Ind. Eng. Chem. Fundam. 1985, 2 4 , 325-330
325
An Improved Potential Theory Method for Predicting Gas-Mixture Adsorption Equilibria Sunll D. Mehtat and Ronald P. Danner. Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
A relatively simple and accurate method based on the potential theory of adsorption has been developed for
predicting gas-mixture adsorption equilibria. The new method has a number of practical advantages compared to other methods while giving results of equal accuracy. Phase compositions and total amounts adsobbed can be predicted over a range of temperature from a single adsorption isotherm for each of the pure components or an isotherm on the same adsorbent for a component of similar chemical nature.
Grant and Manes (1964) have presented a potential theory method for correlating pure-gas isotherms. Their procedure for predicting mixture equilibria is based on the assumption that the adsorption data for all the pure components will fall on the same line when plotted on appropriate coordinates. This is not generally true, however, for adsorption on polar adsorbents such as molecular sieves and silica gel. Even for activated carbon, the assumption does not always hold true. The potential curves for the adsofption of different pure gases on activated carbon are usually close, but they do not coincide. Therefore, a better method of correlating pure-gas isotherms is needed. A number of methods based on the potential theory have been advocated for correlating pure-gas adsorption isotherms. Cook and Basmadjian (1964) presented a review of the most significant of these. On the basis of some preliminary comparison studies, we found the method of Lewis et al. (1950~)to be the most promising of all these methods. Hence, it was singled out for further study. The method of Lewis et al. consists of expressing the adsorption isotherm of a pure component in the following functional form
Introduction The potential theory of adsorption, introduced by Polanyi (1914), has been widely accepted as the basis for correlating the effect of temperature on the adsorption isotherms of pure gases. A number of authors (Dubinin and Timofeyev, 1946; Lewis e t al., 1950c, Maslan et al., 1953; Grant and Manes, 1964) have modified Polanyi's original method of treating pure-gas adsorption isotherms. These modifications attempt to improve the temperature correlation and also to account for the effect of the nature of the adsorbate on the adsorption isotherm for a given adsorbent. Efforts have been made to extend the potential theory for the prediction of gas-mixture adsorption equilibria (Lewis et al., 1950b; Bering et al., 1963; Grant and Manes, 1966; Fernbacher and Wenzel, 1972). According to Sircar and Myers (1973),these extensions of the potential theory are not thermodynamically consistent. Greenbank and Manes (1981), however, point out that if adsorbate nonuniformity (Hansen and Fackler, 1953) is incorporated in the model, the thermodynamic consistency requirement can be met. In any case, the potential theory has had its share of success in predicting mixture equilibria in the ranges normally of interest. It is very difficult to determine the true thermodynamic properties of the adsorbed phase, and the potential theory offers a practical way of estimating mixture equilibria. Of the potential theory methods of predicting gas-mixture adsorption equilibria, the most useful has been the method of Grant and Manes (1966). This method, however, applies only to adsorption on activated carbon and breaks down for adsorption on other adsorbents such as molecular sieves and silica gel. Even on activated carbon, the method gives rather inaccurate results for certain adsorbate mixtures. The method proposed in this work is a modification of the Grant and Manes method but is more accurate and has wider applicability. The discussion of the proposed method is restricted to the adsorption of binary-gas mixtures; the method can, however, readily be extended to multicomponent systems. Correlation of Pure-Gas Adsorption Isotherms The prediction of mixture adsorption equilibria by the potential theory depends on the successful correlation of the pure-gas isotherms. For this reason, the adsorption of pure gases is considered first.
where No,. = moles of pure i adsorbed per unit mass of adsorbent, Vosi = saturated liquid molar volume of pure i at a pressure equal to the adsorption pressure of POi, psi = saturation fugacity of pure i a t the adsorption temperature T , and p i= fugacity of pure i a t pressure Poi and temperature T. The Lewis et al. method is similar to the Grant and Manes (1964) method, the only difference being that the latter method uses a pressure-independent volume VObi (the saturated liquid molar volume at the normal boiling point) instead of the pressure-dependent volume, Vosi. If a successful correlation is achieved, the function f in eq 1 should be independent of the temperature and the nature of the adsorbate. The plot of the left-hand side vs. the right-hand side should then give a potential curve characteristic of the given adsorbent. The method of Lewis et al. was comprehensively evaluated using the pure-component data from an extensive adsorption data bank. This bank contained data for pure, binary, and ternary systems on different adsorbents over wide ranges of temperature and pressure. A complete listing of these data and the results of their evaluation are given by Mehta (1982).
+ Present address: Technology Development Group, Intel Corporation, Livermore, CA 94550.
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1985 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 24, No. 3, 1985
c t
.-
>*
E X P DATA l j ' ~ . O ETHANE ! 2 9 8 K l
ETHANE ! 3 2 3 K ( 3 ETHYLENE ( 2 9 8 K ) A E T H Y L E N E (323 K ) 0
0
ETHYLENE ( 2 9 8 K ) A ETHYLENE ( 3 2 3 K )
0
DANNER AND CHOI ! 19781
DANNER AND CHOI ( 1 9 7 8 )
,63!
0
2
I
2
3
4
5
I
6' 0
I
2
3
4
5
Figure 1. Potential curves for adsorption of pure ethane and ethylene on molecular sieve 13X.
Figure 2. Coalesced potential curve for adsorption of pure ethane and ethylene on molecular sieve 13X.
The method of Lewis et al. was generally found to give a good temperature correlation of single-component data on all types of adsorbents. Also, on the nonpolar adsorbent, activated carbon, the potential curves of many different adsorbates were found to be very close to one another. On polar adsorbents (moleculr sieves and silica gel), however, the potential curves of different adsorbates were widely separated, except when the adsorbates belonged to the same family of hydrocarbons, e.g., paraffins, olefins, or diolefins. This divergence of the potential curves is caused by specific polar adsorbate-adsorbent interactions and by interactions of the polar groups in the adsorbent with multiple bonds in the adsorbate. In many cases, the temperature of interest was above the critical temperature of the pure adsorbate. Thus, the values of Vo,.and f " , had to be extrapolated to the desired conditions in these cases. The modified Rackett equation (Spencer and Danner, 1972) was used to determine the liquid volumes. For supercritical components, Vosiwas taken to be the value at a reduced temperature of 1.0. The fugacities were calculated by using the method of Lee and Kesler (1975). Vapor pressures were calculated by using the Antoine equation. For supercritical components, the vapor pressures were first extrapolated by means of the Antoine equation and the saturation fugacities were then determined at these vapor pressures and the temperatures of interest. A detailed description of the extrapolation techniques used is provided by Mehta (1982). A typical application of the method of Lewis et al. is illustrated in Figure 1, which shows the potential curves obtained for hydrocarbon adsorption on the polar adsorbent, molecular sieve 13X. An excellent temperature correlation is obtained. Similar temperature correlations were observed for all systems studied; however, the available data do not cover systems with both polar adsorbates and polar adsorbents. The potential curves for ethane and ethylene in Figure 1 are distinct. On closer examination of the curves, however, one may observe that they have similar shapes and can be coalesced by an appropriate modification of the abscissa. In this particular case, dividing the abscissa for ethylene by 1.62 yields a single potential curve, as shown in Figure 2. This method of coalescing the potential curves was applied to other adsorbate-adsorbent systems and was generally found to be very successful in bringing the curves for various adsorbates together. In most cases, the curves were found to differ by constant coalescing factors independent of temperature which could be included in the
definition of the abscissa in the potential plots to result in a single curve for all the adsorbates. In other words, the functional representation of eq 1 was modified to
where koi is the coalescing factor for adsorbate i relative to a standard adsorbate. The standard adsorbate could be selected as any of the gases under consideration; it was usually taken to be a paraffin if isotherm data were available. The coalescing factor for the standard adsorbate was assigned a value of unity. A list of the coalescing factors for some representative adsorbate-adsorbent systems is given in Table I. Note that the values of the coalescing factors for the nonpolar adsorbent, activated carbon, are very close to unity. The coalescing factor, as considered so far, is just an empirical divisor to bring together the potential curves of different adsorbates. This factor can be calculated geometrically from a given potential plot. The coalescing factor is clearly related to polar interactions between the adsorbate and adsorbent, but it is difficult to quantify this relation. If correlating pure-gas isotherms were the only purpose, introducing these factors would not be of much use, since they take on a different value for each adsorbate-adsorbent system and cannot be expressed in terms of any readily available property characterizing the system. The coalescing factors, however, prove to be very useful for the prediction of gas-mixture adsorption equilibria. Prediction of Gas-Mixture Adsorption Equilibria G r a n t a n d Manes Method. In their original paper, Grant and Manes (1966) presented the following equation for predicting the phase diagram of a binary adsorbate system on activated carbon:
Here pi is a correlating divisor of the adsorption potential, called the "affinity coefficient". Grant and Manes derived eq 3 by assuming that the Lewis and Randall rule was valid for calculating fugacities in the adsorbed phase and by equating the adsorption potentials of the two pure components at equal adsorbed volumes. They also assumed that a single potential curve
Ind. Eng. Chem. Fundam., Vol. 24, No. 3. 1985 327
Table I. Coalescing Factors ( k o Jof the Potential Curves for Different Adsorbate-Adsorbent Systems adsorbent adsorbate i koi data sources activated methanea 1.00 Costa et al., 1981 carbon ethane 0.95 AC-40 ethylene 1.00 propylene 1.06 activated methane' 1.00 Laukhuf and Plank, 1969; carbon ethane 1.00 Wilson, 1980 BPL propylene 1.07 acetylene 1.14 carbon dioxide 1.00 carbon monoxide 1.13 activated ethane' 1.00 Jelinek, 1953 carbon propane 1.00 Columbia ethylene 1.00 G propylene 1.00 carbon dioxide 0.96 activated propane' 1.00 Lewis et al., 1950d carbon propylene 1.12 GLC activated methane' 1.00 Szepesy and Illes, 1963a,b carbon ethane 1.00 Nuxit-AL propane 1.00 n-butane 0.90 ethylene 1.10 propylene 1.19 acetylene 1.19 carbon dioxide 1.19 activated ethylene" 1.00 Lewis et al., 1950e carbon acetylene 1.00 PCC molecular oxygen' 1.00 Danner and Wenzel, 1969 nitrogen 1.34 sieve 5A carbon monoxide 1.86 molecular oxygen' 1.00 Danner and Wenzel, 1969 sieve 1OX nitrogen 1.21 Nolan et al., 1981 carbon monoxide 1.54 molecular ethane' 1.00 Danner and Choi, 1978; sieve 13X ethylene 1.62 Hyun, 1980 isobutane 0.90 carbon dioxide 2.19 silica gel ethane' 1.00 Jelinek, 1953; Lewis et al., 1.00 1950a,d,e propane ethylene 1.62 propylene 1.62 acetylene 2.32 carbon dioxide 1.78
'Selected as standard adsorbate. would result by applying their previous method (Grant and Manes, 1964) to both pure-component isotherms. That is, in general, they used the following relation (4) where Vobi is the saturated liquid molar volume of pure i a t the normal boiling point. They point out that eq 3 is more general and can be applied to mixtures whose pure-gas isotherms cannot be collapsed to a single correlation curve by the use of Vobias the correlating divisor. Greenbank and Manes (1981) also discuss an abscissa scale factor to b&used in coalescing liquid adsorption isotherms. Hence, for the given adsorption conditions, T, P,, and yi,eq 3 is solved directly for x l , which is the only unknown in the equation. The value of x 2 is calculated by using x1
+ x2 = 1
(5)
For each component i, a pseudoadsorbate pressure, Pi,, is then calculated by means of the following definition:
Pim = Pdi/Xi
(6)
The value of Pi,is used to determine volume of Voa/ for each component, where Vos/ is the saturated liquid molar volume of the pure component i at a pressure equal to Pi,. To calculate the total number of moles adsorbed, either of the equal quantities of eq 3 is used as an abscissa and the corresponding ordinate, V,, is read off the purecomponent potential curve. Then the following equation is used to calculate the total moles adsorbed per unit mass of adsorbent, N,: T7
(7)
This completes the prediction of the mixture adsorption equilibrium by the Grant and Manes method. In practice, the method discussed above fares poorly when applied to the various adsorbate-adsorbent systems encountered in industry. The method was evaluated using the data bank mentioned before, and the results are presented by Mehta (1982). For activated carbon, the predictions are reasonably accurate for binary-gas adsorption, but even on carbons the deviations are rather large in a number of cases (e.g., n-butane-propane adsorption on activated carbon Nuxit-AL). It is also not possible to apply the method to adsorption on polar adsorbents like molecular sieves and silica gel since, for these adsorbents, the potential curves of the pure components do not generally coincide. To summarize, the Grant and Manes method, although useful for providing quick estimates for binary adsorption equilibria on activated carbon, is not reliable enough to be a general predictor for the equilibria of various adsorbate systems on adsorbents of common interests. Proposed Method. For the reasons mentioned above, the Grant and Manes method for the prediction of binary-gas adsorption equilibria needs to be modified to make it more general and to improve its accuracy. A number of observations based on the considerations of the previous section suggested that eq 3 should be written as
where MI2is an interaction parameter for the mixed adsorbate-adsorbent system. In eq 8, Voa/ is still the saturated liquid volume of the pure component i a t the pressure Pi, defined by eq 6. The interaction parameter Mlz should be related to the coalescing factors kol and ko2,which also depend on adsorbate-adsorbent interactions. A detailed investigation was carried out to determine this relation for different structural types of adsorbents. The following conclusions were reached. For nonpolar adsorbents like activated carbon, with minimal adsorbate-adsorbent interactions, the interaction parameter was found to be equal to the ratio of the coalescing factors: MI2
= k01/k02 (nonpolar adsorbents)
(9)
For polar adsorbents such as molecular sieves and silica gel, however, the interaction parameters are stronger functions of the coalescing fadors than eq 9 would indicate. Empirically, a simple relation was determined, which may be asserted as being valid for polar adsorbents in general: MI2 = [ko1/ko2]2 (polar adsorbents)
(10)
Only for the case of nonpolar adsorbents do the interaction
328
Ind.
Eng. Chem.
Fundam., Vol. 24, No.
3, 1985
IO---
/
081-
/
08
/ 06
"
EXPERIMENTAL DATA 0 CH4( I 1- COz( 2 ) -ACTIVAT CARBON B P L 2 9 8 K, 3 4 ATM WILSON ( 1 9 8 0 ) n C&(l)-y.ls(2)-ACTlVATED CARBON AC-40 293K,OIATq COSTA ET A t (1981)
~
ED
'I
0 2 t
-PREDICTED C
Oi
06
?4
1
08
'
4
02
CUPVES S
0 ! ! 40
C&(2) - SILICA GEL 2 9 8 K , I O A T M L E W I S E T A L (1950a) 0 C&( I ) - C&14(2)-MOLECUUq SIEVE I3 X 2 9 8 K , I 4ATM DANNER AND CHOl ( 1 9 7 8 ) 1 0 CzH4 ( I )-
-PREDICTED
CURVES
~
1/1
I
10
02
0
06
04
XI
08
I0
XI
Figure 3. Predicted adsorption phase diagrams on activated carbon.
parameters retain their theoretical significance since, in this caqe, eq 8 can be directly related to the coalescence of the potential curves. In the case of polar adsorbents, eq 10 represents an empirical, but useful, deviation from the potential theory. Equation 8, along with eq 6, can now be used for the prediction of adsorption phase diagrams of binary systems. An iterative method of solution for x1 is required, since the V",,' in eq 8 are functions of the adsorbed-phase mole fraction via eq 6 which defines the pressures at which the values of V",,' are to be determined. After the value of x , (and x 2 , by eq 5) has been determined, the total moles adsorbed can be calculated in the following manner. First, for each component i, by using the computed value of xl, the adsorption potential coordinate E , is calculated:
Figure 4. Predicted adsorption phase diagrams on silica gel and molecular sieve 13X. 25
r
I
I
I
r'----._
-
t
0 IO
x
c
E
0 EXPERIMENTAL DATA
zo
N z - CO- MOLECULAR SIEVE I O X 2 2 7 6 K , I O ATM NOLAN ET A L ( 1 9 8 1 )
5 -
-PREDICTED
0
0
L 02
-
CURVE
J 04
06
08
10
N i TROGEN
The values of El and E2 are then used as abscissas, and the corresponding ordinates Val and V,, are respectively read off the pure-component potential plots. The value of the total moles adsorbed, N,, is then determined by using the equation (12)
This completes the prediction of the adsorption equilibria of the binary system. For a system of n components a t the given adsorption conditions, T, P,, and yi, one would have n - 1 equations of the form of eq 8, along with a mass-balance equation of the form of eq 5. These n equations can be solved iteratively for xl, x 2 , ..., x, by using the definitions of V",,' and Pim(eq 6). Using the calculated value of x1 of each component i, one could determine Eifor that component and hence determine Vaj from the pure-component potential curve. Then an equation similar to eq 1 2 can be used to calculate the total moles adsorbed, N,. Thus, the method presented above can readily be extended to multicomponent systems. Results and Discussion The proposed method, represented by eq 8-12, was applied to the entire set of binary adsorption data in the data bank, and the prediction results were excellent. Complete details of the application of the method to the data bank are presented by Mehta (1982). Some typical phase diagrams obtained by applying the method are
Figure 5. Prediction of total moles adsorbed on molecular sieve 13X.
shown in Figures 3 and 4. These figures show a considerable improvement over the Grant and Manes method in the accuracy of prediction. The proposed method was found to work particularly well in the case of adsorption on activated carbon, as illustrated in Figure 3. In most cases, the average deviation in the calculated adsorbed-phase composition for different values of the gas-phase composition did not exceed 5%. Higher deviations could generally be ascribed to a large scatter in the data. In the case of polar adsorbents like molecular sieves and silica gel, the errors in prediction were only slightly higher. Typical results are shown in Figure 4. Except in a few isolated cases, the average deviation in the adsorbed-phase composition for these two adsorbents was 6-7 90.In view of the general inaccuracies in gas-mixture adsorption measurements, these deviations are not too high. For nearly all the adsorption systems investigated, the value of the total moles adsorbed was predicted to within 10%. A representative case is shown in Figure 5 for adsorption on molecular sieve 1OX. It should be kept in mind, however, that the value of the total moles adsorbed is usually constrained to lie between the adsorption capacities for the pure components at the same temperature and pressure, and thus there is not much variation to be accounted for. Therefore, the degree of success in predicting the total moles adsorbed cannot be used as a sensitive test of a method's ability to predict gas-mixture adsorption equilibria.
Ind. Eng. Chem. Fundam., Vol. 24, No. 3, 1985 329 Table 11. Comparison of the Proposed Potential Theory Method with the Ideal Adsorbed Solution Model (IASM) and the Vacancy Solution Model (VSM) av dev in yl, mol % adsorbent adsorbate system T,K P,, atm IASM VSM this work activated carbon GLC" propane-propylene 298.2 1.0 7.0 7.5 4.3 ethylene-acetylene 298.2 1.0 activated carbon PCC" 3.5 2.6 2.5 molecular sieve 10Xb oxygen-nitrogen 144.3 1.0 3.1 9.6 4.9 oxygen-carbon monoxide 144.3 1.0 18.1 25.8 2.9 nitrogen-carbon monoxide 144.3 1.0 3.1 3.3 6.0 molecular sieve 13Xc ethane-ethylene 298.2 1.4 2.1 2.4 3.2 ethane-ethylene 323.2 1.4 1.9 3.9 4.0 silica gel" propane-ethylene 273.2 1.0 1.1 1.8 2.7 ethylene-propylene 273.2 1.0 4.6 3.1 4.7
"Lewis et al., 1950a,d,e. *Danner and Wenzel, 1969. cDanner and Choi, 1978. The proposed method was quantitatively evaluated against two other methods: one based on the ideal adsorbed solution model (IASM), developed by Myers and Prausnitz (1965), and the other based on the vacancy solution model (VSM), developed by Suwanayuen and Danner (1980a,b). These two methods are commonly used for predicting gas-mixture adsorption equilibria from the isotherm data for the pure adsorbates. The comparison of the prediction accuracy of the proposed method vs. the IASM and VSM methods for some representative systems is shown in Table 11. The complete results of the comparative study lead to the conclusion that the accuracy of the proposed method is equal to that of these two methods. There are certain advantages, however, in using the proposed method in preference to the other two methods. The method developed in this work supplies a desk calculation procedure as does the IASM; but unlike the IASM, it does not require the pure-component isotherms a t very high and very low pressures. Furthermore, the proposed method yields directly the adsorbed-phase composition a t a given gas-phase composition, instead of vice versa, as in the case of the VSM method. Also, in contrast to the VSM method, the proposed method does not require a nonlinear regression for its implementation. The proposed potential theory method provides a direct calculation technique for determining the phase equilibrium diag-ram of a given binary adsorption system. All that is needed for the calculation is the ratio of the coalescing factors of the two pure components. The latter may be obtained from a table or may be estimated from data for other similar systems. For adsorption on activated carbon, or for adsorption of hydrocarbons belonging to the same homologous series, the coalescing factors can generally be set equal to unity. The method may be less satisfactory for systems containing both polar gases and polar solids. Unfortunately, there are inadequate data to test this type of system. The greatest advantage of using the potential theory for predicting gas-mixture adsorption equilibria is that the pure-component potential curves are generally independent of the adsorption temperature. Thus, having determined the isotherms of the pure components at a single temperature, one can predict the mixture equilibria at any temperature and pressure. This may be of immense utility in design calculations, where the data available are minimal. In summary, the proposed method is as accurate as any alternative prediction method and offers many practical advantages over the other methods. The method appears to be reliable as a general predictor of gas-mixture adsorption equilibria.
Acknowledgment Financial support from the Division of Refining of the American Petroleum Institute is gratefully acknowledged. Nomenclature Ei = adsorption potential coordinate of component i, defined by eq 11, atm f = fugacity at the adsorption temperature and pressure, atm f, = saturation fugacity at the adsorption temperature, atm ko, = coalescing factor of pure component i Mij = interaction parameter for adsorbates i and j on the given adsorbent N , = moles adsorbed per unit mass of adsorbent, g-mol/g N , = total moles adsorbed from gas mixture per unit mass of adsorbent, g-mol/g P = adsorption pressure, atm Pi,,, = pseudoadsorbate pressure, defined by eq 6, atm P , = total pressure of adsorption from gas mixture, atm R = universal gas constant, cm3 atm/(g-mol K) T = adsorption temperature, K V , = ordinate of potential curve of pure component i corresponding to an abscissa equal to Ei, cm3/g V , = ordinate of pure-component potential curve corresponding to an abscissa equal to either of the equal quantities of eq 3, cm3/g Vb = saturated liquid molar volume at the normal boiling point, cm3/g-mol V, = saturated liquid molar volume at a pressure equal to the adsorption pressure, cm3/g-mol V0G' = saturated liquid molar volume at a pressure equal to Pim,cm3/g-mol x = adsorbed-phase mole fraction y = gas-phase mole fraction Subscripts a = adsorbed state b = normal boiling point i = component i
m = mixture s = saturated state 1, 2 = components 1 and 2, respectively Superscript O = pure component Greek Letter p = affinity coefficient Literature Cited Bering. B. P.; SerDlnskll. V. V.: Surlnova. S. I . Dokl. Phvs. Chem. (End. Tr'insl.) 1963, '753,949. Cook, W. H.; Basmadjlan, D. Can. J. Chem. Eng. 1964, 42, 146. costa, E.; Sotelo, J. L.; Calleja. 0.; Marron, C. AIChE J. 1981, 2 7 , 5. Danner. R. P.: Choi. E. C. F. I d . Eno. Chem. Fundam. 1978. 77. 240. Danner; R. P.;'Wenzel, L. A. AIChE J.-l989, 75, 515. Dublnin, M. M.; Tlmofeyev, D. P. C . R . (Dokl.) Acad. Sci. URSS 1946, 5 4 , 701.
Fernbacher, J. M.; Wenzel, L. A. Ind. Eng. Chem. Fundam. 1972, 1 7 , 457. Grant, R. J.; Manes, M. I d . f n g . Chem. Fundam. 1964, 3 , 221. Grant, R. J.; Manes, M. Ind. Eng. Chem. Fundam. 1966, 5 , 490. Greenbank, M.; Manes, M. J. Phys. Chem. 1981, 85,3050. Hansen, R. S.; Fackler, W. V. J . Phys. Chem. 1953, 5 7 , 634.
Ind. Eng. Chem. Fundam. 1985, 2 4 , 330-339
330
Hyun, S.H. M.S. Thesis, The Pennsylvania State University, University Park, PA, 1980. Jeiinek, R. J. Ph.D. Thesis, Columbia University, New York, 1953. Laukhuf, W. L. S.; Plank, C. A. J. Chem. Eng. Data 1969, 14. 48. Lee, B. I.; Kesler, M. G. AIChE J . 1975, 2 7 , 510. Lewis, W. K.: Gilliland, E. R.; Chertow. 8.:Bareis. D. J. Am. Chem. Soc. 1950a. 72, 1160. Lewis, W. K.; Gilliland. E. R.; Chertow, B.;Cadogan, W. P. Ind. Eng. Chem. 195Ob. 4 2 , 1319. Lewis, W. K.; Gilliland, E. R.; Chertow, 8.;Cadogan, W. P. Ind. Eng. Chem. IQJOC, 4 2 , 1326. Lewis, W. K.; Gilliiand, E. R.; Chertow, 8.:Hoffman, W. H. J. Am. Chem. SOC.1950d, 72, 1153. Lewis, W. K.; Gilliiand, E. R.; Chertow, 5.; Milliken, W. J. Am. Chem. SOC. I95Oe, 7 2 , 1157. Maslan, F. D.; Altman, M.: Aberth, E. R. J. Phys. Chem. 1953, 5 7 , 106. Mehta. S. D. M.S. Thesis, The Pennsylvania State University. University Park. PA, 1982
Myers, A. L.; hausnitz, J. M. AIChEJ. 1965, 1 1 , 121. Nolan. J. T.; McKwhan. T. W.; Danner, R. P. J. Chem. Eng. Data 1081, 26,
112. Polanyi, M. Verh. Dtsch. Phys. Ges. 1914, 76, 1012. Sircar. S.; Myers. A. L. Chem. Eng. Sci. 1973, 28, 489. Spencer, C. F.; Danner, R. P. J . Chem. Eng. Data 1972, 77, 236. Suwanayuen, S.:Danner, R. P. AICh€ J . 1960a, 2 6 , 68. Suwanayuen, S.: Danner, R. P. AIChE J . 1980b, 26, 76. Szepesy, L.; Illes, V. Acta Chlm. Hung. 1963a, 35, 37. Szepesy, L.; Illes, V. Acta Chlm. Hung. 1963b, 35, 53. Wilson, R. J. M.S. Thesis, The Pennsylvania State University, University Park, PA. 1980.
Received for review September 12, 1983 Accepted January 23, 1985
Predicting Liquid-Vapor Equilibria by a Model of the Interaction Equilibrium in Restrained Molecular Systemst Franclsc A. Gothard Institute for Research. Des@ and Development for Oil Refineries, Ploiepti, Romania
The model of the equilibrium of interactions in molecular systems restrained to the nearest neighbors, proposed by the author for predicting Gibbs excess energy in multicomponent nonelectrolyte mixtures, is considered for correlating binary and predicting ternary liquid-vapor equilibrium data. The performances of the equations derived from the new model in this field are analyzed comparatively with those of the well-known equations given by Wilson and by Renon and Prausnitz. Significantly smaller deviations were obtained according to the Fischer test for most of the 168 binary and 6 ternary data sets by use of the equations derived from the new model.
The Equations for Gibbs Excess Energy and Activity Coefficients according to the New Model In the attempt to overcome the limitations and some consequences of empirical approaches in the development of equations based on the local concentration concept, as discussed by Gothard (1975), Flemr (1976),Kem6ny (1979), and Wilkinson (1979), a simple model was proposed (Gothard, 1980), starting from the mass action law, considered also in earlier approaches by Dolezalek (1908,1910) and Becker et al. (1972, 1980) for nonelectrolyte solutions. The new model considers new parameters, the number of neighbor molecules whose equivalent interactions are effective for the excess property predicted, and leads for the dimensionless Gibbs excess energy to the correlation 1 n ( n - 1) g e / R T = - In -___ K 2 where 2
For binary eq 2 becomes (4)
and eq 3 reduces to 1 n(n - 1) In y1 = - In -K 2
xi xnijx,
- = -ji
J
and K = 4 for systems without miscibility gaps. A c-rivation of the correlation from the theoretical assumptions of the model is given in the Appendix as an alternative for considering eq I and 2 as empirical ones. Using the well-known correlation given by van Ness (1964)for the activity coefficients one obtains starting from eq 1 and 2 Lecture at the Vth Conference for Mixtures of Nonelectrolytes and Intermolecular Interactions, April 18-22, 1983, Halle, GDR.
x,n(2n - 1) I
The model requires two adjustable parameters ni, for binary systems. No additional parameters are required for predicting multicomponent data. For multicomponent systems exhibiting miscibility gaps, K if required as third parameter for the binary subsystems, is the same for all binary subsystems.
0196-4313/85/1024-0330$01.50/00 1985 American Chemical Society
