K.A.M. and Robinson Jr., R.L., 1985b. J. Chem. Eng. Data, 30: 53. Huang, S.H., Lin, H.M., Tsai, F.N. and Chao, K.C., 1988a. Ind. Eng. Chem. Res., 27: 162. Huang, S.H., Lin, H.M. and Chao, K.C., 1988b. Solubihty of methane, ethane, and carbon dioxide in n-octacosane. J. Chem. Eng. Data, 33: 143. Huang, S.H., Lin, H.M. and Chao, K.C.; 1988~. Solubility of methane, ethane, and carbon dioxide in n-eicosane. J. Chem. Eng. Data, 33: 145. Kolbel, H. and Ralek, M., 1980. Catal. Rev. Sci. Eng., 21: 225. Lacombe, R.H. and Sanchez, I.C., 1976. J. Phys. Chem., 80: 2568. Sanchez, I.C. and Lacombe, R.H., 1976. J. Phys. Chem., 80: 2352. Satterfield, C.N. and Stenger Jr., H.G., 1985. I&EC Process Des. Dev., 24: 407. Stern, D., Bell, A.T. and Heinemann, H., 1983. Chem. Eng. Sci., 38: 597. Tsai, F.N., Huang, S.H., Lin, H.M. and Chao, K.C., 1987. J. Chem. Eng. Data, 32: 467. Tsai, F.N., Huang, S.H., Lin, H.M. and Chao, K.C., 1988. Chem. Eng. J., 38: 41. WiIhoit, R.C. and Zwohnski, B.J. (eds.), 1971. Handbook of Vapor Pressures and Heats of Vaporization of Hydrocarbons and Related Compounds. Texas A&M University, College Station, Tex.
179 -
Printed
in The Netherlands
CORRELATION OF SYNTHESIS GAS SOLUBILITY SOLVENTS AND FISCHER-TROPSCH WAXES J.S. CHOU
and K.C. CHAO
School of Chemical Engineering, (Received
IN n-PARAFFIN
Purdue University,
May 27, 1988; accepted
West Lafayette,
in final form November
IN 47907 (U.S.A.)
4, 1988)
ABSTRACT
Chou, J.S. and Chao, K.C., 1989. Correlation of synthesis gas solubility and Fischer-Tropsch waxes. Fluid Phase Equilibria, 46: 179-195.
in n-paraffin
solvents
A correlation of the solubility of Fischer-Tropsch synthesis and product gases in heavy n-paraffin waxes is obtained with the Lacombe-Sanchez equation of state (EOS). The EOS parameters for pure n-paraffins are developed and correlated as functions of carbon number. Interaction parameters with the n-paraffins are determined from solubility data and expressed as functions of carbon number of the n-paraffin solvent for the gases hydrogen, carbon monoxide, methane, carbon dioxide, ethylene and ethane.
INTRODUCTION
The Fischer-Tropsch synthesis is a method of indirect coal liquefaction in which hydrogen and carbon monoxide obtained from coal gasification are reacted to produce hydrocarbons and/or oxygen-containing derivatives. One of the more effective reactor types for Fischer-Tropsch synthesis is the three-phase slurry reactor (Kiilbel and Ralek, 1980). Small particles of the synthesis catalyst are suspended in a heavy wax mixture and the reactants are bubbled through the slurry. Satterfield and Stenger (1985), Albal et al. (1984) and Stern et al. (1983) have concluded that the solubilities of the reactant and product gases in the wax mixture are needed to design these reactors. In this work the Lacombe-Sanchez equation of state is used for the correlation of solubilities of hydrogen, carbon monoxide, methane, carbon dioxide, ethylene and ethane in n-eicosane, n-octacosane, n-hexatriacontane and two Fischer-Tropsch waxes to develop a general description of the solubility of these gases in heavy n-paraffins. 0378-3812/89/$03.50
0 1989 Elsevier
Science Publishers
B.V.
180
THE LACOMBE-SANCHEZ
EQUATION
OF STATE
The Lacombe-Sanchez EOS was chosen to correlate the solubility data because the n-paraffins are chain molecules considered in Guggenheim’s statistical mechanical lattice model from which the EOS is derived. Sanchez and Lacombe (1976) showed their equation to represent polymeric liquid densities effectively. Sanchez and Lacombe (1976) derived an expression for the Gibbs free energy of a pure fluid:
G Nrr *
=.G”= -fi+&+i;
(e-l)ln(l-fi)++
In
where the symbols are as described in the symbols list. The EOS obtained from eqn. (1) is fi* + F + f
[
( 11
In(1- fi) + 1 - + p” = 0
PARAMETERS
(2)
FOR n-PARAFFINS
Only three of the six reducing parameters (either T *, P *, p* or e *, r, are required to characterize a fluid. Each set can be converted to the other by using their definitions and the following relations u*)
RT*p* ----= P*
* *=-- M VP r
(3)
and p*v* -= RT*
1
(4)
Values of r and u* have been determined by Sanchez and Lacombe (1976) for the n-paraffins up to n-C,,. Figure 1 shows the dependence of the molecular core volume ru* on the carbon number N,. A linear dependence is observed and is represented by rv* = lS.OSN, + 18.78
(5)
Extrapolation to heavy n-paraffins by this equation seems reasonable in view of the constant incremental value of rv* for each additional CH, group for the large n-paraffins. Since the reciprocal of ru* is p*/M, eqn. (5) is also a correlation of p* as a function of carbon number. With u* given by eqn. (5), we determine T * and P* for the n-paraffins by fitting saturated vapor pressure values taken from the API 44 tables (Wilhoit and Zwolinski, 1971). Figures 2 and 3 show the dependence of T *
181
400. 3. E >_
300.
E s -
200.
1 100.
I
0. i
5.
0.
I
I
I
I
10.
15.
20.
25.
CARBON NUMBER Fig. 1. Molecular core volume of n-paraffins: (1976); - - -, eqn. (5).
30.
o, parameters
from
Sanchez
and Lacombe
and the product MP* on carbon number. Both these quantities are fitted with the growth equation (Gasem and Robinson, 1985a),
y+_y+;-c*)
exp[ -c4(NC - l)(l
- c2)])‘;“-‘*’
The quantity Y can be identified with either T * or MP *. When Y is T * , the
r
695.
645.
?c i
595.
I
I
I
I
I
I
I
I
40.
80.
80.
4 545.
d
495. 0.
20.
CARBON Fig. 2. T* of n-paraffins;
100
NUMBER
o, parameters
fitted from vapor pressure
data;
-
-
-,
eqn. (13).
182
constants are cr = 417, c2 = -0.405, c3 = 414 and cq = 0.044. When Y is MP*, the constants are c1 = 6.13, c2 = -0.0884, cg = 1.504 and cq = 0.00737. The curves shown in Figs. 2 and 3 represent eqn. (6). Table 1 lists the correlated parameters T *, P * and p* for selected n-paraffins between n-C, and n-C,,. Using these parameters, the Lacombe-Sanchez (LS) equation fits the n-paraffin saturated vapor pressure data reported in the API 44 tables with the average absolute deviations (% AAD) presented in Table 1, where % AAD is defined to be
where n is the number of vapor pressure with an overall average of 1.7% AAD.
TABLE
parameters
and
comparison
of
calculated
vapor
pressures
44 data for n-paraffins
NC of n-paraffin 7
Z-* (K)
P* (atm)
P* (gcme3)
%AAD
501.0
3000.0
0.81
2.8
8
512.0
3000.0
0.82
2.5
9
523.0
3000.0
0.83
1.8
10
533.0
3000.0
0.84
1.5
11
542.0
2990.0
0.85
1.5
12
550.0
2970.0
0.85
1.0
13
558.0
2960.0
0.86
0.33
14
565.0
2950.0
0.86
0.67
17
585.0
2900.0
0.87
1.0
20
601.0
2860.0
0.88
1.6
28
631.0
2730.0
0.90
1.6
36
649.0
2620.0
0.90
1.7
44
660.0
2520.0
0.91
1.3 2.2
50
665.0
2450.0
0.91
55
668.0
2390.0
0.91
1.1
60
670.0
2340.0
0.91
2.5
65
672.0
2290.0
0.92
1.3
70
673.0
2240.0
0.92
1.5
75
674.0
2190.0
0.92
1.7
80
675.0
2140.0
0.92
1.7
85
675.0
2100.0
0.92
2.2
90
676.0
2060.0
0.92
1.6
95
676.0
2020.0
0.92
3.2
Overall
good fits,
1
Lacombe-Sanchez project
data. All cases yielded
% AAD
= 1.7.
with
API
183
250. s 8
200.
A e
150.
% -
100.
s z
50.
'3 0. 0.
20.
40.
CARBON Fig. 3. MP*
of n-paraffins:
60.
80.
100.
NUMBER o, parameters
fitted
from vapor pressure
data;
-
-
-
(13).
Table 2 presents the correlated parameters for the three n-paraffins and two industrial waxes studied along with the light gases to be described in the following section. The industrial waxes, SASOL and Mobil, are primarily mixtures of n-paraffins and have average carbon numbers of 43 and 61, respectively (Chou and Chao, 1989a,b).
TABLE 2 Lacombe-Sanchez Species
parameters
for waxes and gases T* (K)
P* (atm)
P* (gem-3)
600.0 250.0 475.0 550.0 150.0 400.0
2550.0 1400.0 3230.0 3200.0 1000.0 1200.0
1.40 0.68 0.64 0.63 0.045 0.40
Waxes
Carbon dioxide Carbon monoxide Ethane Ethylene Hydrogen Methane
184 PARAMETERS
FOR SIX LIGHT GASES
Although the homologous nature of the n-paraffins enabled the correlation of the characteristic parameters, the six gases studied in this work have quite different physical properties and do not conform to a simple correlation. The characteristic parameters for each gas must be determined on an individual basis. Parameters for the light gases obtained from fitting vapor pressure data yielded poor fits of the gas solubility data. The liquid states of the light gases are at temperatures far below those of the solubility phenomenon of interest. Parameters obtained from fitting their pure liquid states do not describe the The characteristic parameters were finally desolubility phenomenon. termined by directly fitting the solubility data in all solvents. The final parameter values for all six gases are presented in Table 2. GAS-PARAFFIN
INTERACTION
PARAMETERS
The correlation of the gas solubilities in the waxes requires the mixture form of the LS equation of state (Sanchez and Lacombe, 1976). Before the mixture equation is presented, some of the new mixture variables will be introduced. The number of sites a molecule occupied on the lattice has been represented by Y. For the mixture calculations, the number of sites occupied in the pure state will now be designated by r,“, and the molecular core volume by r,“u,%. The molecules of component i in the mixture now occupy r, sites where r, = r,”
(7)
(u*/v*)
u* is now the average close-packed volume of a segment in a mixture can be expressed in terms of a pure-state volume fraction ~$0
v* = c+pv*
which
(8)
(9) The average number r = xxi’;
of sites occupied
by molecules
in the mixture
is 00)
185
The volume fraction for the mixture is then
The expression for the Gibbs free energy of the mixture resembles the expression for the free energy of a fluid in the pure state, with the exception of an added term G”= -fi+PC+F
i
(8-l)ln(l-@)++
lnp+C:
I ( III In 2
i
02)
The reduced quantities take the same form as for pure components, but the characteristic parameters for the mixture are calculated from mixing rules. The characteristic energy c * for the mixture is obtained from the mixing rule E* = c c+&Cl”;
(13)
The characteristic volume u* is given by eqn. (8). The cross terms ei”;.are related to the pure fluid energies by Ei”;= (1 - kij)(Ciej)1’2
(14)
The calculated gas solubility crucially depends on the interaction energy between the gas and the wax. In the correlation of the solubility data the cross interaction coefficient kij is adjusted for the best fitting of the data by the EOS calculation. The equation of state for the mixture obtained from eqn. (12) is fi2+j+li
[
ln(l-p”)+
i
1-i
)I
p =0
(15)
This equation is identical in form with eqn. (2), the equation of state for the pure fluid. The difference lies in the mixture parameters used in the reducing variables. The expression for chemical potential derived from eqn. (12) is
(16)
186
where
A,, = -A,, = +
+ (@I- @&
- * 1
2
c.=T/q* Fi = P/P** At equilibrium, the chemical potential in the vapor and liquid phases for each component are equal lu’l = jJk
(17)
This is the basis for the VLE gas solubility calculations. Using the parameter values in Table 2, optimal kjj values were determined to correlate the solubility data of Huang et al. (1988b,c), Tsai et al. (1987, 1988) and Chou (1988). Tables 3-7 show the optimal kjj values obtained. The ki, values were correlated as a function of carbon number by least-squares fitting.
TABLE 3 Comparison of the calculated gas solubilities in n-eicosane with experimental data Gas
t (“C>
Optimal k,,
%AAD
Correlated ki,
%AAD
HZ
100 200 300
0.37 0.31 0.32
2.6 1.6 1.5
0.37 0.33 0.31
2.6 4.2 1.2
co
100 200 300
0.35 0.35 0.38
3.6 3.8 3.1
0.35 0.35 0.37
3.6 3.8 2.8
CH,
100 200 300
0.16 0.20 0.25
1.0 1.7 0.89
0.16 0.20 0.24
1.0 1.7 1.7
CO,
100 200 300
0.30 0.35 0.41
2.0 1.5 1.1
0.30 0.36 0.41
2.0 2.1 1.1
200 300
0.24 0.25 0.27
0.80 0.90 1.2
0.24 0.25 0.26
0.80 0.90 2.0
100 200 300
0.28 0.29 0.31
0.75 1.5 2.1
0.28 0.29 0.31
0.75 1.5 2.1
C2%
C,H,
100
187 TABLE 4 Comparison of the calculated gas solubilities in n-octacosane with experimental data Gas H2
co
CH,
co*
C2H6
C,K,
t (“Cl
Optimal kij
%AAD
Correlated kij
SAAD
100 200 300
0.47 0.43 0.39
2.1 2.5 1.7
0.48 0.42 0.40
4.6 2.9 2.4
100 200 300
0.42 0.41 0.43
4.6 2.7 2.1
0.42 0.42 0.44
4.6 3.8 2.5
100 200 300
0.17 0.22 0.26
2.1 1.8 1.8
0.18 0.22 0.27
2.4 1.8 3.0
100 200 300
0.33 0.39 0.45
2.9 1.5 2.8
0.33 0.39 0.44
2.9 1.5 2.8
100 200 300
0.28 0.28 0.30
2.2 0.70 2.3
0.28 0.28 0.30
2.2 0.70 2.3
100 200 300
0.31 0.32 0.34
1.8 0.75 1.5
0.31 0.32 0.34
1.8 0.75 1.5
Linear forms were found to fit the data, and the parameters are listed in Table 8. As an illustration, Fig. 4 shows the kij of hydrogen and of ethane interacting with the waxes at 200 o C. Tables 3-7 also list for each kii value the error between the calculated and experimental gas solubilities expressed as percentage average absolute deviation (W AAD), defined as
n
&Ml=+
IXiexp-XtcalcI
n I.I I
’
Xi,exp
’
I x 100 1
where n is the number of data points along the isotherm. In addition, Tables 3-7 present the correlated kijs obtained from the carbon number, and their associated 5%AADs. In a few cases, the error for the correlated k,, value is less than that for the optimal value. This occurs because the optimization was performed on the objective function Q
=
C
[1n(xexp,i/xcalc,i)]2 i
rather than % AAD.
188 TABLE 5 Comparison of the calculated gas solubilities in n-hexatriacontane with experimental data Gas
t (“Cl
Optimal kjj
%AAD
Correlated k,,
%AAD
100 200 300
0.60 0.53 0.50
5.1 4.3 3.5
0.60 0.52 0.49
5.1 4.5 3.3
co
100 200 300
0.50 0.50 0.51
4.1 3.7 2.5
0.50 0.48 0.50
4.1 4.8 2.6
CH,
100 200 300
0.20 0.24 0.30
1.8 1.4 1.9
0.20 0.25 0.30
1.8 2.9 1.9
CO,
100 200 300
0.36 0.42 0.48
0.97 2.3 1.7
0.36 0.42 0.48
0.97 2.3 1.7
C,H,
100 200 300
0.31 0.32 0.33
1.9 2.8 2.2
0.31 0.32 0.34
1.9 2.8 2.6
100 200 300
0.34 0.35 0.37
1.7 1.6 2.2
0.34 0.35 0.37
1.7 1.6 2.2
H2
C2H4
0. 10.
I
I
I
I
I
20.
30.
40.
50.
60.
CARBON Fig. 4. k,,
70.
NUMBER
vs. carbon number for H2 and C,H,
at 200°C:
o, H,;
+, C,H,.
189 TABLE 6 Comparison of the calculated gas solubilities in SASOL wax with experimental data Gas
t (“Cl
Optimal kij
%AAD
Correlated ki j
SAAD
HZ!
200 260
0.61 0.55
4.0 1.5
0.60 -
8.1 _
300
0.55
2.8
0.57
3.8
200 260
0.52 0.52
1.2 1.1
0.54 _
4.7 -
300
0.53
0.50
0.56
5.7
CH,
200 260
0.27 0.30
1.0 0.97
0.27 _
1.0 -
300
0.32
0.84
0.33
3.0
CO,
260
0.48
0.80
-
_
300
0.50
0.67
0.51
2.2
200 260
0.35 0.35
2.1 2.1
0.35 _
2.1 -
300
0.37
2.4
0.37
2.4
200 260
0.37 0.39
2.0 1.9
0.37 _
2.0 -
co
C,H,
C,H,
TABLE 7 Comparison of the calculated gas solubihties in Mobil wax with experimental data Gas H2
co
CH,
co2
C2H6
t (“C>
Optimal ki,
190 TABLE Fitting
8 parameters
for k,, correlations
(k = b + mlv,)
t CC)
Gas
Slope, m
Intercept,
100
H2 co CH, CO2 C2H, C,H,
1.44x10-2 9.38 x 1O-3 2.50x10-3 3.75x10-3 4.38 x lo- 3 3.51x10-3
0.0775 0.161 0.107 0.225 0.154 0.221
200
H2 co CH, CO2 C,H, C,H,
1.18 x10-’ 8.21 x~O-~ 3.22~10-~ 3.82 x lop3 4.44x10-3 3.51 x 1o-3
0.0922 0.185 0.131 0.279 0.159 0.221
300
H2 co CH, CO2 C,H, C2H,
1.10x10-* 7.99x10-3 4.26x1O-3 4.28~10~’ 4.93 x 1o-3 3.75 x 10-3
0.0917 0.211 0.150 0.325 0.163 0.235
b
0.21 xi
0.16
q z
0.15
B !: v3
0.12 0.09 L
E
z 2
0.03 0.06 1 0.0 L0
i 10
20
30
PRESSURE
40
50
60
(ATM)
Fig. 5. Comparison of calculations with solubility data for hydrogen n-C2s ; o, n-C,,; X, n-C,,; a, n-C,,; - - -, n-C,, (predicted).
at 300 o C: + , n-C,,;
*
191 0.28 , ,’
/’ /’
0.24
/’
^M
I
+. 0.20 ;; vE 0.16
ic
2 z 3
0.08 -
E: 0.04 -
0
n
I’
x
_i;ji-l: ,’
/’
/’
x
,’
/’
0.12 -
I’
/’
I’
x
*
30
40
*
/’
/’
/’
10
20
50
60
(ATM) Fig. 6. Comparison of calculations with solubility data for carbon monoxide at 300 o C: n-Go; *, n-C,,; 0, n-C,,; X, n-C,,; A, n-C,,; - - -, n-C,, (predicted).
+ ,
The experimental uncertainty was estimated by determining the 95% confidence limits on the measured solubilities. These confidence limits were compared against the average solubility for each experiment and expressed as percentages. These percentage errors were averaged over each isotherm, and ranged from 0.9% to 2.8%, with an average error over all isotherms of 1.4%. This estimation of the experimental uncertainty is generally smaller than the deviations obtained from the equation of state calculations; therefore it is expected that the errors reported for the calculations arise from the equation rather than from experimental error. Figures 5 and 6 compare the solubilities predicted by the correlation against the experimental solubilities for hydrogen and carbon monoxide in several n-paraffins at 300” C. The correlation performs well at the lower pressures, but gives low solubility estimates at higher pressures. The amount of under-prediction tends to increase for heavier n-paraffins. Predicted solubilities where no experimental data exist for hydrogen in n-C,, and carbon monoxide in n-C,, are shown in the figures. Based on the deviations obtained from the optimal kijs, the LS equation correlates the gas solubiliy data very well. With the exception of one isotherm all isotherms fitted to within 5.3% AAD. Of the 85 isotherms, 82 fitted within an AAD of 5% or less. However, the LS equation does not perform as well when the correlated kijs are used. Although employing the correlated kij values decreases the performance in some cases, the overall fitting quality is still very good: 68 of the 73 isotherms fitted to an AAD within 5%.
192
I
1
I
20.
30.
40.
I
0.
10.
PRESSURE Fig. 7. Comparison dioxide: +, n-C,,; TABLE
50
(ATM)
of calculations with Gasem *, n-C2s; o, n-C,,; x , n-C,.
and Robinson’s
data
for carbon
9
Comparison of performance of the Lacombe-Sanchez equation for n-C,c, n-C,, and n-Cs6 Gas
(1985b)
t (“C)
equation
n-C,,
n-C,0
with
the modified
n-C,,
LS %AAD
RKS % AAD
LS %AAD
RKS %AAD
LS % AAD
RKS %AAD
HZ
100 200 300
2.6 4.2 1.2
3.8 3.0 2.6
4.6 2.9 2.4
5.8 3.7 4.3
5.1 4.5 3.3
4.7 5.2 5.1
co
100 200 300
3.6 3.8 2.8
3.3 4.7 4.2
4.6 3.8 2.5
5.1 4.3 3.8
4.1 4.8 2.6
4.0 4.1 4.1
(334
100 200 300
1.0 1.7 1.7
2.7 2.4 2.8
2.4 0.9 1.6
2.7 3.4 4.5
1.8 2.9 1.9
2.5 2.7 4.0
CO,
100 200 300
2.0 2.1 1.1
3.9 3.1 1.9
2.9 1.5 2.8
3.6 3.4 4.2
1.0 2.3 1.7
4.8 3.5 3.8
C,H,
100 200 300
0.8 1.4 2.0
1.8 0.7 2.0
2.2 0.7 2.3
2.5 0.9 2.5
1.9 2.8 2.6
1.8 2.8 3.3
100
0.8 1.5 2.1
4.9 3.2 3.8
1.8 0.8 1.5
4.1 1.8 2.4
1.7 1.6 2.2
2.3 2.9 4.0
GH4
200 300
RKS
193 COMPARISON
OF RESULTS
WITH
DATA OF GASEM
AND ROBINSON
Gasem and Robinson (1985b) studied the solubility of carbon dioxide in heavy n-paraffins. A comparison of the Lacombe-Sanchez correlation with their data is presented in Fig. 7. For all four n-paraffins the correlation works well at low pressures, but tends to under-predict the solubilities at higher pressures (above 20 atm.).
COMPARISON
WITH
HUANG’S
CORRELATION
Huang et al. (19SSa) have developed a correlation for gas solubilities in heavy n-paraffins based on an extended form of the Redlich-Kwong-Soave (RKS) equation of state and a new mixing rule based on polymer solution theory. Although the modified RKS equation was able to satisfactorily correlate the gas solubility data, the LS equation yielded better results, particularly for the higher molecular weight industrial waxes (see Tables 9 and 10). The better performance of the LS equation may come from the direct development of the LS equation of state from Guggenheim’s long-chain statistical mechanical lattice model, in contrast with the modified RKS equation which accounts for the effects of the long-chain n-paraffins by a modification of the mixing rules for the attractive energy term.
TABLE
10
Comparison of performance of the Lacombe-Sanchez equation for SASOL and Mobil wax Gas
1 (“Cl
SASOL wax
equation
with
the modified
Mobil wax
LS %AAD
RKS %AAD
LS %AAD
RKS %AAD
H2
200 300
8.1 3.8
5.0 4.8
5.0 4.2
5.5 5.5
co
200 300
4.7 5.7
5.5 3.2
3.8 4.0
4.3 4.6
CH,
200 300
1.0 3.0
4.0 1.9
3.2 3.2
6.3 5.1
co2
300
2.2
1.7
2.8
5.4
C,H,
200 300
2.1 2.4
2.9 6.6
5.3 6.3
6.6 7.3
C,H,
200
2.0
2.2
_
RKS
194
In both correlations the interaction coefficient k,, is temperature dependent. Though the dependence is not strong, a slight variation of kjj can cause a significant variation in the calculated solubility. Interpolation between the tabulated temperatures will be necessary, if the temperature of interest is not one of the tabulated. ACKNOWLEDGMENT
Financial support for this study was provided by the Department Energy through Contract DE-AC22-84PC70024.
of
SYMBOLS
G k A4
N P
P*
P Y
R T T*
If u ll*
u”
V V*
Gibbs free energy Boltzmann constant molecular weight number of molecules pressure characteristic pressure ( 6 */u * ) reduced pressure ( P/P * ) number of molecular segments gas constant temperature characteristic temperature (6 */k) reduced temperature (T/T * ) specific volume core volume per molecular segment reduced volume (V/V * = l/p”) volume characteristic volume ( N( YU* ))
Greek letters E* P P*
b w
total interaction energy per molecular segment mass density characteristic density (l/ V * ) reduced density ( p/p* ) molecular size/symmetry/flexibility parameter
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195
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