i39
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 139-142
pointed out in the earlier work of Wang and Tien (1982). The use of a single Freundlich expression for the single species isotherm data of the individual adsorbate in the present work was merely for the purpose of convenience and clarity in presenting the basic idea of species grouping. Acknowledgment
2 k, n 9 qT S
U
z ti
a %
i in S
Nomenclature aP C
Pb PP
defined by eq 10 bulk density of adsorbent density of adsorbent particles spreading pressure corrected time
Subscripts
This study was performed under Grant CPE 83 09508, National Science Foundation. A
Greek Letters
4i
Freundlich coefficient radius of adsorbent particles adsorbate concentration in the solution phase total adsorbate concentration in the solution phase liquid-phase mass-transfer coefficient particle-phase mass-transfer coefficient reciprocal of Freundlich exponent adsorbate concentration in the adsorbed phase total adsorbate concentration in the adsorbed phase quantity defined by eq 9 superficial velocity axial distance mole fraction of the ith adsorbate in the adsorbed phase
ith adsorbate inlet condition adsorbent-solution interface
Superscript 0
single-speciesstate
L i t e r a t u r e Cited Calllgarls, Mary Beth: Tien, C. Can. J. Chem. Eng. 1982, 6 0 , 772. Dobbs, R. A.; Cohen, J. M. EPA-600/8-80-023, 1980. Hsieh, J. S. C.; Turian, R. M.; Tien, C. AIChE J. 1977, 23, 263. Larson, A. C.; Tien, C. Chem. Eng. Commun. 1984, 2 7 , 339. Mehrotra, A. K.; Tien, C. Can. J. Chem. Eng. 1984, 6 4 , 632. Ramaswaml, S. M. S. Thesis, Syracuse University, Syracuse, NY, 1984. Vermuelen, T.; Klein, G.; Helster, N. K. I n “Chemical Engineers Handbook”, 5th ed.;McGraw-HIII: New York, 1973; p 1. Wang, S . G . : Tien, C. AIChE J. 1982, 2 8 , 565.
Received for review October 15,1984 Accepted May 31, 1985
Group Contribution Method To Predict Critical Temperature and Pressure of Hydrocarbons Joseph W. Jalowka and Thomas E. Daubed’ Department of Chemical Englneering, The Pennsylvanle State University, University Park, Pennsylvania 16802
A group contribution model was developed to predict critical pressures and critical temperatures of hydrocarbons using second-order, Benson-type groups. The critical temperature model utilizes the normal boiling point and the groups present In the molecule as parameters. When compared to established methods published by Lydersen and Ambrose, the model produces more accurate results for all families of hydrocarbons except alkanes. The critical pressure model uses the normal boiling point, the critical temperature, and the groups present in the molecule as parameters. The results again are more accurate than the Ambrose or Lydersen models using either an experimental critical temperature or a critical temperature estimated by using the model developed in this study.
Critical properties are important parameters in many calculations involving phase equilibria and thermal properties. These properties are difficult to measure experimentally since equipment capable of rapidly and accurately producing and maintaining high temperatures and pressure is required. Also, for those compounds which contain high numbers of single bonds, the problem of thermal decomposition is a major obstacle to measurement of critical properties. Thus, prediction methods are important and in many cases are the only means by which these properties can be determined. Some of the most successful prediction methods employ group contribution models requiring the structure of the molecule to estimate critical properties. Lydersen’s (1955) model to predict critical temperature is given by Tb
- = 0.567
+ (CAT)- (CAT)2
(1)
TC where Tb = normal boiling point in kelvins, T,= critical temperature in kelvins, and CAT= summation of group
0196-4305/86/1125-0139$01.50/0
increments using T,and Tb in the specified units. Lydersen also developed a similar model to estimate the critical pressure by developing an equation which has the form
(y)”’ + = 0.34
CAP
where MW = molecular weight, P, = critical pressure in atmospheres, and CAP= summation of group increments. Both models work extremely well for a wide variety of organic compounds. Another method which can be generalized to a wide range of organic compounds and produces accurate results was developed by Ambrose (Ambrose et al., l974,1978a,b; Ambrose, 1978). The major difference between the Ambrose and Lydersen model is the inclusion of the Platt number. The Platt number is the number of carbon atoms three bonds apart and is an indicator of the degree of branching in the molecule (e.g., the Platt number of an n-alkane is equal to the number of carbon atoms minus three). Ambrose developed equations of the following form 0 1985 American Chemical Society
140
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
for estimating both critical temperature and pressure. for critical temperature
T, =
Tb
[
1+
1
y
+ CAT- 0.023A(Platt no.)
]
Table I. Critical Temperature Models Analyzed" model %AFD AAD.K 2.50 14.45 Tc = En14 1.50 7.84 T , = al + Cn,A, Tc = ai + Zn,A, + ~ 5 ( I 3 n , A , ) ~ 1.48 7.70 T , = a1 QZTb+ xn,Az 0.45 2.53 0.43 2.56 T , = QST, + x n , A , TC= QSTb+ ( x n L A z ) Q5(CnLAL)' 0.44 2.57 0.53 3.03 Th/Tc = a1 + ~ ~ L A L l'b/Tc = ai + + as(In,A,)* 0.54 3.12 Tc = Tt,/(Qi + x n l A , ) 0.56 3.17 Tc = Tb/[a,+ x n , S , + as(zt~,S,)'] 0.57 3.18 (Tb/Tc)*= a, + zn,A, 0.50 2.88 T b / ( T d T J Z= a, + I n A 0.49 2.95
~
(3)
where y = 1.242 for all compounds except perfluorocarbons, and monohydrogen-substituted perfluorocarbons, T, = critical temperature in Rankine, Tb = normal boiling point in Rankine, CAT= summation of temperature increments, and A(P1att no.) = difference in the Platt number of any alkyl chains from the Platt number of the nalkane with the same number of carbon atoms for critical pressure 14.7MW P, = (4) (0.339 + CAP- 0.026A(Platt no.))' where P, = critical pressure in psia, MW = molecular weight, and CAP= summation of pressure increments. The models developed by Lydersen and Ambrose are quite accurate despite their generalized nature. The major limitations of these methods are (1)the nearest-neighbor effects present are neglected, (2) isomerization is not taken into account, and (3) only a few inorganic compounds are included. Many other group contribution methods are specific to certain families of organic compounds. The purpose of this study was to determine whether including the next nearest-neighbor effects can improve on the critical properties predicted by using first-order groups. The groups utilized in this study were formulated by Benson and Buss (1958) and Benson (1976) for the prediction of ideal-gas heat capacity. The carbon atoms are categorized as follows: C = any single bonded carbon, Cd = any double bonded carbon, C, = any triple bonded carbon, Cb = a benzene carbon (any carbon atom which is a member of an aromatic ring), and C, = an allenic carbon (carbon atom with two double bonds =C=). The groups are defined by using the following notation: The central carbon atom is listed first, followed by a bond which indicates that the central carbon atom is bonded to the following ligands. All monovalent ligands are then listed followed by any other polyvalent ligands. Parentheses are placed around the ligands not bonded to one another. A numerical subscript indicates how many of each ligand is attached to the central carbon atom (e.g., C-(C)dH)(O)). In this formulation, any carbon atom appearing in an aliphatic ring is considered equivalent to a nonring carbon. However, a ring-correction group is included to correct for possible stress-strain effects in such molecules. A ciscorrection group is introduced to take care of isomerization in alkene compounds. For aromatic compounds, an ortho correction is included, and the formulation assumes that effects from meta or para isomers are negligible. Development of Model In this study (Jalowka, 1984), a data base of compounds for which experimental critical temperatures and pressures are available was obtained from the American Petroleum Institute Data Book research group at Penn State. The data base contained 186 hydrocarbons for which experimental critical temperatures are available and 178 hydrocarbons for which experimental critical pressures are known. A generalized group contribution equation of the form shown below was used to develop the second-order group contribution model. Q = a1 + UzMW + U 3 T b + U ~ C ? Z ~+Aa5(ZniAi)' ~ (5)
"AD = absolute deviation = ]experimental value - predicted value(. AAD = average absolute deviation = ZAD/number of points of data. % AFD = average % absolute deviation = 100 x X(AD/experimental value)/number of points of data.
Table 11. Comoarison of Critical TemDerature Models no. of method % AFD AAD,K compds 0.71 4.20 Ambrose et al., 1974, 1978a,b 187 0.82 5.05 187 Lydersen, 1955 0.77 2.64 187 Soulie and Rey, 1980 Twu,1983 0.95 5.51 187 Nokay, 1959 0.84 2.80 187 Fishtine, 1980 1.06 6.27 187 Forman and Thodos, 1958, 1960 1.61 9.98 187
where Q = property, MW = molecular weight, Tb= normal boiling point, Ai = contribution of the ith group, n, = number of occurrences of the ith group, and a,-a, = regression constants. Two problems were encountered in attempting to regress group contribution values. First, colinearity problems developed with the following four groups: C-(H),(C) C-(H),(Cd), C-(H13(Ct),and C-(H)3(Cb). To circumvent this problem, these groups were combined into one group, as previously done by Benson (1976). Second, colinearity problems were encountered when attempting to regress group values for smaller segments of the data base. This occurred because the contribution values of similar groups, such as the four shown above, were functions of one another and could not be separated reliabily. Thus, all groups were regressed together. Since only 13 groups appear more than 10 times, it was concluded that the entire data base could be regressed together without affecting the statistical significance of the group values obtained. In this study, a2 was set equal to 0 and a4 was set equal to 1when regressing critical temperature increments, and a3 was set equal to 0 when regressing critical pressure increments. For all the models considered, the linear form (a, = 0) was considered first, and the group increments generated were used as initial guesses in the nonlinear model. The first model analyzed was the simplest possible, setting al = a2 = u3 = 0 and Q = T,; thus, T, = CniAi. Values for the A's for each model shown in Table I were derived by using a linear regression routine. Statistical summaries are also shown in Table I. Including the regression constant a in the model improved the results, but when compared to other methods as reviewed by Elliott and Daubert (1984)and shown in Table 11,the results were not as accurate. It should be noted, however, that the simple model of Table I involving only al and increments would be quite useful if a normal boiling point was unavailable. As adding a nonlinear term gave only slight improvement, it was concluded that the groups alone could not predict the critical temperature. The normal boiling point was then included as a parameter. This was accomplished in two different ways: by
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
Table 111. Critical Pressure Models Analyzed method
P, = EniA, P, = a, + X.n,A,
(MW/P,)1/2 = a,
+ E:n,A,
P, = MW/(al + X n , A J 2 P, = T2/Tb2(al + X?t,A,)a
P, = T2/TbZ(al+ Cn,Ai)*
% AFD
AAD,MPa
no. of compds
6.57 3.50 2.73 2.91 1.95 2.30
0.205 0.103 0.081 0.082 0.058 0.065
178 178 178 178 178 178
a With experimental critical temperature known. predicted from the model developed in this study.
changing the objective function to Tb/Tcor by setting u3 = 1 and keeping T, as the objective function. As Table I illustrates, the inclusion of the normal boiling point significantly improved the results. Using Tb on the right-hand side of the equation gave the best results and also gave group values which had the lowest standard errors. The linear model given by
T, = a,
+ U3Tb + Enid;
Table V. Group I~CrementBUsed in the Prediction of Critical TemDeratures of Hydrocarbons
AT
groups
19.953 19.585 3.030 17.481 8.450 5.340 25.078 11.407
Using T ,
Table IV. Comparison of Critical Pressure Models AAD, no. of method % AFD MPa compds Ambrose et al., 1974, 1978a,b 2.23 0.071 178 Lydersen, 1955 3.65 0.108 178 Fishtine, 1980 4.49 0.149 178 Twu, 1983 4.78 0.164 178 Forman and Thodos, 1958, 1960 4.30 0.125 178
(6)
gave the best results, but since the constant ul was shown to be statistically insignificant, it was discarded in favor of (7)
A similar procedure was followed for the development of a critical pressure model. The same general equation was considered; however, u3 was set equal to 0, and u4 was set equal to 1. The simplest linear model was considered first, by setting a, = u2 = u5 = 0; thus, P, = CniAi. The models and statistical results are shown in Table 111. When compared to other established methods, shown in Table IV and evaluated by Elliott and Daubert (1984),the simple additivity model was shown to be insufficient for accurate predictions. The addition of a regression constant, a,, in the model improved the results, but it became apparent that the groups alone are not sufficient to correlate critical pressure. The objective function was changed to (MW/PC)lI2, which improved the results marginally. A model derived by Soulie and Rey (1980) which allows calculation of the critical pressure of a compound using the critical temperature, normal boiling point, and the group increments as parameters was considered. The equation is developed in the following manner: the difference between the internal energy of a real gas and that of an ideal gas is given by
where AU = difference in internal energy, P = pressure, T = temperature, and dB/dT = derivative of the second virial coefficient with respect to temperature. An empirical expression which gives the second virial coefficient as a function of the reduced temperature and critical pressure is then substituted into the above equation. The AU is expressed as the energetic contribution of the atoms, AU = f(ai),where the ai’s are increments attributed to the atoms or group of atoms of which the
141
cyclopentene cyclohexane cyclobutane cyc1opentane cyclohexane
groups
AT
Ci-C Cb-H
13.022 12.962 24.394 cb-c 17.699 Cb-Cb Cb (fused ring carbon) 17.816 15.235 C, -2.444 cis-alkene ortho-substitution -6.988
Ring Corrections -29.593 cycloheptane -33.978 cyclooctane -23.757 cyclononane -22.000 cyclodecane -30.810
-30.053 -28.406 -8.326 -8.041
Table VI. Group Increments Used in the Prediction of Critical Pressures of Hydrocarbons groups C-(H)dC), C-(H),(Cb), c-(H)3(cd), C-(H)a(Ct) C-(H)z(C)z C-(H)(C)3 C-(C), C-(H)z(Cd)(C) C-(H)(Cd)(C)z C-(H)dCb)(C)
c-(cb)(c)3
C-(H)dCb)z C-(H) (Cb)(c)z Cd-(H)z cyclobutane cyclopentene cyclohexane cycloheptane
AP groups 65.441 cd-(H)(c) Cd-(c)Z 47.049 cd-(H)(cd) 28.004 Cd-(H)(Cb) 5.911 Ct-H 52.223 Ct-C 30.689 Cb-H 57.975 cb-c 25.056 Cb-Cb 20.585 Cb (fused ring carbon) 41.175 cis-alkene 56.334 ortho-substitution
Ring Corrections 7.598 cyclooctane 10.524 cyclononane 21.686 cyclodecane -9.650
AP 40.880 28.953 45.036 42.041 44.949 31.557 40.938 25.831 2.685 29.085 -0.633 -4.078 -8.466 -53.288 -70.872
organic molecule is constructed. By combining the two equations, the following model is developed:
P, =
T,3 T b 2 (ai 4- CniAp)
(9)
where P, = critical pressure in megapascals, T, = critical temperature in kelvins, Tb = normal boiling point in kelvins, a, = regression constant in kelvins/megapascal, and Ap = group increment in kelvins/megapascal. When this model was applied to the critical pressure data base, the results obtained were superior to other established methods. One apparent liability would appear to be the inclusion of the critical temperature as a parameter. Therefore, the next step was to apply the model by using the critical temperature predicted in this work. The results obtained were comparable to the Ambrose model but more accurate than those obtained from other methods. Since either an experimental or predicted value of the critical temperature could be used in the model, two sets of group increments were generated. One set was regressed by using experimental critical temperature values while the other was regressed by using predicted critical temperature values. The results obtained by using the predicted increments and experimental critical temperatures were comparable to using both experimental increments with experimental critical temperatures. The same was
142
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1980
Table VII. Percent Error in the Prediction of Critical Temwrature of Pure ComDonents for Different Homoloaous Series alkanes cycloalkanes alkenes cycloalkenes dienes alkynes aromatics overall 0.769 0.379 this study 0.299 0.393 0.483 (0.0) 0.723 (o.o)= 0.463 0.177 1.493 (0.321) (0.379) Ambrose 2.863 0.964 0.713 0.523 1.173 0.781 (0.497) Lydersen (0.603) 3.671 0.958 0.820 ( ) = fewer than five data points available.
Table VIII. Percent Error in the Prediction of Critical Pressure of Pure Components for Different Homologous Series alkanes cycloalkanes alkenes cycloalkenes dienes alkynes aromatics overall this study 1.367 2.731 0.638 1.954 1.951 an (0.0) (0.O)C 3.982 1.291 2.300 bb 1.871 (0.0) 1.800 (0.0) (5.387) 3.489 2.233 3.930 2.069 (1.169) Ambrose 0.759 1.834 (1.224) (4.158) 4.535 3.650 3.375 3.524 Lydersen a Usine the exDerimenta1 critical temDerature in the model. than five data points available.
Using the predicted critical temperature obtained in this work.
true when using experimental increments with a predicted critical temperature. Thus, it was decided to use the experimental increments for all cases. Results Using the experimental critical temperature data base which contained 186 hydrocarbons, optimal values for 34 second-order groups and ring corrections were obtained. The model utilizes the normal boiling point and the groups present in the molecules as parameters. The model selected has the functional form where T, = critical temperature in kelvins, Tb= normal boiling point in kelvins, a, = 1.806, dimensionless regression constant, n, = number of occurrences of group i, and C A T = sum of temperature group increments in kelvins. The values for the group increments are given in Table V. An experimental critical pressure data base which contained 178 hydrocarbons was utilized to develop a model to predict critical pressure. All the groups were regressed together, and the model selected has the functional form
P, =
Tc3 Tb2 (ai iCn,Ad
(11)
where P, = critical pressure in megapascals, Tb = normal boiling point in kelvins, T, = critical temperature in kelvins, al = 43.387 kelvins/megapascal, n, = number of occurrences of group i, and CAP= summation of group increments in kelvins/megapascal. Values for the group increments are given in Table VI. Comparison of Results The models developed in this work were compared for various homologous series of hydrocarbons with the Lydersen and Ambrose models. LeMieux (1983) showed that regressing Lydersen’s groups using an expanded hydrocarbon data base gave an improvement of only 0.03% (0.3 K) for critical temperature prediction for hydrocarbons although improvements were more substantial for nonhydrocarbons. Thus, Lydersen’s original increments were used in this analysis. Also, the data base used by Ambrose in the mid-1970s is essentially the same as the one used in this study, allowing the original Ambrose increments to be used for this comparison.
( ) = fewer
Table VI1 compares three methods for the prediction of critical temperature by corresponding family. The model developed in this study produced superior results, especially for aromatic compounds. The only exception is alkanes, which are predicted slightly more accurately by the Ambrose model. No quantitative comparisons can be made for cycloalkene or diene families as so few experimental values exist for these compounds. A similar comparison is made for the critical pressure model and is shown in Table VIII. Once again, the Amb r a e model is slightly more accurate for predicting critical pressures of alkanes. All three methods produce equivalent results for cycloalkanes, but the proposed method produces more accurate results for alkene and aromatic compounds. In s “ y , the models developed can be applied to any class of hydrocarbons and produce equal, and in most cases superior, results to established methods. The increase in the number of groups enables us to predict critical properties with increased accuracy, but the increase in the groups is not so large (e.g., only an addition of 19 groups over Lydersen’s method for critical temperature prediction) that the calculation process becomes tedious. Literature Cited Ambrose, D.; Broderick, B. E.; Townsend. R. Appl. Chem. 6i0techn0l. 1974, 24, 359. Ambrose. D. Natlonal Physics Lab ReDort Chemical No. 92, Middlesex, England, Sept 1978. Ambrose, D., Sprake, C. H. S.; Townsend, R. J. Chem. Thermodyn. 1978, 6 , 693. Ambrose, D., Counseil, J. F.; Laurenson. K. J.; Lewis, G. B. J. Chem. Thermodyn. 1978, 10, 1033. Banson, S. W.; Buss. J. H. J. Chem. Phys. 1958, 29, 546. Benson. S.W. “lhermochemlcalKinetics”, 2nd ed.;Wiley: New York, 1976. Elliott, J. R.; Daubert. T. E. The Pennsylvania State University, private Communlcatlon, 20 April 1984. Fishtine. S. H. 2.phvs. Chem. New Fo&e 1980, 123, 39. Forman, J. C.; Thodos, G. A I C E J. 1958, 4 , 358. Forman, J. C.; Thodos, G. AIChE J. 1950, 6 , 206. Jalowka, J. W. M. S. Thesis, The Pennsylvania State University, University Park, 1984. LeMieux, M. A. M. S. Thesis, The Pennsylvania State University, University Park, 1983. Lydersen, A. L. “Estimation of Critical Properties of Organic Compounds”; University of Wisconsin: Madison, 1955. Nokay, R. Chem. Eng. 1959, 66(4), 147. Soulie. M. A.; Rey, J. Bun. Soc. Chim. Fr. 1980, 3 - 4 , 1-117. Twu, C. H. FiuM Phase EquHb. 1983, 1 1 , 65.
Received for review January 8, 1985 Revised manuscript received May 28, 1985 Accepted June 9, 1985
