Ind. Eng. Chem. Res. 1991,30, 1612-1617
1612
Table IV. Summary of Regmrrion of Observed Vapor Pmrrure VI Predicted from Ear 12 and 16 eq R' slope av error MacKay (eq 12) 0.99 1.21 (*0.008) 0.47 (10.27) this work (eq 16) 0.99 1.04 (*0.008) 0.18 (10.15)
It is believed that the improved estimates of ASB,ASM, ACm,and ACm will lead to a better understanding of the vaporization process and to better estimates of vapor
of the slope is closer to the theoretical value of unit for eq 16 compared to 1.2 for MacKay's equation. The prediction is improved for each of the 72 compounds. The mean of the absolute error, lobserved minus predictedl, is reduced to 0.18 log units compared to 0.47 log units for eq 12. It should be noted that, unlike empirical equations described above, eq 16 uses no adjustable parameters. Furthermore, all numerical coefficients are obtained from non vapor pressure data. Although eqs 13-15 are at present only approximate, they clearly show that symmetry and flexibility are important factors in the evaluation of the parameters that govern vapor pressure, These equations will be finalized as more reliable data are acquired.
Literature Cited
pressure for complex molecules.
Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Gases; Wiley: New York, 1968. Mackay, D.;Bobra, A.; Chan, D. W.; Shiu, W . Y. Environ. Sci. Technol. 1982,16,645-649. Mishra, D.S.;Yalkowsky, S. H.Znd. Eng. Chem. Res. 199Oa, in press. Mishra, D.S.; Yalkowsky, S . H. Chemosphere 1990b, 21,111-117. Shaw, R. J . Chem. Eng. Data 1969,14,461-465. Tsakaniias, P.; Yalkowsky, S.H.Toxicol. Enuiron. Chem. 1988,17, 19. Yalkowsky, S.H.;Mishra, D.S.; Morris, K. Chemosphere 1990,21, 107-110. Received for review May 14, 1990 Accepted November 26,1990
Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified Peng-Robinson Equation of State. 3. Thermal Properties. A New Significant Characterizing Parameter m Marek &galski* Laboratoire de Thermodynamique Chimique et Appliquie, ZNPL-ENSZC, 54001 Nancy Cedex, France
Bruno Carrier and Andr6 Pheloux Laboratoire de Chimie-Physique, Faculti des Sciences de Luminy, 13228 Marseille Ceder 9,France
Enthalpies of vaporization and isobaric liquid heat capacities of hydrocarbons were calculated by using a previously developed cubic equation of state. The enthalpy of vaporization predictions were in good agreement with the available experimental data and compared favorably with those obtained by using predictive methods published in the literature. The results obtained with liquid heat capacities were satisfactory in most cases. Characterizing parameters proposed by Carrier et al. were used to develop general correlations for enthalpies of vaporization and vapor pressures of hydrocarbons. On the basis of the results obtained, m can be said to be a significant parameter that can be useful for correlating thermodynamic properties.
Introduction Recently Carriet et al. (1988) proposed a modified version of the Peng-Robinson equation (CRP model) with which it is possible to accurately represent the vapor pressures of hydrocarbons between the triple and the normal boiling point. The encouraging results obtained with vapor pressures led us to study the thermal property predictions yielded by this model. In the present study, predictions of enthalpies of vaporization and isobaric liquid heat capacities obtained with the CRP model were studied and the results were compared with experimental data. In the second part of this paper, a component characterization proposed by Carrier et al. (1988) is considered. According to the CRP model, the Peng-Robinson equation of state (EOS),which is valid in the low-pressure range, can be established for every hydrocarbon for which the normal boiling temperature and the parameter m are known. The physical significance of the parameter m is
discussed with regard to the possibility of using it as an alternative to the acentric factor. Expressions giving the enthalpy and the isobaric heat capacity derived from the CRP model are given in Table I. Full details concerning the formulation and the use of eq T.1 were given in the paper by Carrier et al. (1988).
Enthalpies of Vaporization and Isobaric Liquid Heat Capacities All the experimental enthalpies of vaporization used here were from the recent compilation by Majer and Svoboda (1985),covering data published in the literature up to that date. The selected data base contained the enthalpies of vaporization of 80 hydrocarbons between 260 and Tb+ 20 K. The liquid isobaric heat capacities of 35 hydrocarbons that are below the normal boiling temperature were from TRC tables (1975-1986). Calculations were performed by using eqs T.l-T.12 established with the covolume calcu-
OSsS-SsSS/ 91/ 2630-1612$02.50/0 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1613 Table I. Basic Formulas for Calculating Thermal Propertier T.l P RTI(D - 6) - a/iND + ~6). with Y = 4.82843 T.2 T.3 T.4 T.5 T.6
T.7 T.8 T.9 T.10 T.ll T.12 T.13 T.14 T.15 T.16 T.17
a(T) = as = a [l + m d l - T:/*)12
a, = 0.45724R8T,"/Pc (&/aT)s = -a,psTr-1/2 [ l + m (1 - T?/2)l/Tc Bs = ac(l + ms)[l + ms(1 - T,'b1 = -a,pS(l + ~s)T;'/*/~T,
Table 11. Comparison between Enthalpies of Vaporization and Isobaric Liquid Heat Capacity Predictions and Experimental Data av % dev of O(T)
no. of compounds no. of data m mP For T < Tb + 20 and a(T) Calculated with Equation T.7 80 410 0.66 0.72 AHV 1.78 ACPl 35 258 1.79
For T > Tb + 20 and o ( T ) Calculated with Equation T.13
AHV
ACPl
12 3
46 31
1.11 1.93
lated with a group contribution method. The value of parameter m was either directly determined with experimental vapor pressure data (m)or calculated with a group contribution method (mJ. The predictions and experimentaldata are summarized in Table 11. It can be noted that the predictions obtained with m and mp were very similar. The value obtained with the group contribution method, m , was therefore used in further calculations. The overalr average deviation of about 0.7% (0.28 kJ) obtained with the enthalpies of vaporization of 80 hydrocarbons can be said to be satisfactory and corresponds roughly to the experimental uncertainty level. Experimental data are often subject to considerable dispersion and systematic errors, especially with heavy compounds and at high temperatures (at 298 K, the experimental data on n-dodecane are dispersed to within 2.3 kJ and those on n-hexadecane to within 10.7 kJ). Thus it is very difficult to separate experimental errors from those due to the model. An overall average deviation of about 2% (Table 11)was obtained with the isobaric liquid heat capacities. With many compounds, systematic deviations were observed in the vicinity of 300 K. This can be partly attributed to the errors on the low-temperature ideal gas heat capacities calculated by using a cubic polynomial with parameters reported by Reid et al. (1987). Satisfactory results (about 1%) were usually obtained at higher temperatures. Several exceptions should be noted, however. Considerable deviations were found with benzene, cyclopentane, cyclohexane, and 3,3-diethylpentane. It should be mentioned that the vapor pressures of these compounds were not perfectly represented by the CRP model (Carrier et al., 1988). In the literature, special attention has been paid to the enthalpies of vaporization at 298 K and Tb.In Table 111,
predictions obtained with the CRP model (using m,) at these temperatures are compared with the values recommended by Majer and Svoboda (1985). At 298 K, only small random deviations were observed with most of the compounds studied. The overall average deviation of about 0.5% compares favorably with the deviation obtained by using the group contribution method recently published by Ducros et al. (1980) (1.2%). At the normal boiling temperature the corresponding deviation was slightly higher, i.e. 0.8%. With the same set of data, three correlations recommended by Reid et al. (1987, methods proposed by Riedel, Chen, and Vetere and reported at pp 226-227) were tested. The overall average deviation in AH+,obtained with Riedel's method was 1.2%. The corresponding deviation obtained with the methods by Chen and Vetere was 1.1%. The error distribution obtained with the present method was more homogenous. Vetere's correlations (the other two give similar results) are very accurate with the low-boiling hydrocarbons and only roughly correct with the high-boiling ones. In the same table, the values of the Cpl at 298 K given for 59 hydrocarbons in the compilation by Domalski et al. (1984) are compared with CRP predictions. The overall mean deviation of 2.1% was due to large errors observed with several compounds (see above). This result is comparable to the 1.8% yielded by the Rowlinson-Bondi method, which is recommended by Reid et al. (1987) in the case of nonpolar compounds (with the same set of data and with the same ideal gas heat capacity values). These results confirm the utility of the CRP model for thermal property calculations. The CRP model is not valid at temperatures of more than 20 K above the Tb. In this case it is recommended (Rogalski et al., 1990) that a(T) be established with the Soave type function (Soave, 1972). Corresponding expressions for the enthalpy and the isobaric heat capacity are listed in Table I. High-temperature thermal data are rather scarce. The few enthalpies of vaporization data published by Majer and Svoboda (1985) and isobaric heat capacities given in TRC tables (1975-1986) were compared with predictions obtained by using Soave's model. The results are summarized in Table 11.
.
Parameter m Significance and Applications Both function a(T) in the CRP model and the Soave function can be said to be the one-parameter forms of the expression proposed by Gibbons and Laughton (1984): m = a,[l + ml(l - Tr1I2)+ m2(l - T,)] (1) Soave's function is obtained from eq 1by substitution with ml = ms(ms + 1) -m2 = m1/2 - ms (2) The comparison with the CRP function, eq T.7, is less direct because the latter was defined in terms of the normal boiling and not the critical conditions. It is evident however that this difference does not modify the functional relation of eq 1 in T. The CRP function can be obtained from eq 1, defined in terms of the normal boiling conditions, by taking ml = 7.1562m - 1.9829 -m2 = m1/2 - mS (3) It is interesting to note that the relationship between the parameters m land m2 is the same with both functions studied. Consequently parameters m and ms are given with the same expression defined under the appropriate reference (normalizing) conditions as m = ( N n a)/d(ln T ) ) u , ~ . . ~ b (4) ms = (Wn a)/a(ln T)~,,T..T, (6)
1614 Ind. Eng. Chem. Res., Vol. 30,No.7,1991
Table 111. Comparison between AH,b, AH* and C,,,Predictions and Experimental Data T Tb To comDound AH" SH.,.% AH" SH,,% 2,2-dimethylpropane 2-methylbutane n-pentane cyclopentane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylpentane 3-methylpentane hexane methylcyclopentane 2,2-dimethylpentane benzene 2,4-dimethylpentane cyclohexane 2,2,3-trimethylbutane 3,3-dimethylpentane 2,3-dimethylpentane 2-methylhexane cis-l,3-dimethylcyclopentane 3-ethylpentane heptane methylcyclohexane ethylcyclopentane 2,2-dimethylhexane 2,5-dimethylhexane 2,4-dimethylhexane toluene 3,3-dimethylhexane 2,3,4-trimethylpentane 2,3-dimethylhexane 2-methyl-3-ethylpentane 2-methylheptane 4-methylheptane 3,4-dimethylhexane 3-methyl-3-ethylpentane 3-ethylhexane 3-methylheptane
trans-l,4-dimethylcyclohexane 1,l-dimethylcyclohexane
cis-l,3-dimethylcyclohexane 1-methyl-1-ethylcyclopentane 2,2,4,4-tetramethylpentane trans-1,2-dimethylcyclohexane 2,2,5-trimethylhexane cis-I,4-dimethylcyclohexane trans-1,3-dimethylcyclohexane n-octane isopropylcyclopentane
cis-l,2-dimethylcyciohexane propylcyclopentane ethylcyclohexane ethylbenzene p-xylene m-xylene o-xylene 3,3-diethylpentane nonane isopropylbenzene ieopropylcyclohexane propylcyclohexane propylbenzene 1,3,5-trimethylbenzene tert-butylbenzene 1,2,4-trimethylbenzene isobutylcyclohexane tert-butylcyclohexane isobutylbenzene sec-butylbenzene decane 1,2,3-trimethylbenzene indene butylcyclohexane butylbenzene 2-methyldecane 2-methyldecane
22.74 24.69 5.79 27.30 26.31 27.38 27.79 28.06 28.85 29.08 29.23 30.72 29.55 29.97 28.90 29.62 30.46 30.62 30.40 31.12 31.77 31.27 31.96 32.07 32.54 32.51 33.18 32.31 32.36 33.17 32.93 33.26 33.35 33.24 32.78 33.59 33.66 32.56 32.51 32.91 33.20 32.51 32.96 33.65 33.28 33.39 34.41 33.56 33.47 34.70 34.04 35.57 35.67 35.66 36.24 34.61 36.91
-0.61 -1.42 -0.80 -1.19 -0.82 -0.78 -0).76 -0.87 -0.64 -0.39 -0.12 -1.31 -0.32 -1.05 -1.07 -0.38 -0.41 -0.67 -0.65 -0.14 -0.38 -0.51 -0.52 0.46 0.40 0.24 -1.35 -0.41 -0.78 0.28 -0.41 -0.79 -0.53 -0.15 -0.13 -0.05 0.02 -0.66 -0.61 0.21 -0.12 -0.43 -0.63 0.89 0.08 0.36 -0.53 -1.18 -1.00 0.09 -0.90 -0.88 -1.32 -1.59 -1.48 -1.55 -0.73
37.01
-1.71
38.75
-2.18
39.63
-1.10
38.87 38.87 40.70
-4.29 -4.29 -0.62
21.84 24.85 26.43 28.52 27.68 29.12 29.89 30.28 31.56 31.64 32.42 33.83 32.88 33.01 32.05 33.03 34.26 34.87 34.20 35.22 36.57 35.36 36.40 37.28 37.85 37.76 38.01 37.53 37.75 38.78 38.52 39.67 39.69 38.97 37.99 39.64 39.83 37.90 37.92 38.26 38.85 38.49 38.36 40.16 39.02 39.16 41.49 39.44 39.70 41.08 40.56 42.24 42.40 42.65 43.43 42.00 46.41 45.13 44.02 45.08 46.22 47.50 47.71 47.93 47.54 46.96 47.86 47.98 51.38 49.05 48.79 49.36 51.36 51.36 53.76
-0.80 -1.41 -0.79 -0.69 -0.59 -0).83 -0.45 -0.69 -0.40 0.55 0.27 0.55 0.09 0.33 -0.30 -0.06 -0.07 0.22 0.42 -0.34 -0.15 0.28 0.27 1.00 0.59 0.23 0.14 -0.54 0.01 0.28 -0.40 0.20 0.23 -0.10 -0.03 -0.23 0.07 -0.44 0.62 0.18 0.22 0.72 -0.87 1.61 0.48 0.78 -0.10 -0.46 0.09 0.34 0.59 0.54 -0.31 0.00 -0.15 -0.98 -0.03 0.93 0.54 0.46 -0.04 -0.18 0.40 -0.91 0.77 0.53 -0.98 -0.92 0.17 -0.92 0.65 -0.59 0.73 0.73 0.53
298.15 K
C",
6C.1, ?&
164.85 167.99 128.83 188.74 188.74 193.70 190.83 194.97 158.70 221.12 135.76 224.22 156.31 213.51 211.70 215.90 222.92
0.12 -0.74 5.95 1.06 1.27 -0.02 -0.89 -0.61 4.02 3.02 9.44 3.48 6.09 1.54 0.77 0.20 0.59
219.58 224.74 184.51
0.21 0.30 2.58
249.20
1.87
156.50 246.60 247.32
3.32 1.34 2.60
251.58 251.08
1.41
349.66 210.25 209.24 209.37
0.54 1.06 3.87 0.89
209.41
0.01
212.09 212.84 253.89
2.12 2.47 0.61
210.20 216.27 211.79 185.81 183.76 183.18 187.82 278.20 281.20 215.40
1.61 3.47 1.10 3.27 2.31 1.64 0.92 6.95 0.14 3.68
242.04 214.72 209.33 238.11 213.00
0.46 1.62 0.40 4.05 0.58
312.50 216.44
1.01 0.84
217.04 243.34 243.34
0.45 1.26 1.25
1.11
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1616 Table 111 (Continued)
T
T, 298.15 K bH,,% CPl
TL - Y
~
AHV
compound undecane dodecane tridecane tatradecane pentadecane hexadecane mean dev
bH,,46
PJi, 56.43 61.51 66.43 71.30 76.11 81.38
0.50 0.87 1.01 1.11 1.19 1.91 0.51
0.76
1.0 7
bCp1,
345.05 375.93 406.89 438.44 469.95 501.45
%
2.03 2.51 2.96 3.47 3.96 4.41 2.07
I?. 0
I
0 O
i
I
4
o
0 0
0 0 0 0 0 0
0.0
D 0.4
1 0.8
5
1.2
Figure 1. Acentric factor aa a function of parameter m.
Thus it can be expected that a relationship exists between the parameters m and m* Parameter ms is usually determined by using two vapor pressure data, namely, the critical pressure and the vapor pressure corresponding to the reduced temperature T,= 0.7. It is therefore not surprising that it is related to the acentric factor. The usual method for establishing mg is based on this relationship. Parameter m determines low vapor pressure behavior, and it cannot be hoped to establish a perfect relationship between the acentric factor and m. A rough relationship among the two parameters exists however, as illustrated in Figure 1. The analytical representation of this plot is as follows: In o = 4.6225 - 4.9128m-1/8 (6) Equation 6 makes it possible to calculate the acentric factors of 90 hydrocarbons to within 5 % . In view of discrepencies among the o reported by various authors and the lack of reliable prediction methods, this result can be said to be significant. The significance of parameter m can be discussed in connection with Figure 2. In this figure the values of ACp at the n o d boiling temperature are plotted as a function of m. The linear relationship observed makes it possible to attribute to m the behavior of AC ,which is related to the variation in the degrees of freedom of the molecule passing from the liquid to the vapor phase. It is therefore to be expected that m will be useful as a characterizing parameter in thermodynamic property correlations. This possibility was studied with several examples. All the correlations presented in this chapter were obtained by using m established with a group contribution method (m,& Enthalpies of Vaporization. First, we dealt with the correlation of the entropy of vaporization. As was pointed out by Trouton in 1884, the entropy of vaporization of simple compounds at T b is nearly constant. On the basis of this observation numerous empirical correlations have
20
,
,
m
1
->
1616 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991
parameters w and n can be determined with eqs 9,lO. This method was used with 80 hydrocarbons considered in the previous chapter. The overall mean deviation of 0.93% obtained is only slightly greater than that obtained with the CRP method (0.76%). Thus, eqs 8-10 offer a reliable predictive method for calculating the hydrocarbon enthalpies of vaporization. Vapor Pressures. Finally, the results of the vapor pressure prediction will be discussed. In this case, parameters of equations currently used for vapor pressure representation were adopted in a general form that is valid for hydrocarbons. Two equations were selected: the Antoine equation, which was mostly used in the range 7-200 kPa and the Frost-Kalkwarf equation (1953), which was suitable for a wide range of pressures. The following general form of the htoine equation valid with hydrocarbons was obtained: In (P/bar) = A - B / ( C + )'2 (11) A = In (Po)+ B / ( C + Tb) with Pa = 1.01325 bar (12)
B = -154.523
+ 14.8202Tb0.9'+ 1764.31m1s5- 69.6147&0.59(13)
C = -0.00499Tb''56
(14)
The parameters of eqs 13 and 14 were obtained by fitting experimentalvapor pressures to eq 11. The data base used was the same as that in the paper by Carrier et al. (1988). Only vapor pressures between 0.07 and 1.1 bar were considered. The overall average deviation obtained with 122 compounds and 2148 experimental data was 0.41%. Equations 11-14 can be used with every hydrocarbon, the normal boiling temperature, Tb, and the molec_ularstructure of which are known. Parameters m and b were calculated by using the group contribution methods developed by Carrier et al. (1988). To represent the full range of vapor pressures, the Harlacher and Brown (1970) modification of the FrostKalkwarf equation was chosen. In (P/bar) = AH(l/Tr - 1) + BH In (T,) + 0.4218(Pr/T,2 - 1) (15) AH = -6.91972 - 11.691m2
(16)
BH = [In (Po/p,)- AH(Tc/Tb - 1) -
O - ~ ~ ~ ~ ( P O T -~l)l/[ln / P C /(Tb/Tc)] T ~ ~ (17) Parameter BH was determined by taking the normal boiling temperature. Parameter AH was established as a simple function of m (eq 16). Constants of eq 16 were obtained by fitting experimental vapor pressures to eq 15. Vapor pressures of hydrocarbons considered in the paper of Rogalski et al. (1990) in the range 0.002 k P Cwere taken. The overall average deviation obtained with 47 compounds and 2069 experimental data was 0.46%. In all the correlations described in this paper, m turns out to be a significant characterizing parameter. Ita practical utilization is facilitated by the fact that m is established with a group contribution method. The results reported in this paper are summarized in Table IV.
Conclusions A previously proposed modified version of the PengRobinson equation (CRP method) was tested with experimental enthalpies of vaporization and isobaric liquid heat capacities. The satisfactory results obtained confirmed ita utility for petroleum industry numerical modeling. In view of the accuracy of the vapor pressure and
Table IV. Results Obtained with Correlations Using Parameter m property no. of compounds av mean, dev, % mvha
A H 2
AH"(77 w
Ps(Antoine) Pa(Har1acher)
128 128
80 90 122 47
0.13 0.26 0.93 4.90 0.41 0.46
"Correlation waa established by using data calculated with eq T.2.
volumetric and thermal property predictions, this equation can be said to be a reliable means of determining thermodynamic properties of hydrocarbons. This application seems to be of particular interest in the case of heavy compounds. Finally, the physical significance and the possible applications of parameter m were discussed. It is worth noting that the use of m leads to a substantial improvement in the empirical correlation of the thermodynamic properties of hydrocarbons. Acknowledgment We thank G. Auxiette for helpful discussions during the work on this paper. This research was sponsored by Total-Compagnie Franqaise de PBtroles. Nomenclature a, b = cubic equation of state parameters GT, = value of parameter a calculated at Tb b = pseudocovolume related to the cubic EOS C = isobaric heat capacity If=enthalpy AHv = enthalpy of vaporization at T m = parameter defined by eqs T.8 and T.9 ms = parameter of the Soave function, eq T.13 P = pressure T = absolute temperature ii = noncorrected volume related to the cubic EOS
x = ~~(1001Xj=P - XpCl/Xpp)/n ax%= average absolute deviation of X calculated from n experimental determinations
Subscripts 0 = value at 298.15 K b = Value at Tb c = value at Tc g = gas gr = calculation using group contribution method 1 = liquid s = saturation property Superscripts = ideal gas property
*
Literature Cited Carrier, B.; Rogalski, M.; PBneloux, A. Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified PengRobinson Quation of State. Znd. Eng. Chem. Res. 1988,27,1714. Domalski, E. S.; Evans, W. H.; Hearing, E. D. Heat capacities and entropies of organic compounds in the condensate phase. J. Phys. Chem. Ref. Data 1984, 13. Ducroe, M.; Gruson, J. F.; Sannier, H. Estimation des enthalpies de vaporisation des compo& organiques liquides. Thermochim. Acta 1980,36, 39. Frost, A. A.; Kalkwarf, D. R. Semiempirical equation for the vapor pressure of liquids aa a function of temperature. J. Chem. Phye. 1963,21, 264.
Gibbons, G.; Laughton, G. An equation of state for polar and nonpolar substances and mixtures. J. Chem. SOC.,Faraday Trans. 2 1984,80, 1019.
1617
Ind. Eng. Chem. Res. 1991,30, 1617-1624 Harlacher, E. A.; Brown, W.G. A four parameter extension of the theorem of corresponding states. Znd. Eng. hocess Des. Dev. 1970,9, 479.
Majer, V.; Svoboda, V. Enthalpies of vaporization of Organic Compounds. Blackwell Scientific Publications: Boston, 1985. Peng, D.-Y.; Robinson, D. B. A new two constant equation of state.
Znd. Eng. Chem. Fundam. 1976,15,59. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids. McGraw-Hill Book Co.: London, 1987. Rogabki, M.; Carrier, B.; Solimando, R.; PBneloux, A. Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified Peng-Robinson Equation of State. 2 Representation
of the Vapor Pressures and the Molar Volumes. Znd. Eng. Chem.
Res. 1990,29,659. Soave, G. Equilibrium constante from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 1972,27,1197. TRC Thermodynamical Tables, Hydrocarbons. The Texas A&M University System: College Station, TX, 1975-1986; Vol. IX (loose-leaf data sheets). Trouton, 1884. Cited by Barrow, G. M. Physical Chemistry; McGraw-Hill Book Co.: New York, 1961; pp 396-397.
Received for review July 23, 1990 Accepted February 12,1991
Model with Temperature-IndependentParameters for the Viscosities of Liquid Mixtures Hongqin Liu and Wenchuan Wang* Department of Chemical Engineering, Beijing Institute of Chemical Technology, 100029, Beijing, China
Chien-Hou Chang Department of Chemical Engineering, Tianjin University, 300027, Tianjin, China
On the basis of our previous reviaed free-volume theory, a new model with temperature-independent parameters is proposed for the calculation of viscosities for liquid mixtures. It is used to estimate and predict the viscosities for diverse binary and ternary mixtures, including aqueous and partially miscible systems. For 70 binaries (2033 data points), the grand absolute average deviation percent (AAD) is 1.60 from use of the parameters regressed from the data a t a constant temperature. Moreover, for 23 ternaries (560 data points) the AAD is 2.76. A rigorous comparison with the Teja and Rice method indicates that our method is a good improvement.
Introduction Much interest has been shown in establishing a predictive model for the liquid viscosity of mixtures. Over 50 empirical or semiempirical models have been put forward (Reid et d., 1987; Teja and Rice, 19811, and these models can be divided into three classes: (1)Predictive models: Only models for ideal solutions can be used to predict the viscosities of mixturea from the properties of pure substances. In contrast, for real solutions, moet of them need some information of the mixtures. Wei and Rowley (1985) proposed a local composition model, but vapor-liquid equilibrium (VLE) and heat of mixing data are needed. The group contribution method has been investigated by some researchers (Isdale et al., 1986; Wu, 1986; Reid et al., 1987). However, this method sometimes predicta the properties of mixtures with unexpected errors, especially for the strongly associated solutions, e.g., aqueous solutions. Furthermore, a considerable amount of binary experimental data is necessary for regressing the group interaction parameters (Wu, 1986). (2) Correlation models with temperature-dependent parameters, which have been developed by many workers: The models of McAllister (1960) and Dizechl and Marschall (1982a) and the local composition model (Tong and Li, 1981; Zhong, 1980) are representative of them. Almost all of these models are based on the absolute reaction rate theory of Eyring (Glasstone et al., 1941). Some of them (e.g., the model of Dizechl and Marschall, 1982a) can correlate liquid viscosities for various systems, include aqueous solutions. Since the model parameters change significantly with temperature, these models can be used only to correlate the experimental data at a constant temperature. (3) Models with temperature-independent parameters: A notable advantage over those in claes 2 is that the model
parameters regressed from the data at one temperature can be adopted to predict the viscosities at other temperatures. A successful model, the generalized corresponding states principle (GCSP) method, is proposed by Teja and Rice (1981). This model (see Appendix) has one parameter and well correlates the viscosities for nonaqueous solutions over a certain temperature range. But it is not adequate for strongly associated systems, e.g., aqueous solutions, where a maximum occurs in mixture viscosities (Liu et al., 1989). Besides, this method needs the critical volume, which unfortunately is unavailable for some substances. As mentioned above, there are two problems in the calculation of mixture viscosities: a proper description of the temperature dependence, and the capability for aqueous solutions. It is worth noting that except the GCSP method, almost all of the previous models are based on the Eyring theory (Glasstone et al., 1941). In this theory the activation free energy is an important parameter; the temperature dependence, however, cannot be solved easily. Hilderand et al. (1977) denied the existence of an "activation energy" in liquids and suggested the use of the free-volume theory. But their model cannot be applied to strongly polar substances. Few efforta have been made to extend the free-volume theory to mixtures. Recently we proposed a revised free-volume model for liquids (Liu, 1988) and furthermore a correlation equation for the liquid viscosity of pure substances (Zhang and Liu, 1990). In this paper, on the basis of our previous works, we propose a new model for liquid mixtures whose parameters can be considered independent of temperature to a certain extent. The model is used to calculate the liquid viscoeitiea of binary and ternary mixtures for diverse systems, including aqueous solutions. All the binary parameters are regressed from the experimental data at a
0888-5885/91/2630-1617$02.50/0Q 1991 American Chemical Society
