Correlation of Vapor-Liquid Equilibria of c 3 - C ~Hydrocarbons Using Solubility Parameters Paul Barton,* R. E. Holland,' and Robert H. McCormick Department of Chemical Engineering, The Pennsylvenia State University. Universfty Park, Pennsylvania 76802
A consistent set of apparent solubility parameters that account for compositional dependence of liquid
phase activity coefficients was developed for fourteen C3 to Cs hydrocarbons. These parameters were obtained as a function of temperature from binary and multicomponent vapor-liquid equilibrium data using regular solution theory. T h e correlative and predictive capabilities of these parameters, used with pure component physical properties and appropriate equations of state, to generate relative volatilities are good, with average deviations of less t h a n 2 % for 563 data points. Accurate equilibrium vaporization K values were predicted, with average deviations of 2%.
Introduction For nonpolar compounds, regular solution theory provides a means of representing the activity coefficients of multicomponent systems in a simple manner using equations of the van Laar type. Deviations from ideal liquid solutions are calculated by using only pure component properties including apparent solubility parameters. One objective of this study is to develop a consistent set of solubility parameters to represent the effect of liquid phase composition on the vapor-liquid equilibria of hydrocarbons. Fourteen hydrocarbons with three to five carbon atoms are represented, including paraffins, monoolefins, diolefins, and an acetylene. Another objective is to provide engineers with a means to calculate vapor-liquid equilibria for wider range multicomponent hydrocarbon systems with more accuracy than previously possible, without having to supply binary constants for each binary in the system. For quick desk calculations, the equilibria are presented as K charts which include the effects of temperature, pressure, and composition, or alternatively as charts of relative volatility (from which knowledge of system pressure is not available). However, these desk calculations are limited to the particular systems and temperatures for which the plots are presented. For more universal applications, liquid phase activity coefficients are calculated from solubility parameters using regular solution theory; then the equilibrium relationships are calculated using equations of state that account for nonperfect gas, nonideal vapor mixtures, and pressure effects on liquid fugacity. The approach to maintaining accuracy in the development of apparent solubility parameters in this work was to use selected data, preferably recent, rather than every data point published. This meant using isomeric multicomponent data obtained from multistage equilibrium units, when available, in order to establish solubility parameter differences within a close-boiling isomeric group. Multistage data have the advantage of reduced errors due to analytical uncertainties (McCormick, et al., 1962). It also meant excluding systems with experimental data which gave activity coefficients less than unity beyond the bounds of reasonable experimental error. Having established the solubility parameter differences within each isomeric group, the solubility parameter differences between the isomeric groups were established using selected binaries. Absolute magnitudes were assigned to the solubility parameter of each compound by relating the data to Mobil Research and Development. Paulsboro, N. J. 08066
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Ind. Eng. Chern., Process Des. Develop.. Vol. 13, No. 4 , 1974
published solubility parameters for propane (Edmister, 1960).
Theory For a vapor-liquid system a t a specified temperature and pressure, the criterion for equilibrium is that the fugacity of each component is the same in each phase.
Usually, the goal is to express the composition of the vapor phase as a function of the composition of the liquid phase a t equilibrium. This is achieved by introducing activity coefficients and pure component fugacities, which are evaluated a t the system temperature and pressure. In
terms of system pressure p , vapor pressure p L o ,and pure component fugacity coefficients in the vapor d L , vand in the liquid $i,L, eq 2 becomes
The activity coefficients account for deviations from ideal solution and the fugacity coefficients account for deviations from perfect gas. In addition, ~ L , accounts L for pressure effects on the pure component liquid fugacity. The vaporization equilibrium ratio is defined upon rearrangement of eq 3.
The relative volatility of component j with respect to component i is represented by
(5) For mixtures of isomeric hydrocarbons, the vapor phase activity coefficient yi,v is taken as unity (Lewis assumption). For hydrocarbons in mixtures having constituents with different numbers of carbon atoms yi,v d i , v is combined into a mixture fugacity coefficient $i,v. The fugacity coefficients are evaluated from the relations
In
1
@i,L
= -
RT
P o
(Vi,v - y ) d P -t
0
I
PP
The physical properties used in evaluating these coefficients are based on the best available, yet convenient, correlations. The saturated liquid molar volumes V,,L are calculated using the Francis equation (Francis, 1957; American Petroleum Institute, 1966). Pure component vapor pressures are calculated from equations of the Antoine type. For hydrocarbons in isomeric mixtures, the vapor molar volume V,,V of pure i is evaluated from the pressure explicit form of the virial equation. For hydrocarbons in nonisomeric mixtures the partial molar volume Y,,V of i in the vapor mixture is evaluated from the Cheuh-Prausnitz modified form of the Redlich-Kwong equation (Prausnitz, 1969), and the pure component V,,V is evaluated from the Redlich-Dunlop modified form of the Redlich-Kwong equation (Redlich and Dunlop, 1963). Fugacity coefficients evaluated by the two methods, virial us. RedlichKwong equation, agreed within several per cent when checking mixtures of isomeric compounds. The Lewis assumption was thus verified for isomeric systems, and credence was given to using the generalized Redlich-Kwong equation for other systems. The liquid phase activity coefficients are calculated using regular solution theory. -
Vi L In y i , L = ( b i - 6)2 RT
The average solubility parameter 8 of the liquid mixture is calculated from the pure component solubility parameters 6i by -
N
z,h,
fi =
(10)
i.1
where z, is the volume fraction of i in the liquid phase. zi
=
XiVi, L N
It is apparent from eq 9 that regular solution theory always predicts activity coefficients greater than unity.
where N is the number of components, M is the number of data points, a is the experimental volatility of component i relative to base component 1, and a’ is the calculated relative volatility using the equations in the previous section. Once the parameters for the C3, C4, and C5 isomeric systems were established, the same technique was used on systems containing components of different numbers of carbon atoms in order to interrelate all components. Having established the solubility parameters as a function of temperature, they were used to reproduce the phase equilibrium data for the systems used in the regression analysis and, in addition, to predict the equilibria for several other systems. The results are presented (Holland, 1971) as charts of relative volatility, product of vaporization equilibrium ratio and system pressure ( K p ) , and activity coefficient us. average solubility parameter of the liquid mixture. Some typical results and comparisons to the experimental data are included in this paper. Literature data were found insufficient for relating the CS hydrocarbons to other hydrocarbons covered in this work. The propane-2-methylbutane data of Vaughn and Collins (1942) were judged the best available, but they did not give solubility temperature relationships for C5 compounds consistent with those established for C3 and C4 hydrocarbons. Thus, it became necessary to establish an experimental data point, near room temperature, for a binary system containing 2-methylbutane and a C3 or Cq hydrocarbon. Butane was selected as the second component, and the data point was used in conjunction with the higher temperature data of Vaughn and Collins to establish the temperature dependencies of the C5 solubility param eters. The experiment was performed in a 250-cm3 equilibrium cell installed in a constant temperature bath. This apparatus, when tested with known systems, provided one theoretical stage within a few per cent accuracy. The cell was equipped with a sight glass, calibrated thermocouple, calibrated pressure gauge, stirrer entering through a stuffing box, and heated sample valves. The unit was evacuated and charged with butane and 2-methylbutane having purities greater than 99.5 mol %. The charge was stirred for 15 min and then allowed to stand for 30 min before sampling. Samples of the vapor and liquid were flashed directly into the gas sampling valve on a gas chromatograph, immediately injected, and analyzed. The chromatograph was equipped with a hexamethylphosphoramide-Chromosorb column and a thermal conductivity detector and was calibrated using samples of pure butane and 2-methylbutane. Multiple samples were withdrawn and analyzed, giving reproducible results, the averages of which are given in Table I.
Correlation a n d Experimental Procedures Vapor liquid equilibrium data in the form of T, p , y,, x , were selected from predominantly recent literature for hydrocarbons in the C3 to Cg range. Corresponding vapor pressures, molar liquid volumes, and fugacity coefficients were then evaluated using recent data, much of it being obtained from the API Data Book (American Petroleum Institute, 1966). Details can be found in the original reference (Holland, 1971). The apparent solubility parameters were determined by regression analysis. For each isomeric system and temperature, solubility parameters were selected and adjusted to minimize the following residual
Solubility Parameters Solubility parameters for each component were derived a t each temperature level by minimizing the residual described previously. The base solubility parameter is that of propane. It was arbitrarily taken from the work of Edmister (1960) as being linearly dependent on temperature. The values for all the compounds were also found t o vary linearly with temperature, as shown in Figures 1, 2, and 3. The agreement is good, with the largest discrepancy being 1.8%. These lines are recommended for correlating vaporliquid equilibria. To facilitate analytical computations, the coefficients of the equations of the lines are presented in Table II. A few of the points in Figure 1 were not used in the least-square fits of the lines. They were omitted due to Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
379
.
12t I I 260
1
I
,
1
I
1
,
I
I
1
,
300
I
,
1
~
l
,
l
I
3 40
l
/
I
3 80
TEMPERATURE , ' O K
Figure 1. Apparent solubility parameters of alkanes.
DROPYNE
t 14 2 80
1
1
1
1
~
1
300
8
1
1
(
320
TEMPERATURE,
1
340
O K
Figure 3. Apparent solubility parameters.
Table 11. Coefficients of Solubility Parameter: Temperature Equation 6 = mT b
t I -BUTEN< N
+
-
Temp range, Compound
2- ME THY L PROPENE -
~
-
Propane Propene Propadiene Propyne Butane 2-Methylpropane 1-Butene 2-Methylpropene trans-2-Butene cis-2-Butene Pentane 2-Methylbutane 2-Methyl-2-butene 2-Methyl-1,3-butadiene
m
m
3
_1
I2260
3 00
TEMPERATURE,
3 40
380
O K
Figure 2. Apparent solubility parameters of alkenes.
Table I. Experimental Butane-2-Methylbutane Vapor-Liquid Equilibrium Data
Temperature, O K Pressure, N / m 2 Composition, mol % butane Vapor Liquid Activity coefficient Butane 2-Methylbutane
298.2 236,000 96.14 91.17 1.ooo
1.037
discrepancies in the original vapor-liquid equilibrium data. These discrepancies occurred in binary systems of different carbon numbers and resulted in activity coefficients less than unity. The parameters increase with molecular weight and also in the direction paraffin-monoolefin-diolefin-alkyne. The parameter of a normal isomer is larger than that of a singly branched isomer. Insufficient data were available to determine the effect of conjugated us. nonconjugated bonds. The solubility parameters determined in this work a t 298.2"K are compared to literature values in Table III. 380
Ind. Eng. Chern..
Process Des. Develop.,Vol.
13, No. 4 , 1974
OK
- 18.0224 18,806 255-370 - 27.1829 - 40.7715
-37.2242 - 20,4734 - 20.4734 - 30.8110 - 29.0561 - 27 ,2922 - 36.0131 - 24,1247 - 23.9815 - 24.6772 - 23 ,3881
55 7
b
22,762 28,476 29,292 20,384 20,272 24,909 24,014 23,581 26,928 22,388 22,270 23,372 23,833
283-333 283-333 283-333 311-378 266-378 311-378 328-378 311-378 328-378 325-336 273-348 325-336 325-336
Differences of u p to 13% are found when the literature values are adjusted to a propane solubility parameter of 13.4 (J/m3)Ij2. Good agreement with the Chao-Seader values (1961) is not expected since they assumed the solubility parameters of all isomers of a given compound to be equal. The differences in the values determined from thermal or vapor pressure data (Blanks and Prausnitz, 1964; Hildebrand and Scott, 1950) and the values determined from vapor-liquid equilibrium data indicate that the source of data and deviations from regular solution theory are important. Therefore, combining of published solubility parameters with the apparent solubility parameters developed here should be done judiciously when attempting to predict vapor-liquid equilibria, keeping in mind that it is the difference between solubility parameters, not their absolute magnitudes, that is important. Correlations of Equilibria The temperature smoothed solubility parameters were used to generate vapor-liquid equilibria for the Ca to Cg hydrocarbon systems used to establish the parameters, and the calculated results were compared to experimental data. The systems studied include propane-propene-propadiene-propyne a t 283-333°K (Hill, et al., 1962); butane -2-methylpropane-1 -butene-2-methylpropene-trans-2-
Table 111. Comparison of Solubility Parameters a t 298.2"K
Solubility parameter, 6 X Hildebrand and Scott (1950) 12.3 ... 13.7 14.3 12.8 13.7 13.7 14.7 14.4 13.8 ... 15.2
Propane Propene Butane trans-2-Butene 2-Methylpropane 2-Methylpropene 1-Butene cis-2-Butene Pentane 2-Methylbutane 2-Methyl-2-butene 2-Methyl-1,3-butadiene
2
Chao and Seader (1961)
(J/m3)I/*
Blanks and Prausnitz (1964)
13.1 13.2
Edmister (1960)
12.7 12.5 13.5 14.3 12.8 13.7 13.7 14.7 14.4 13.8 ... 15.2
13.8
13.8 13.8 13.8 13.8 13.8 14.4 14.4 14.4
...
This work
13.4 ...
13.4 14.7 14.3 15.4 14.1
14.0
...
... ... ... ... 14.7 ...
15.3
15.7 16.2 15 2
...
15.1 16 . O
...
16.9
116
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2 - M E THY L - 2 - B U T E N E 080
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1
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LJERAGE SOLdBlLlTY 1
I5 0
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AVERAGE SOLUBILITY P A R A M E T E R
I
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MIXTURE, Sx (J/m3)"2 Figure 4. Comparison of experimental relative volatilities at 3243°K with lines calculated using regular solution theory.
butene-cis-%butene a t 327.6"K (Carmichael, et al., 1962); p e n t a n e-2 - m e th y 1b u t ane-2 -m e t hyl-2-butene-2-methyl1,3-butadiene a t 325-336°K (Peiffer, et al., 1973); propane-2-methylpropane a t 267-367°K (Hipkin, 1966); propane-2-methylbutane at 273-348°K (Vaughn and Collins, 1942); and butane-2-methylbutane a t 298°K (this work). A few of the results are illustrated here, followed by a summary of the accuracies attained for all the systems. The relative volatility, K p product, and activity coefficients are shown a s functions of average solubility parameter a t 3243°K for the multicomponent C5 system in Figures 4, 5 , and 6, respectively. The excellent fit of the calculated lines with the experimental points demonstrates the capability of regular solution theory for describing such a multicomponent system. Average deviations of 0.5% and 3% are noted for the relative volatilities and K p products, respectively. Excellent agreement is also noted for the activity coefficient correlation. The agreement of the calculated and experimental values of the relative volatilities, K p products, and activity coefficients of propane-2-methylpropane a t 366.5"K and propane-2-methylbutane a t 348.2"K is shown in Figures 7 and 8, respectively. Average deviations of relative volatilities and K p products are less than 2% for propane2-methylpropane and less than 3% for propane-2-methyl-
1
I
1
,
1
1
I5 5
PARLME-ER
I
1
I
1
I6 0
OF
S X I O - ~J, - 3 ) '2 Figure 5. Comparison of experimental K p products at 324.8"K with lines calculated using regular solution theory. WYTJRE
butane. The fit of the activity coefficient data is reasonable. Similar good agreement between calculated and experimental values was found for all of the systems studied. The average deviations in relative volatility for the multicomponent Cs, C4, and C5 systems range from 1.2 to 3.270, 0.7 to 5.7%, and
