Ind. Eng. Chem. Process Des. Dev. 1904, 23,
73
73-79
Thermodynamics of Alcohol-Alcohol Mixtures. 2. Prediction of Vapor-Liquid Equilibrium Data from Heat of Mixing Data for Alcohol-Alcohol Mixtures Concepclon Pando,+Juan A. R. Renunclo,t Richard W. Hanks, and James J. Chrlstensen' Department of Chemical Engineering and Contribution 293 from the Thermochemical Institute, Brigham Young Universm, Provo, Utah 84602
An expression for the heat of mixing (hE)of binary mixtures based on the assumption of the continuous linear association of both components is fitted to experimental hEdata of alcohol-alcohol binary mixtures to determine the adjustable parameters. By use of these parameters, either isothermal or isobaric vapor-liquid equilibrium data are predicted following the method of Hanks, Gupta, and Christensen for 21 binary systems for which VLE and hEdata have been found in the literature. The predictions obtained agree well with the experimental vapor-liquid equilibrium values even when the temperature difference between VLE and hEdata is large.
Introduction Vapor-liquid equilibrium (VLE) data are of great interest in industrial design and basic research. Although large amounts of these data have been reported in the literature, it is not always possible to find data for a certain binary system at the required conditions of temperature or pressure. Data for multicomponent systems are even more scarce. Many estimation methods for VLE data have been proposed over the years (Prausnitz, 1969). Most of these methods seem to fail when they are applied to binary mixtures formed by two alcohol molecules since there is not a theoretical or semiempirical equation to represent the particular behavior of these systems. In the preceding paper, part 1,we have proposed an equation able to represent the excess Gibbs energy and heat of mixing of the alcohol-alcohol mixtures. This equation assumes continuous linear association of both alcohol molecules and has been shown to be able to represent the excess Gibbs energy of these systems. A method for estimating VLE data from heat of mixing, hE, data has been described by Hanks et al. (1971). This method is especially useful in those cases where it is difficult to obtain good VLE data experimentally, but heat of mixing data are available (Christensen, 1982) or can be measured easily (Christensen, 1981). This method involves deriving an expression for the heat of mixing from an equation for the excess Gibbs energy by application of the Gibbs-Helmholtz relations. The parameters of the hE model are determined by curve-fitting the expression for hE to a set of experimental heat of mixing data. These parameters are then used in the original expression for the excess Gibbs energy to calculate VLE equilibrium data. In this paper, the method of Hanks, Gupta, and Christensen (1971) (hereafter called the HGC method) for predicting VLE data from heat-of-mixing data is applied to the new equation for alcohol-alcohol systems described in the preceding paper, part 1. Prediction Method The usual approach to VLE correlation or prediction is Departmento Quimica Fisica, Universidad Complutense, Madrid-3, Spain.
to obtain experimental information such as total pressure, p , liquid composition, x , and very often vapor composition, y, at constant temperature or pressure. From these data the liquid-phase activity coefficients, rj,may be calculated and then the excess Gibbs energy is computed using the well-known equation n
gE = R T C x j In y j j=l
(1)
The excess Gibbs energy data are then curve-fitted to some semiempirical model, gE ( x j , Cl...Ck)where Ckare the adjustable parameters, which are usually assumed to be temperature independent (Prausnitz, 1969; Marina and Tassios, 1973; Abrams and Prausnitz, 1975; Fredenslund et al., 1975). The excess enthalpy, hE,is related to gE by the GibbsHelmholtz relation
Frequently, a model is not able to correlate gEand hE data simultaneously. If gE data are used to determine the parameter values, CK,there is an error magnification effect inherent to the differentiation process of eq 2. Hanks et al. (1971) have proposed a prediction method in which an algebraic equation for the heat of mixing, hE ( x j , C,...Ck),is derived from a given gEmodel by application of eq 2. The values for Cl...Ck are determined by curvefitting of hE data. These Ckvalues are then used in the original gE model to calculate the activity coefficients, y j , from which the x-y data are finally predicted. It was found that x-y data computed in this manner agree well with available isobaric or isothermal experimental data for a number of binary hydrocarbon mixtures formed by alkanes, alkenes, alicyclic, aromatic, or arenes (Hanks et al., 1978),alcohol-hydrocarbon mixtures (Hanks et al., 1979), and hydrocarbon ether and hydrocarbonaldehyde mixtures (Pando et al., 1983). Some multicomponent systems have been also studied (Tan et al., 1977, 1978). When the HGC method is applied to the model for gE derived in the preceding paper (Pando et al., 1983), the following expression for the heat of mixing is obtained.
0196-4305/84/1123-0073$01.50/00 1983 American Chemical Society
74
Ind. Eng.
Chem. Process Des. Dev., Vol. 23, No. 1, 1984
- 15i1 E
20 I
0
(1 + 4KA@A)1/2 - (1 + 4KJ1’*
2KA
1-
Aho = 7%(R In K)/dT
(4)
where K may be either KAor KB. The value of Aho has been assumed to be -25000 J mol-’ (Renon and Prausnitz, 1967a,b). Results Table I lists the components and reference sources for the 21 alcohol-alcohol systems used in this study for which VLE data and hE data are available. These data were obtained from compilations by Christensen et al. (1982) (hE)and by Gmehling and Onken (1977) (VLE).These systems may be comprised of two molecules of a similar size (for example, systems XII, XIV,XV,XVI,XVII, XVIII,XIX),two molecules of different size (for example, systems V,VI,VII,XI),two small molecules (for example, systems I, 11, 111, VIII, IX),or two large molecules (for example, systems XX,XXI). A few other systems for which both hEand VLE data were available were discarded because of the insufficient number of experimental points or because the experimental data were very badly scattered. Heat-of-mixingdata were fitted to eq 3 with a nonlinear regression computer program (Pennington, 1970). Table I1 shows the temperature at which hE data have been measured, the values of the parameters KA,KB, and p’ for eq 3, the standard deviation u between experimental and calculated values of hE,and the percentage ( % ) of this standard deviation with respect the highest absolute value of hE. The subscript A in Table I1 refers always to the first component of the system. The total pressure of the mixture and the composition of the vapor phase have been calculated using the well-known relation
EXPERIMENTAL
- CALCULATED
il
T=298.15 K
7
where each symbol is defined in the preceding paper as well as in the list of nomenclature. There are three adjustable parameters in eq 3, two association-equilibrium constants, KA and KB, and one interaction-energy parameter p’. The molar volumes, U A and uB, also influence the curve fitting. The enthalpy of a hydrogen bond, Aho,is related to the equilibrium constants by
1 0
P - 1 0 1 . 3 5 KPa
0
0.2
0.4
0.6
0.8
1.0
XA
Figure 1. System methanol-ethanol: (a) fit of hE data vs. composition; (b) experimental and predicted values of the y-x diagram.
this equation were taken from Gmehling and Onken (1977). When the temperature interval for these coefficients was smaller than that of the experimental data, the Antoine coefficients compiled by Reid et al. (1977) were used. Molar volumes of pure components have been taken from Timmermans (1950,1965)or estimated by means of the method of Gunn and Yamada (1971). Values for the critical constants and acentric factors required to apply the Redlich-Kwong equation of state were taken from Reid et al. (1977). Table I11 shows the results of the predictions of YA and/or p . Temperatures and pressures for which the VLE data are valid are indicated in this table. For isothermal data, the column of pressures shows the interval of pressures in which the experimental data were taken. For simplicity only integer values are given for the interval, while the temperature is given with the precision reported by the authors. For isobaric data, the value of the pressure is given in the pressure column (only one value) and the temperature column shows the temperature range of experimental points. Values for the relative or percent standard deviation may lead to erroneous conclusions when one or two experimental data points are in a low mole fraction region for the component chosen for comparison. Therefore, we have chosen to report the comparison between experimental and calculated values of yAin terms of the mean absolute deviation defined by i=N
mean deviation = ( C lyl(exptl) - y,(calcd)l)/N (6) i=l
where y is the activity coefficient of component i, f is the vapor-phase fugacity coefficient, y is the vapor mole fraction, p is the total pressure, x is the liquid mole fraction, and fo is the pure-liquid fugacity coefficient. The activity coefficients y i have been calculated by substituting the values obtained for the parameters, KA, KB, and p’ into eq 21 and 22 of the preceding paper. The fugacity coefficient has been calculated with Wilson’s modification of the Redlich-Kwong equation of state (Wilson, 1969). The vapor pressures of pure components were calculated with the Antoine equation. Values for the coefficients of
where N is the number of data points. We have also chosen to use the same criterion for the total pressure. The maximum value of the deviation in the range of the data is also reported in order to avoid the erroneous conclusions that may be derived from consideration of only the mean deviation. When dashes appear in they mean and maximum deivation columns, no experimental values have been reported for the vapor composition. A few characteristic alcohol-alcohol systems were chosen in order to illustrate the results given in Table 111. Figure 1shows results for system I, which is formed by two small molecules and exhibits moderate endothermic heats of mixing. Figure 2 shows results for system XXI formed by two large molecules which gives rise to larger magnitude
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table I. Source Reference for the Alcohol-Alcohol Systems system
ref for h E
ref for VLE
I1 I11
+ ethanol methanol + 1-propanol methanol + 2-propanol
IV
methanol
+
1-propanol, 2-methyl
Polak et al. (1970a)
V VI VI1
methanol methanol methanol
+ + +
1-butanol 1-pentanol 1-dodecanol
Pope et al. (1967) Sun et al. (1980) Diaz PeEa et al. (1964)
VI11
ethanol
+
1-propanol
Ramalho and Rue1 (1968)
IX
ethanol
+
2-propanol
Parks and Kelly (1925)
X XI
ethanol ethanol
+
+
1-butanol 1-pentanol
Pope et al. (1967) Ragaini et al. (1968)
XI1
1-propanol + 2-propanol
Polak et al. (1970b)
XI11 XIV
Pope et al. (1967) Geiseler et al. (1973)
xv
1-propanol + 1-butanol 1-propanol, 2-methyl + 2propanol, 2-methyl 1-propanol, 2-methyl + 1-butanol
XVI XVII
1-propanol, 2-methyl + 2-butanol 2-propanol, 2-methyl + 1-butanol
Geiseler et al. (1973) Geiseler et al. (1973)
XVIII
2-propanol, 2-methyl + 2-butanol
Geiseler et al. (1973)
XIX
1-butanol + 2-butanol
xx
1-hexanol t 1-octanol 1-octanol + 1-decanol
Murakami and Benson (1973), Geiseler et al. (1973) Pope et al. (1967) Pope et al. (1967)
Pavlov et al. (1968)
methanol
XXI
1 0
g
25
75
Slobodyanik and Babuskhina (1966), Schmidt (1926) Schmidt (1926) Verhoeye and De Schepper (1973), Freshwater and Pike (1967) Lesteva and Khrapkova (1972), Udovenko and Frid (1948), Jaenecke (1933) Hill and Van Winkle (1952) Hill and Van Winkle (1952) Diaz PeKa and Sotomayor (1971) Ochi and Kojima (1969), Udovenko and Frid (1948) Ballard and Van Winkle (1952) Gay (1927) Hellwing and Van Winkle (1953) Ballard and Van Winkle (1952) Gay (1927) Geiseler et al. (1973)
Pflug et al. (1968) Taylor and Bertrand (1974)
'
Geiseler et al. (1973), Tamir and Wisniak (1975) Geiseler et al. (1973) Quitzsch et al. (1969), Geiseler et al. (1973) Morachevsky and Popovich (1965), Geiseler e t al. (1973) Quitzsch et al. (1969)
Murakami and Benson (1973)
L
EXPERIMENTAL
- CALCULATED
Rose and Supina (1961) Rose et al. (1958)
1
EXPERIMENTAL
0
r
7
g
20
120
15
u '
c
10
u
5 0.2
I '!
1 .o 0
0.4
0.6 '
0.8 '
1u.O
Oo
XA EXPERIMENTAL
0.2
0.4
0.6
0.8
,
1.0
0.8
P = 3 9 . 9 9 KPa
L
o.2 00
0.2
0.4
0.6
1
i 0.8
1.0
XA
#
0.6 o
.
~" 0 1 . 3~5
KPa
1
EXPERIMENTAL
0.2 0
4
1.0
- CALCULATED
o
0.2
0.6
0.4
0.8
1.0
XA
Figure 2. System 1-octanol + 1-decanol: (a) fit of hE data vs. composition; (b) experimental and predicted values of the y-x diagram.
Figure 3. System methanol + 2-methyl-1-propanol: (a) fit of hE data vs. compositions;(b) experimental and predicted values of the y-x diagram.
heats of mixing. Figures 3 and 4 illustrate two systems (IV and VIII) formed by molecules of different size, and Figure 4 also illustrates hE fits at different temperatures. The x-y predictions are made with the parameters corresponding to the closest temperature to that of the VLE data. Figures 5, 6, and 7 show three examples of systems having exothermic heats of mixing. Figure 5 gives the results of system I11 formed by XVI and XIX molecules of different
size while Figures 6 and 7 give two examples of systems consisting of isomer molecules. Discussion The results reported in Table I1 show that eq 3 is able to fit hE curves of very different shapes and very different magnitudes, endothermic or exothermic. The standard deviations are always of the order of a few joules per mole, which represents less than 5% deviation for most of the
76
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table 11. Calculated Values of the Parameters and Comparison of Experimental and Calculated hE Data parameters of eq 3 std dev,' system TIK K , x 103 K , x 103 p'/J cm-3 o/J mol-' -7.199 -2.541 0.25 I 298 4.743 11.64 2.489 0.61 I1 298 6.244 -15.89 6.9 13.00 55.23 I11 298 4.938 2.2 10.04 21.46 IV 298 1.7 26.02 4.198 V 298 9.681 35.06 7.810 4.9 VI 298 9.084 10.40 9.2 30.03 171.5 VI1 298 9.712 -~1.744 0.77 VI11 293 7.535 --3.894 0.36 12.596 15.977 298 -3.909 1.01 12.382 15.955 303 5.1 8.953 -5.743 IX 2 98 5.444 2.216 0.37 0.8832 2.453 X 2 98 3.779 0.47 2.742 5.871 XI 293 1.788 0.71 6.933 13.61 298 0.90 20.45 0.1804 9.789 303 0.39 17.65 -7.770 XI1 298 15.91 0.16 7.105 -1.611 XI11 298 5.737 -35.00 9.3 72.88 115.89 XIV 299 -- 2.0 70 0.55 9.675 9.959 xv 298 3.9 48.26 -13.44 XVI 2 99 43.10 -- 2 5.8 5 5.3 51.15 9.553 XVII 299 1.8 12.07 -8.457 19.98 XVIII 299 0.8 5 22.07 - 9.446 18.33 XI x 298 0.31 9.261 0.7568 xx 2 98 8.123 0.31 7.600 -0.4708 XXI 298 6.538 a Standard deviation between experimental and calculated hE values. highest absolute value of h " .
100
%b
__
5.4 0.8 8.8 1.4 1.1 2.3 1.8 3.3 1.8 5.6 9.5 0.8 0.5 0.8 1.1 0.8 2.9 2.3 4.8 5.5 1.2 1.7 0.9 1.2 1.4
Percentage of standard deviation with respect to
I
0293 -20
o EXPERIMENTAL -- CALCULATED
I
C
4.
6 01
A
-3
'u
c
0
0
4:
0.2
I
'P
0.61
A 0.4
0.6
0.8
P=101.35KPa
:::, ; 1
9
EXPERIMENTAL
-CALCULATED
1.0
II
-60
-100 0 1 .o 0.8
0.2
0.6
0.4
1.0
0.8
EXPERIMENTAL c u L A TE D D ~ '
c
~- AL
F
b
3p'
of
i
I
T=328.15 K
J
0 0
0.2
0.4
0.6
0.8
1.0
XA
0
0.2
0.4
0.6
0.8
1.0
XA
Figure 4. System ethanol + 1-pentanol: (a) fit of hE data vs. composition at several temperatures; (b) experimental and predicted values of the y-x diagram. Parameters at 303.15 K were used to make predictions.
Figure 5. System methanol + 2-propanol: (a) fit of hE data vs. composition; (b) experimental and predicted values of the y-x diagram.
systems (only 5 systems have deviations ranging from 5 to 10%). These standard deviations are, for most of the systems, of the same order of magnitude as the standard deviations reported by authors when the raw experimental data are fitted to an empirical equation (Christensen et al., 1982). This accuracy is equivalent to that of other studies using the HGC method reported previously (Hanks et al., 1971, 1978, 1979; McFall et al., 1983; Pando et al., 1982, 1983). The highest standard deviations in Table I1 correspond to strongly exothermic systems when 0' has to adopt large negative values in order to neutralize the effect of the second and third terms on the right-hand side of eq 3. For these conditions, the molar volumes of each component
become the main factors in the fitting of the hE curve. If the molar volumes are identical or very similar, the curve is nearly symmetrical (minimum near the equimolecular mixture). This is the case of systems I11 (Figure 5) and IX, for which the standard deviations with respect to the maximum value of hE are 8.8 and 9.5%, respectively. It is important to notice that for most of the systems studied, heat-of-mixing data are given at 298.15 K while most of the VLE data are at higher temperature. It may be observed in Table I11 that better predictions correspond to systems when the temperature difference between hE and VLE data is small. One reason for this may be that the KA's do not seem to exhibit the correct temperature dependency as discussed in the preceding paper, part 1.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table 111. Comparison of Experimental and Calculated VLE Data vapor compn system
____ I
I
I11 IV
V VI VI1 VI I1
IX X XI XI1 XI11 XIV
xv XVI XVII
XVIII XIX
xx XXI
TIK ~ 273.15 293.15 313.15 323.15 333.15 353.15 373.15 339-351 273.15 283.15 293.15 303.15 313.15 323.15 333.15 328.15 340-3 52 323.15 333.15 343.15 338-370 344-375 339-395 298.15 323.15 333.15 343.15 353.15 353-367 351-355 353-388 353-405 356-369 373-388 313.15 313.15 381-389 313.15 313.15 314-340 335-361 347-374 355-383 313.15 358-368 313.15 328-340 352-364 363-377 372-386 3 55-3 65 365-375 376-394 393-411 409-427 441-459
_
PlkPa
-
_
2-4 6-12 20-33 32.52 51-79 114-162 237-318 101.35 1-4 2-7 3-12 6-20 10-33 17-52 27-79 33-66 101.35 14-52 22-77 34-115 101.35 101.35 101.35 4-16 14-28 24-45 38-70 58-104 101.35 101.35 101.35 101.35 101.35 101.35 5-13 3-4 101.35 4-6 3-13 13.31 39.99 66.67 93.36 6-14 101.35 3-6 13.31 39.99 66.67 93.36 4.00 6.69 2.69 6.69 13.31 39.99
_
mean dev
_
_
~
-
max dev
~
-
0.0098
-0.0361
0.0320 0.0211 0.0283 0.0306 0.0321 0.0165 0.0619
0.0580 0.0428 0.0942 0.0958 0.0960 0.0545 0.1925
0.0552
0.20 54
0.022c 0.0230 0.0193 0.0211 0.0118 0.0101 0.0090 0.0179 0.0160 0.0077 0.0470 0.0211 0.0036 0.0173 0.0339 0.0381 0.0413 0.0433 0.0468 0.0048 0.0592 0.0235 0.0143 0.0163 0.0179 0.0191 0.0134 0.0087 0.0242 0.0244 0.0070 0.0223
0.0492 0.0515 0.0413 0.0428 0.0247 0.0168 0.0189 0.0675 -0.0270 0.0135 0.1431 0.0394 0.0092 -0.0339 0.1010 0.1048 0.1129 0.1141 0.1228 0.0102 0.1068 -0.0628 -0.0279 -0.03 7 5 -0.0362 -0.0416 0.0324 0.01 70 0.0456 0.0406 0.0136 0.0408
Predictions of the vapor mole fraction, y, for VLE data ranged from excellent to adequate as indicated in Table 111. Values for the mean deviation of y range from 0.01 to 0.04 for most of the systems studied. These deviation values are essentially equivalent to those reported when experimental VLE are directly fitted to one of the commonly used expressions for GEsuch as Margules, van Laar, Wilson, NRTL, or UNIQUAC models (Gmehling and
77
total pressure/Wa
---___
mean dev
max dev
0.083 0.13 0.66 1.17 2.00 2.76 5.65 6.21 0.083 0.13 0.23 1.45 1.45 2.34 1.38 4.34 9.10 3.72 5.65 8.62 5.24 11.38 17.58 3.72 1.45 1.93 4.34 5.58 4.69 3.311 1.71 7.98 4.64 1.44 1.43 0.17 2.21 0.24 0.6 1 1.68 5.22 8.69 12.61 0.11 4.62 0.33 0.50 1.79 2.84 3.94 0.25 0.21 0.10 0.11 0.30 1.63
0.14 0.28 0.90 1.79 2.62 3.79 7.72 8.83 -0.14 0.18 0.40 1.86 1.93 3.38 2.34 6.97 11.17 5.45 7.93 12.07 11.24 16.13 31.51 4.96 2.14 2.90 5.86 8.07 6.76 4.55 2.48 11.51 7.58 1.93 1.93 0.28 2.90 0.41 1.03 3.59 8.55 14.06 20.13 0.21 6.41 0.48 0.83 2.62 4.62 6.34 0.28 0.35 0.14 0.21 0.76 2.83
Onken, 1977). Predictions of the total pressure for a majority of systems also show a degree of accuracy similar to that of the y predictions except for a few systems (VI11 at 343.15 K, XVII at 13.31 KPa, and XXI at 6.69 and 13.31 KPa) where high deviations of y correspond to low deviations in p (or vice versa). This may be due to different uncertainties of the experimental variables. It must be pointed out that similar trends in the mean deviations
78
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 0 \
-20
"
T = 299.15 K
\
0
-
0
EXPERIMENTAL CALCULATED
-100 0.2
0
0.4
0.6
0.8
10
0.6
0.8
1.0
T=313.15 K
0.2 0
0.2
0
0.4 XA
+
F i g u r e 6. System 2-methyl-1-propanol 2-butanol: (a) fit of hE d a t a vs. composition: (b) experimental and predicted values of t h e y-r diagram.
0 -20
- CALCULATED
3
'-
-40
E
T= 2 98.1 5 K
-60
7
\
W
-80 -100
0.4
0.2
0
0.6
0.8
1.0
P=93.36 KPa 0
0.6-
2
EXPERiMENTAL
-CALCULATED /
0.4-
b
P
r '
0.2 0
/6
5
1
/1
0.2
0.4
0.6
0.8
I
1.0
XA
+
F i g u r e 7. System 1-butanol 2-butanol: (a) fit composition; (b) experimental a n d predicted values gram.
of he d a t a vs. of t h e y-x dia-
appear when other models for GE are applied directly to these three sets of data (Gmehling and Onken, 1977). There seems to be no correlation between the goodness of fit of hE data and the fit of the VLE data. Generally, predictions of isobaric sets of data are better than those of isothermal data. This happens in systems 111, IV, VIII, XV, and XIX with the only exception being system XVII. This may be connected to the greater experimental uncertainties of the vapor composition determined by an isothermal method of measurement. The results of this study suggest that the model for the gE proposed in the preceding paper is able to represent both gE and hE data simultaneously and may be reliably used in conjunction with the HGC method for predicting VLE data from isothermal or isobaric hE data for alcohol-alcohol systems. Acknowledgment This work was partially funded by U.S. Department of
Energy Contract No. DE-AC02-82ER13024. C. Pando wishes to acknowledge the Board of Foreign Scholarships and the Spanish Mnistry of Education for their support through a Fulbright/MUI award. Nomenclature C = coefficients of a model f = fugacity coefficient gE = excess Gibbs energy hE = excess enthalpy K = equilibrium constant n = number of components N = number of data points p = pressure R = constant of gases T = temperature u = molar volume x = molar fraction in the liquid phase y = molar fraction in the vapor phase p' = physical interaction parameter y = activity coefficient 4 = volume fraction d* = volume fraction in the pure component u = standard deviation % = ratio of standard deviation over the maximum value of heat-of-mixing Subscripts A = first component of the mixture B = second component of the mixture i = data points J = components of a mixture Literature Cited Abrams, D. S.; Prausnitz, J. M. AIChEJ. 1975, 2 7 , 116. Baiiard. L. H.;Van Winkle, M. Ind. Eng. &em. 1952, 4 4 , 2450. Christensen, J. J.; Hanks, R. W.; Izatt, R. M. "Handbook of Heat of Mixing"; Wiley: New York. 1982. Christensen, J. J.; Hansen, L. D.; Izatt, R. M.; Eatough, D. J. Rev. Sci. Instrum. 1981, 52, 1226. Diaz PeAa, M.; Fernandez Martin, F. An. Real SOC.ESP. f i s . Quim. 1984, 808, 9. Diaz Peiia, M.;Sotomayor, C. P. An. Real SOC.ESP.Fis. Quim, 1971, 678, 233. Fredensiund, A.; Jones, R. L.; Prausnitz, J. M. AIChE J . 1975, 2 7 , 1089. Freshwater, D. C.; Pike, K. A. J . Chem. Eng. Data 1987, 72, 179 Gay, L. Chim. Ind. 1927, 18, 167. Geiseler, G.; Siihnei, K.; Quitzsch, K. Z . Phys. Chem. (Leipzig) 1973, 254, 261. Gmehiing, J.; Onken, U. "Vapor-Liquid Equlilbrlum Data Collection", Chemistry Data Series; Dechema: Frankfurt, West Germany, 1977; Vol. 1. Parts 2a and 2b. Gunn, R. D.; Yamada, T. AIChE J . 1971, 77, 1341. Hanks, R. W.; Gupta, A. C.; Christensen, J. J. Ind. Eng. Chem. Fundam. 1971, 70, 504. Hanks, R. W.; Tan, R. L.; Christensen, J. J. Thermochim. Acta 1978, 2 7 , 9. Hanks, R. W:; O'Neiil, T. K.; Christensen, J. J. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 408. Heilwing, L. R.; Van Winkle, M. Ind. Eng. Chem. 1953, 4 5 , 624. Hili. W. D.; Van Winkle, M. Ind. Eng. Chem. 1952, 4 4 , 205. Jaenecke, E. 2.Phys. Chem. 1933, 764, 401. Lesteva, T. M.; Khrapkova, 2 . I . Zh. Fir. Khim. 1972, 46, 612. McFali, T. A.; Hanks, R . W.; Christensen, J. J. Thermochim. Acta 1983, 60, 327. Marina, J. M.; Tassios. D. P. Ind. Eng. Chem. Process Des. Dev. 1973, 12, 67. Morachevsky, A. G.; Popovich, 2 . P. Zh. Prikl. Khim. 1983, 38, 2129 Murakami, S.; Benson, G. C. Bull. Chem. Soc.Jpn. 1973, 46, 74. Ochi, K.; Kohima. K. Kagaku Kogaku 1989. 3 3 , 352. Pando, C.; Renuncio, J. A. R.: Hanks, R. W.: Christensen. J. J.. Thermochim. Acta, 1983, 62. 113. Pando, C.; Renuncio, J. A. R.; Hanks, R. W.; Christensen, J. J. Ind. Eng. Chem. Process Des. Dev. 1983; preceding paper In this issue. Parks. G. S.: Keliv. K. K. J . Phvs. Chem. 1925. 2 9 . 727. Pavlov, S. Yu.; Karpacheva, L.*L.; Serafimov, L. A.; Kofman, L. S. Zh. Fiz. Khim. 1988, 4 2 , 73. Pennington, R. H. "Introductory Computer Method and Numerical Analysis". 2nd ed.; Mcmlllan: London, 1970. Pfiug, H. D.: Pope. A. E.; Benson, G. C. J . Chem. Eng. Data 1988, 13. 408. Poiak, J.; Murakami, S.; Lam, V. T.; Pflug, H. D.; Benson, G. C. Can. J. Chem. 1970a, 48, 2457. Polak, J.; Murakami, S.; Benson, G. C.; Pfiug. H. D.; Can. J. Chem. 197Ob, 48, 3782. Pope, A. E.; Pfiug, H. D.; Dacre, B.; Benson, G. C. Can. J . Chem. 1967, 4 5 , 2665. Prausnh, J. M. "Molecular Thermodynamics of Fluid-Phase Equilibria"; Prentice-Hail: New York, 1969.
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 79-87 Quitzsch, K.; Koehler, S.; Taubert, K.; Gelseler, G. J. frakt. Chem. 1969, 311, 429. Ragaini. V.; Santi, R. Carra, S. Lincei-Rend. Sc. Fis. Mat. e Nat. 1968, 4 5 , 540. Ramalho, R. S.; Ruel, M. Can. J. Chem. Eng. 1988, 46, 456. Reid, R. C.; Prausnitz, J. M.; Sherwocd, T. K. “The Properties of Gases and Liquids”, 3rd ed.; McGraw-Hill: New York, 1977. Renon, H.; Prausnitz, J. M. Chem. Eng. Sci. 19678, 22, 299. Renon, H.; Prausnitz, J. M. Chem. Eng. Sci. I987b, 22, 1891. Rose, A.; Papahronls, B. T.; Williams, E. T. Chem. Eng. Data Ser. 1958, 3 , 216. Rose, A.; Suplna, W. K. J. Chem. Eng. Data 1961, 6 , 173. Schmidt, G. C. Z . fhys. Chem. 1926, 121, 221. Slobodyanik, I.P.; Babuskhina, E. M. Zh. frikl. Khim. 1966, 39, 1555. Sun, H. H.; Christensen, J. J.; Izatt, R. M.; Hanks, R. W. J. Chem. Thermodyn. 1980, 12,95. Tamir, A.; Wisniak. J. J. Chem. Eng. Data 1975, 2 0 , 391.
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Received for review September 7 , 1982 Revised manuscript received March 7 , 1983 Accepted March 25, 1983
Permeation of Pure Gases under Pressure through Asymmetric Porous Membranes. Membrane Characterization and Prediction of Performance Ramamurtl Rangarajan, M. A. Mazld, Takeshl Matsuura, and S. Sourlrajan’ Division of Chemistry, National Research Council of Canada, Ottawa, Canada, K 1A OR9
Permeation rates of hydrogen, helium, methane, nitrogen, ethylene, and argon through asymmetric porous cellulose acetate, hydrolyzed cellulose acetate propionate, and polysulfone membranes have been measured at room temperature. The experimental data have been analyzed by a transport equation which incorporates terms for a pore size distribution on the membrane surface. The mechanism of transport involves simultaneous Knudsen, slip, viscous, and surface flow through the porous structure. The analysis is useful for membrane characterization and predictlon of performance with respect to a reference gas.
Introduction Sorption as well as unsteady- and steady-state permeation of different gases in a variety of membranes have been studied in great detail in view of the potential applications in gas separations and also to elucidate the nature of transport prevailing in such systems. As this communication is concerned only with the mechanisms of transport as it relates to the characterization of a given membrane and the prediction of membrane performance, it is deemed adequate to cite only the major accomplishments in the understanding of the nature of such transport as reported in some of the recent publications. An in-depth understanding of the different approaches to the subject may be pursued through the references cited therein. A historic perspective to transport of gases through synthetic polymer membranes is given by Stannett (1978). Reports of pertinent literature by Barrer (1967), Hopfenberg (1974), Hwang and Kammermeyer (1975), Stern (1976), Stannett et al. (1979), and Stern and Frisch (1981), based on their extensive researches, provide the necessary background material for the understanding of vapor and gas transport through membranes. From the mechanistic point of view of gas and vapor transport through “nonporous” membranes, the free volume theory and the dual sorption theory have been highly successful in the analysis of sorption, unsteady-, and steady-state permeation in such membrane systems. The concept of free volume as developed by Cohen and Turnbull (1959) for the case of self-diffusion in a liquid of hard spheres suggests that the permeant diffuses by a cooperative movement of the permeant and the polymer segments, from one “hole” to the other within the polymer, the creation of a 0196-4305/84/ 1123-0079$0 1.50/0
“hole” itself caused by fluctuations of local densiy. Based on the concept of redistribution of free volume to represent the thermodynamic diffusion coefficient (Fujita et al., 1960),and standard reference state for free volume (Frisch, 1970), Stern and Fang (1972) interpreted their permeability data for nonporous membranes, and Fang, Stern and Frisch (1975) extended the theories to the case of permeation of gas and liquid mixtures. The dual sorption model invokes the existence of two thermodynamically distinct populations of the penetrant gas, namely, molecules dissolved in the polymer by an ordinary dissolution mechanism (obeying Henry’s law) and molecules residing in a limited number of preexisting microcavities in the polymer matrix (obeying Langmuir type of isotherm) with rapid exchange between these two populations. The development and the illustration of the applicability of the dual sorption model for permeation through glassy polymers are the results of extensive investigations by Paul, Koros, and their associates, Vieth and his associates, and Stern and co-workers. The work of Koros et al. (1977) on sorption and transport of gases in polycarbonate, Chan et al. (1978) on hydrocarbon gas sorption and transport in ethyl cellulose, and on CO,, CHI, A, and N2 transport through polysulfone by Erb and Paul (1981) and the relaxation of immobilization assumption by Petropoulos (1970) resulted in a coherent approach to the understanding of the transport behavior of gases through glassy polymer membranes. Two exhaustive reviews by Vieth et al. (1976) and Paul (1979) furnish detailed information pertaining to different aspects of dual-sorption model. In addition to the free volume theory and dual sorption models, a few molecular models have also been proposed. Published 1983 by the American Chemical Society
