Ind. Eng. Chem. Fundam. 1984, 23, 97-100
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Generalized Correlation of Latent Heats of Vaporization of Coal Liquid Model Compounds between Their Freezing Points and Critical Points Alwarappa Slvaraman, Joe W. Magee, and Rlkl Kobayashl Department of Chemical Englnwrlng, George R. Brown School of Engineering, Rice University, Houston, Texas 7725 1
Based on Pitzer's thresparameter corresponding states principle (1955), we have developed a correlation of the latent heat of vaporization of aromatic coal liquid model compounds. Temperature ranges from the freezing point to the critical point. An expansion of the form L = L *(o) o'L is used for dimensionless latent heat of vaporization. This model utilizes a nonanalytic functional form based on results derived from renormalization group the0 of fluids in the vicin of the critical point. A simple expression for the latent heat of vaporization, L = ~ ~ ~ 0 7+ 3 3D32 E0.8333+ D 2083 El€ iE3t3is cast in a corresponding states principle correlation for coal liquid compounds. Benzene, the basic constituent of the functional groups of the multi-ring coal liquid compounds, is used as the reference compound in the present correlation. This model works very well at both low and high reduced temperatures approaching the critical point (0.02 < E = ( T , - T ) / T , < 0.69). About 16 compounds, including single, two, and threering compounds, have been tested and the percent root-mean-square deviations in latent heat of vaporization reported and estimated through our model are 0.42 to 5.27 % . Tables of the coefficients of L *(o) and L *(1) are presented. The contributing terms of the latent heat of vaporization function are also presented in a table for small increments of E.
+
2
+
+
Introduction
The latent heat of vaporization is an indispensable physical property for the design and development of industrial processes at elevated temperatures. Data are generally available for low-molecular-weighthydrocarbons. However, as molecular weight increases, data become more scarce, especially at high temperatures. It is a rather new development to use both the renormalization group theoretical approach (Wilson and Fisher, 1972) and the phenomenological scaling hypothesis (Widom, 1965; Kadanoff, 1966; Griffiths,1967) to predict pure-componentproperties in the vicinity of the critical point. In this work a correlation is developed for coal liquid model compounds including one, two, and three-ring aromatic compounds. We apply the corresponding states principle to predict latent heat of vaporization of a coal liquid compound between the freezing point and the critical point. An expression due to Torquato and Stell (1982) has been used in the present model to represent the latent heat of vaporization. Originally, the model parameters were system-dependent. This treatment may be generalized by using the corresponding states principle. I t is usually not convenient to determine all required data experimentally. Hence a correlation is needed to estimate values or to extend or extrapolate limited available data from the freezing point all the way to critical temperature. A number of methods have been proposed by various researchers (Giacalone, 1951; Riedel, 1954; Chen, 1965; Vetere, 1973) for estimating the latent heat of vaporization. However, none of the generalized vapor pressure correlations is very accurate for aromatic compounds, especially for multi-ring coal liquid compounds. Previous correlations were unable to represent latent heat data accurately over the broad domain of values in which we are interested here. Coal liquid model compounds are formed basically from benzene as one of the functional groups. Since most of the required information and data are readily available for benzene, we used it as a reference compound in our model.
The principle of corresponding states frequently is useful for prediction of properties of a large class of compounds from a knowledge of the properties of a few compounds. C o r r e l a t i o n Development
We propose a latent heat of vaporization correlation based upon the form for the dimensionless latent heat
L* = L*(O)+ w* L*(1)
(1)
where L* = L/RTc
and (3)
where R is the gas constant (8.3145 J/(mol K)), T , is critical temperature of the substance (K), w is the acentric factor of the substance under investigation w = -log
PI (at TI = 0.7) - 1.0
(4)
P, = PIPc where P is the vapor pressure at T I = 0.7, wo = 0.212, P, is the critical pressure, wo is the acentric factor of benzene, the reference substance. In addition o1= 0.461 is the acentric factor of carbazole, the other refernce substance. The basis for choosing carbazole as a reference compound is that most of the coal liquid compounds have w values between 0.21 and 0.46. The factor w gives a measure of the deviation of a complex fluid from simple fluid behavior. The deviation is due to the formation of complex molecules from simple molecules and functional groups. The acentric factor can be obtained accurately from a minimum amount of normally available data. This is an important parameter which is assumed to apply to all the coal liquid model compounds. L*(o),and L*(l) are the reduced latent heats of vaporization taking into consideration its singular (near critical)
0196-4313l8411023-0097$01.50/00 1984 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
Table I. Physical Constants of Coal Liquid Compounds Studied substance
mol wt
w
benzene toluene ethylbenzene o-xylene p-xylene m-xylene fluorene 2-methylnaph thalene n-butylbenzene diphenylmethane acridine carbazole
78.108 92.134 106.160 106.160 106.160 106.160 166.230 142.200 134.212 168.240 179.22 167.210
0.212 0.257 0.301 0.314 0.324 0.331 0.334 0.371 0.392 0.438 0.450 0.461
___P,, atm 0 0.1807 0.3574 0.4096 0.4498 0.4779 0.490 0.6386 0.7229 0.9076 0.9558 1.0000
562.60 593.95 619.55 631.59 618.15 619.15 870.00 761.00 660.95 770.20 891.10 901.80
Table 11. Coefficients of Curve Fit of Latent Heat of Vaporization --
D, benzene toluene ethylbenzene o-xylene p-xylene m-xylene fluorene 2-methylnaphthalene n-butylbenzene diphenylmethane carbazole acridine
69.9844 15.9982 42.7044 51.7973 22.1277 37.1990 -4044.9956 892.9590 -0.4860 -944.3278 -14527.5081 3448.9855
D,
-1248.6766 -2121.6613 296 3.20 79 616.7414 612.1521 -1144.5860 -604.4192 -1500.6937 1803.2369 -692.3865 -1404.0792 1847.3024 644.7805 865.1490 -1333.5350 -129.8145 -598.3529 598.9994 116916.2900 141688.2726 -238269.3142 -30663.4917 -41227.2720 65997.2260 1184.2 508 1040.8836 -2115.4310 31547.8398 41692.2859 -56726.9599 414879.0802 515743.2683 -853151.3248 -96201.0901 -114750.4876 195193.7122
and analytic contribution (away from the critical) (Torquato and Stell 1982) as L*(o)= Ale’ + A 2 F A + A ~ E ~ -+~ B+ ’~ +EB ~ E+ ’ B3e3 (5)
+
where
= (T, - n / T C
(7)
a,p, and A are critical exponents given by a = lI8;p = lI3; A (Wegner’s first “gap” exponent) = ‘ I 2 obtained from
renormalizationgroup theoretical calculations. In eq 5 and 6 , A l , A,, AS,B1,B2, Bat and AI*, A2*, A3*, B1*, B2*, &* are the system-independent constants. Results and Discussion The reduced latent heat of vaporization (L*)of each coal liquid compound was interpolated from a fit of the model L = D1t0,3333D 2 D3 t1,2083 E l € Ezc2 E3e3
+
+
+
+
+
(8)
at various E , namely 0.02,0.06,0.08, 0.10,0.20, 0.23,0.26, 0.35,0.38,0.42,0.45,0.475,0.50,0.525,0.55,0.60,0.62,0.65, 0.69 and plotted against w* (varying from 0 to 1)as shown in Figure 1. Apparently the curves are linear and they tend to flatten out (parallel to x axis) at E = 0, because the latent heat of vaporization tends to 0 when T tends to T,. About 14 coal liquid compounds whose latent heats of vaporization are precisely known are plotted in Figure 1. Benzene, toluene, ethylbenzene, o-xylene, p-xylene, mxylene, n-butylbenzene latent heats of vaporization are from (Vargaftic’s tables (1975) and fluorene, acridine, and carbazole data are from this laboratory (Sivaraman and Kobayashi, 1982; Sivaraman et al., 1983a,b). 2-Methylnaphthalene data are also from this laboratory (Wieczorek and Kobayashi, 1981). Table I presents the physical properties used in this work and Table I1 presents the coefficients found by curve fitting. Interpolated and experimental latent heats of vaporization were compared and their average percent of deviation is given in Table 11.
E,
E3
587.4095 -248.2035 -50.5704 -6.9103 455.1850 -153.3696 340.2681 -89.3493 -237.5644 112.2836 228.1847 -96.8468 -20199.7137 4113.1624 5937.3959 -668.9784 -16.4311 -35.7952 -7274.9025 1941.9285 -81659.8581 20348.9714 14619.4760 -2179.4372
av % dev 0.050 0.067 0.077 0.044 0.022 0.041 0.024 0.022 0.034 0.023 0.017 0.002
Table III. Terms of Latent Heat of Vaporization Correlation Function for Various Reduced Temperatures, E
E
L*(1) = AI*@ A 2 * ~ @ + + AA3*e1-DI+@ + B1*e + B2*c2+ &*e3 (6)
48.3 40.6 35.6 36.8 35.0 34.7 46.4 34.6 28.5 28.2 31.72 30.9
_ l _ _ l l _ _
E,
D3
278.65 178.1 5 178.20 247.92 286.30 225.28 389.65 307.73 185.15 296.15 384.15 520.65
0 0.02 0.06 0.08 0.10 0.20 0.23 0.26 0.35 0.38 0.42 0.45 0.475 0.50 0.525 0.55 0.60 0.62 0.65 0.69
L*(O) 0 2.1970 3.4590 3.8408 4.1648 5.3436 5.6062 5.8473 6.4273 6.6220 6.8530 7.0367 7.1540 7.2640 7.3510 7.5125 7.7142 7.7903 7.8811 7.9823
L*(l) _ _ 0 0.1388 0.4457 0.6854 0.7852 1.2207 1.2824 1.4019 1.7614 1.7750 1.8830 1.8940 1.9770 2.0570 2.1860 2.2345 2.3854 2.4473 2.5439 2.6662
Table IV. Coefficients of Least-Squares Analysis of L*co, by US^ of Eq 5 A, ‘4, ‘43
B, B2 B3
6.536924 -2.466698 -77.52141 59.63435 36.09887 -14.60567
Table V. Coefficients of Least-Squares Analysis of L*(l) by Use of Eq 6 A*, A*, A*, B*l B* 2 B* 3
-0.132584 -28.21525 -82.95820 99.00008 19.10458 -2.795660
~
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Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
98
Table VI. Application of Latent Heat of Vaporization Correlation to Other Coal Liquid Model Compounds compound no. of points Lrms, % data sources 22 0.91 Wieczorek and Kobavashi (1981) thianaphthane (C,H,S) 0.64 Vargaftic tables (1975) ' naphthalene (C ,,H ) 35 1-methylnaphthalene (C llH lo) 18 0.88 Wieczorek and Kobayashi (1981) bicyclohexyl (C IzHl,) 1.09 Wieczorek and Kobayashi (1981) 17
IO
9t
i
8
-
"P 6 X Y
E = (Tc-T)/TC
1
Figure 3. Dependence of latent heat of vaporization correlation functions L*(,)and L*(,)upon reduced temperature (e).
Figure 1. Dependence of reduced latent heat of vaporization (L*) upon acentric factor (a*)at several reduced temperatures (e).
--E
I-METHYLNAPTHALENE
l -
+5+ 2 i
0 200
300
400 TEMPERATURE, T
5co
600
W
o
-
BICYLOHEXIL
I
(OK)
Figure 2. Predicted latent heat of vaporization values with eq 8 compared to experimental data (Vargaftic, 1975) as a function of temperature. Solid line denotes the predicted values and black circles the measured values.
Fairly good curve fits were obtained with eq 8. Benzene data are shown as an example for comparison in Figure 2. A least-squares linear fit of L* vs. o* yielded the intercept L*(o)and the slope L*(l)for various e values. The correlation functions L*(o)and L*(l) for this work are tabulated at different e values ranging from 0 to 0.69 in Table 111and their respective coefficients of least squares analysis in Tables IV and V. Figure 3 presents the deupon e. These values were pendence of L*(o)and substituted to calculate the predicted latent heats of vaporization, Ld, for various compounds. The comparison of the experimental data and the predicted values by this correlation gives the percent root-mean-square deviation in latent heats of vaporization 10012/n)1'2 L,, = ( [ ( c & ~ p t l - Ldcd)/Lexptl)
a
-11
-2 1
405.15
no I
I 705.15
I
505.15 605.15 TEMPERATURE, T
J
(OK)
Figure 4. Percent residual latent heat of vaporization as a function of temperature.
for benzene, toluene, ethylbenzene, o-xylene, p-xylene, m-xylene, fluorene, 2-methylnaphthalene, n-butylbenzene, diphenylmethane, acridine, and carbazole used for the development of correlation are 0.91,1.04,0.40,0.42,1.07, 1.07, 4.26, 3.7, 0.50, 1.19, 5.26, and 2.39% respectively. Then the correlation was applied to other new compounds, namely thianapthene, naphthalene, l-methylnaphthalene, and bicyclohexyl. A comparison of available data for their latent heats of vaporization with the predicted values using this model gave L , values 0.91,0.64, 0.88, and 1.09%)respectively, as listed in Table VI. The percent residual latent heats of vaporization have been plotted against their respective temperatures for all the four compounds in Figure 4. In the Appendix the pro-
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Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
cedure for calculation or prediction of latent heat of vaporization is shown as an example for one compound, naphthalene. Conclusions We have developed a correlation for the estimation of latent heats of vaporization of coal liquid compounds from their freezing points to their critical temperatures. If one knows a compound's acentric factor and its critical temperature precisely from vapor pressure data, then its latent heat of vaporization can be calculated with the L*(o)and L*(') from Table 111. The acentric factor, w , is a measure of the increase in the entropy of vaporization over that of a simple fluid. Pitzer et al. (1955) have shown that w depends upon the core radius of a globular molecule and the length of an elongated molecule. It is required because the intermolecular force in complex molecules is a sum of interactions between various parts of adjacent molecules, not just between their centers. Moreover, it is known that many complex coal liquid compounds are formed from benzene and similar functional groups, layered or attached to each other. One can see from Table I that there is a change in w when one goes from a simple benzene molecule to complex acridine or carbazole. To our knowledge there has been no good single correlation which covers a broad temperature region in coal liquid compounds. One of the merits of our present model is that it covers low temperatures (especially above e = 0.50, where most of the compounds are solids) and high temperatures close to T,. Efforts are in progress to extend this correlation to regular hydrocarbon series and other classes of compounds. Acknowledgment The authors thank Phillips Petroleum Company and the United States Department of Energy for their continued financial support for carrying out this project. Appendix Example. Calculate the latent heat of vaporization for naphthalene a t 553.15 K. Solution: napthalene; critical temperature, T, = 751.35 K; w = 0.302. From eq 7 Tc - T 751.35 - 553.15 e=--= = 0.2638 TC 751.35 From eq 5, knowing 6 and the coefficients of least-squares analysis of L*(o)and listed in Table IV, and L*(') = 5.8621 and corresponding to e = 0.2638 are = 1.4347; w = 0.302 (0.302 - 0.212) a* = = 0.3614 from eq 3 (0.461 - 0.212) From eq 1 L* = L*@)+ w* L*(') = 5.8621 + 0.3614 X 1.4347 = 6.3806 L = L**R*T, = 6.3806 X 8.3145 X 751.35 = 39.86 K J mol-'
Reported Lerptl= 39.82 K J mol-'. The error is 0.04 or 0.10%.
Nomenclature AI, A2, A , = constants in eq 5 AI*, A2*,A3* = constants in eq 6 B1,B2,B, = constants in eq 5 B*, B2*,B3* = constants in eq 6 D,, D2,D3 = constants in eq 8 E , E2,E, = constants in eq 8 L = latent heat of vaporization, K J mol-' Lcdcd = calculated latent heat of vaporization Lerptl= experimental latent heat of vaporization data L* = reduced latent heat of vaporization P = pressure, atm P, = critical pressure, atm P, = reduced pressure, PIP, R = gas constant (8.3145 X lo3 J/kmol K) T = temperature, K T , = critical temperature, K T , = reduced temperature, T I T , Tf = freezing point, K Greek Letters = scaling exponent = scaling exponent A = scaling exponent w = acentric factor of the substance under investigation wo = acentric factor of reference compound, benzene wi = acentric factor of reference compound, carbazole (Y
Registry No. Benzene, 71-43-2; toluene, 108-88-3;ethylbenzene, 100-41-4;o-xylene,95-47-6;p-xylene, 106-42-3;m-xylene, 108-38-3;fluorene, 86-73-7; 2-methylnaphthalene,91-57-6;n-butylbenzene, 104-51-8;diphenylmethane, 101-81-5;acridine, 26094-6; carbazole, 86-74-8;thianaphthene, 95-15-8;naphthalene, 91-20-3; 1-methylnaphthalene,90-12-0;bicyclohexyl, 92-51-3.
Literature Cited Chen, N. H. J . Chem. Eng. Data 1965, 70, 207. Giacaione, A. Gazz. Chim. Ita/. 1951,87, 180. Griftihs, R. B. Phys. Rev. 1967, 158, 176. Kadanoff, L. P. Physlcs 1966,2, 263. Pitzer, K. S.:Lippmann, D. L.; Curl, R. F., Jr.; Huggins, C. M.: Petersen, D. E. J . Am. Chem. SOC. 1955, 7 7 , 3433. Riedel, L. Chem. Ing. Tech. 1954,26, 679. Sivaraman, A.; Kobayashi, R. J . Chem. Eng. Data 1982,2 7 , 264. Sivaraman. A.: Martin, R. J.: Kobayashi, R. Fluid Phase Equllib. l983a, 72, 175. Sivaraman, A.; Kobayashi, R. J . Chem. Thermodyn. 1963b (accepted for publication). Torquato, S.; Steil, G. R. Ind. Eng. Chem. Fundam. 1982,27, 202. Vargaftic, N. B. "Tables on thermophyslcai Properties of Gases and Liquids", 2nd ed.; Hemisphere Publication: New York, 1975. Vetere, A. "New Generalized Correlation for Enthalpy of Vaporization of Pure Compounds", Laboratori Ricerche Chimica Industraie, SNAM PROGETTI, SanDanato Milanese, 1973. Widom, B. J . Chem. Phys. 1985,4 3 , 3989. Wieczorek, S.A.; Kobayashi, R. J . Chem. Eng. Data 1881, 26, 11. Wilson, K. G.: Fisher, M. E. Phys. Rev. Left. 1972,2 8 , 248.
Received for review May 5, 1983 Accepted October 3, 1983
