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Correlation for prediction of latent heat of pure components incorporating renormalization group formulations with corresponding-states principle
Fluid Pbose Equilibria, 16 (1984) l-12 Elsevier Science Publishers
B.V., Amsterdam
-
Printed
in The Netherlands
CORRELATION FOR PREDICTION OF LATENT HEAT OF PURE COMPONENTS INCORPORATING RENORMALIZATION GROUP FORMULATIONS WITH CORRESPONDING-STATES PRINCIPLE A. SIVARAMAN,
J. MAGEE
and RIKI
KOBAYASHI
George R. Brown School of Engineering, Department of Chemical Engineering, Rice University,
P.O. Box 1892, Houston, TX 77251 (U.S.A.) Received
August
8, 1983; accepted
in final form September
23, 1983)
ABSTRACT Sivaraman, A., Magee, J. and Kobayashi, R., 1984. Correlation for prediction of latent heat of pure components incorporating renormalization group formulations with correspondingstates principle. Fluid Phase Equilibria, 16: 1-12. A corresponding-states correlation has been developed using renormalization group theory, phenomenological scaling theory and Pitzer et al’s (1955) three-parameter corresponding-states principle to predict latent heats of vaporization for simple fluids (aliphatic and atomatic hydrocarbons, including complex coal-liquid model compounds) from the freezing point to the critical point. The resulting correlation consists of an expansion in the acentric factor which is truncated after the second term, in accordance with the linear dependence observed for experimental data. The dimensionless latent heat of vaporization defined as L* = L/RT, is given by L* = LX + w*L,+,), where L’ is a nonanalytic function (Torquato and Stell, 1982). Subsequent tests of the ability of this correlation to predict latent heats of vaporization over a board domain of reduced temperatures for a large number of different types of compounds have confirmed the validity of the present approach. The results show root-mean-square deviations between reported and predicted latent heats of vaporization in the range 0.46-6.9358 for aliphatic and aromatic hydrocarbons including one-, two- and three-ring coal-liquid model compounds in the range of reduced temperatures 0.02 i L = (T,
- T)/T,
< 0.69.
INTRODUCTION
In the design of coal-conversion processes it is necessary to know the physical properties of the products, including their latent heats of vaporization. These are difficult to measure at elevated temperatures, since thermal degradation occurs for most coal-liquid model compounds_ Hence it is necessary to have a general correlation to predict latent heats with the least possible uncertainty over a wide temperature range. Although several meth037%3812/84/$03.00
0 1984 Elsevier Science Publishers
B.V.
2
ods have been proposed for estimating latent heats of vaporization (Riedel, 1954; Tamir et al., 1983; Chen, 1965) none of them is capable of predicting latent heats over the broad domain of values of interest here. Utilization of the renormalization group-theoretical approach (Wilson and Fisher, 1972) and the phenomenological scaling hypothesis (Widom, 1965; Griffiths, 1967) together to predict pure-component properties in the vicinity of the critical point is a new development. This work presents a simple correlation developed to predict latent heats of vaporization of coal-liquid model compounds (one-, two- and three-ring aromatics), and extended to other hydrocarbons and normal fluids, including, of course, simple fluids. The corresponding-states principle is applied to develop a generalized correlation. A theoretically derived expression (Torquato and Stell, 1982) is used in the present model to represent the latent heat of vaporization. However, the model parameters are treated as compound-independent by using the corresponding-states principle. As stated above, although the correlation was developed and tested for predicting latent heats of vaporization of coal-liquid model compounds, it has been extended to accommodate all other hydrocarbons and normal fluids in general. The principle of corresponding states (Pitzer et al., 1955) is frequently useful for predicting the properties of a large class of compounds from a knowledge of the properties of a few of its members. CORRELATION
FORMULATION
The proposed corresponding-states correlation heat consists of a two-term expansion:
for the dimensionless
L* = L*(0) + i&FL* (1) where
latent
(1)
L* = L/RT,,
(2) w* = w/o, R is the gas constant (8.3145 J mol-’ K-l); and w the acentric factor of the substance defined by Pitzer (1955) as w=
-logP,-1.0
(3)
T,(K) is the critical temperature under
investigation,
the latter
(4)
with P,= P/P,,where P is the vapor pressure at T,= 0.7; w, = 0.490; and PC is the critical pressure. The acentric factor can easily be obtained accurately
3
from a minimum amount of available data; it is an important parameter which correlates all the coal-liquid model compounds studied in this work. L& and L.$, are the reduced latent heats of vaporization taking into consideration the singular (near-critical) and analytic background contributions (Torquato and Stell, 1982): L$,, = A#
+ A@+*
+ Agc’-a+a + B,c + B,c* + B3c3
+ B;r’ L1;, = A;@ + A;@+* + A;c’-~‘+~ + B*16 + B*c* 2
(5) (6)
where c = (T, - T)/T,
(7)
The critical exponents (Y, p and A are given by (Y= l/8, j3 = l/3 and A = l/2 (Wegner’s first “gap” exponent), obtained from renormalization group-theoretical calculations. In eqns. (5) and (6), A,, A,, A,, B,, B,, B, and A:, AT, AT, B:, Bz, B: are system-independent constants. RESULTS AND DISCUSSION
The reduced latent heats of vaporization model compounds as well as the hydrocarbon from a fit of the model
L* of each of the coal-liquid compounds were interpolated
L = D,rO.“” + D2~o.8333+ D3<1.2083 + E,E + E,c* + E,c’
(8)
at various values of E, namely, 0.02, 0.06, 0.08, 0.10, 0.20, 0.23, 0.26, 0.35, 0.42, 0.45, 0.50, 0.525, 0.555, 0.60 and 0.69, and plotted against w* (varying from 0 to 1) as shown in Fig. 1. The curves obtained are linear. Their slope tends towards zero as the temperature tends to T,. Results for some thirteen different compounds, including various hydrocarbons and coal-liquid aromatic compounds for which the relevant data are known precisely, are plotted in Fig. 1. Experimental latent heat of vaporization data for benzene, toluene, n-hexane, ethylbenzene, m-xylene, n-heptane, n-butylbenzene, n-decane and isobutane were taken from Vargaftic’s tables (1975). Data for fluorene and 1-methyhraphthalene were obtained in the authors’ laboratory (Sivaraman and Kobayashi, 1982; Wieczorek and Kobayashi, 1981). Data for methane and ethane were obtained from Hestermans and White (1961) and Goodwin et al. (1976). Table 1 presents the physical properties of the simple fluids considered. Table 2 presents the coefficients of the curves fitted to the latent heats of vaporization. Comparison of interpolated and experimental latent heats of vaporization gave the average deviations also reported in Table 2. Good fits are obtained using eqn. (8) as a model. As an example,
o
0.2
0.4 W”:
0.6 0.8 w/w,
Fig. 1. Dependence of reduced latent several reduced temperatures C.
TABLE
1
Physical
constants
heat of vaporization
L* upon
acentric
factor
o* at
of simple fluids studied
Substance
Mol. wt.
w
w*
T. WI
Methane Ethane Isobutane Benzene Toluene n-Hexane Ethylbenzene m-Xylene Fluorene l-Methylnaphthalene n-Heptane n-Butylbenzene n-Decane
16.043 30.070 58.124 78.108 92.134 86.178 106.160 106.160 166.230 142.200 100.205 134.212 142.286
0.008 0.098 0.176 0.212 0.257 0.296 0.301 0.331 0.334 0.340 0.351 0.392 0.490
0.016 0.200 0.359 0.433 0.524 0.604 0.614 0.675 0.682 0.694 0.716 0.800 1.000
190.55 305.33 408.13 562.60 593.95 507.85 619.55 619.15 870.0 772.00 540.16 660.95 619.15
P, (MPal 90.70 89.90 113.55 278.65 178.15 177.80 178.20 225.28 389.65 251.15 182.60 185.15 243.50
4.599 4.884 3.647 4.894 4.114 2.969 3.607 3.516 4.702 3.567 2.736 2.888 2.117
D1 - 3.1728 8.1906 19.1006 69.9844 15.9982 -6.0044 41.7044 37.1990 - 4044.9956 2979.0877 37.3461 - 0.4860 40.1308
Compound
Methane Ethane Isobutane Benzene Toluene n-Hexane Ethylbenzene m-Xylene Fluorene l-Methylnaphthalene n-Heptane n-Butylbenzene n-Decane
489.9883 336.9943 - 93.8664 - 1248.6766 616.7414 2151.9841 - 604.4192 - 129.8145 116916.2900 - 88601.5530 - 219.2860 1184.2508 49.0697
4
716.6062 452.1371 - 352.0698 - 2121.6613 612.1521 3970.0380 - 1500.6937 - 598.3529 141688.2726 - 110129.8878 - 641.4468 1040.8836 - 355.5899
4
Coefficients of curves fitted to latent heats of vaporization (eqn. (8))
TABLE 2
E2
- 156.6397 - 74.0431 92.2219 587.4095 - 50.5704 1188.4977 455.1850 228.1847 -20199.7137 15189.7441 213.8806 - 16.4311 173.0617
4
- 1082.3986 - 716.2352 387.0218 2963.2079 - 1144.5860 - 5297.2873 1803.2369 598.9994 - 238269.3142 18315.6775 732.1476 -2115.4310 217.0293
50.9335 9.9884 - 23.0364 - 248.2035 - 6.9103 463.0202 - 153.3696 - 96.8468 4113.1624 - 2342.1463 - 85.8480 - 35.7952 - 58.9371
E,
0.150 0.197 0.103 0.050 0.067 0.882 0.077 0.041 0.024 0.012 0.245 0.034 0.144
Average deviation (%)
6
METHANE -
Predicted o Experimental
T/K
Fig. 2. Latent heats of vaporization reported for methane, as a function
predicted using eqn. (8) compared of temperature.
to experimental
data
the predicted and experimental values of the latent heat of vaporization of methane are shown for comparison in Fig. 2. Least-squares linear regression of L* against w* for various e values yield TABLE
3
Terms of correlation c
function
for latent heat of vaporization
at various
c
4)
=?I,
0 0.02 0.06 0.08 0.10 0.20 0.23 0.26 0.35 0.42 0.45 0.50 0.525 0.555 0.600 0.690
0 1.6206 2.5195 2.8744 3.1007 4.0280 4.2369 4.4247 4.8970 5.1668 5.2666 5.4525 5.4916 5.5688 5.7382 6.0267
0 1.0646 1.6512 1.9034 2.1351 2.8080 2.9676 3.1180 3.5206 3.8568 4.0091 4.1946 4.3330 4.4625 4.5964 4.7937
reduced
temperatures
1
TABLE 4 Coefficients of least-squares analyses Coefficient
Value
Lb @P. (511 Al A, A, Bl B, B,
- 0.932980 275.553255 416.646872 - 617.167986 - 94.438858 29.557315
LX, (ew. (6)) A: At A; Bf Bt B;
10.494541 - 351.097613 -617.139173 854.731448 155.934841 - 50.592504
8 I
Fig. 3. Dependence of correlation functions L,,* for latent heat of vaporization upon reduced temperature c
a
values of L& as the intercepts and of L$, as the slopes. The terms of the correlation functions L& and L1;, obtained are tabulated for various E values ranging from 0 to 0.69 in Table 3, and the coefficients of the respective least-squares analyses in Table 4. Figure 3 presents the dependence of L& and L$, upon E. The newly predicted latent heats of vaporization were evaluated by substituting into eqn. (1) L&, L$, and w* at temperatures approaching the vicinity of the critical point T,. The percentage root-mean-square deviations in the latent heat of vaporization given by L rms =
(m L
talc.
-
2/n)“2
Lexp. )/Lexp. x 1001
for methane, ethane, isobutane, benzene, toluene, n-hexane, ethylbenzene, m-xylene, fluorene, 1-methylnaphthalene, n-heptane, n-butylbenzene and n-decane were 0.99, 1.11, 1.39, 2.83, 0.54, 2.11, 0.53, 1.12, 4.17, 0.81, 0.88, 0.46 and 0.55, respectively. The present correlation was also applied to other compounds to test its validity. Fourteen different compounds including various hydrocarbons and coal liquids were tested, namely, propane, n-pentane, n-octane, n-nonane, naphthalene, o-xylene, p-xylene, diphenylmethane, a&dine, thianaphthene, bicyclohexyl, 2_methylnaphthalene, 9-fluorenone and carbazole. A close comparison of available data for latent heats of vaporization (Goodwin and
TABLE 5 Application
of correlation
for latent heat of vaporization
Compound
Number Temperature (K) L,, of points
Propane n-Pentane n-Octane n-Nonane Naphthalene o-Xylene p-Xylene Diphenylmethane Acridine Tbianaphthene Bicyclohexyl 2-Methylnaphthalene 9-Fluorenone Carbazole
22 17 32 34 35 35 29 22 20 22 17 20 20 12
110-346 153-313 223-533 223-553 363-703 253-593 293-573 425-635 465-625 425-635 425-585 425-605 505-635 525-635
1.03 0.84 0.86 1.77 0.99 0.57 1.16 0.80 5.29 1.20 1.58 2.12 6.93 2.42
to other compounds
(W) Data sources Goodwin and Hayes (1982) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Wieaorek and Kobayashi (1981) Sivaraman and Kobayashi (1983) Wieczorek and Kobayashi (1981) Wieczorek and Kobayashi (1981) Wieczorek and Kobayashi (1981) Sivaraman et al. (1983) Sivaraman et al. (1983)
Haynes, 1982; Sivaraman and Kobayashi, 1983; Sivaraman et al., 1983) with the values predicted using this model gave L,,, values of 1.03, 0.84, 0.86, 1.77, 0.99, 0.57, 1.16, 0.80, 5.29, 1.20 1.58, 2.12, 6.93 and 2.42, respectively, as listed in Table 5. Appendix A illustrates the procedure for calculating the latent heat of vaporization for three compounds, namely, naphthalene, n-octane and thianaphthene. The variation of the percentage error in the latent heat of vaporization with temperature is shown for these three compounds in Fig. 4.
CONCLUSIONS
From a comparison of experimental latent heats of vaporization and the values predicted by the present model, it can be seen that the proposed generalized correlation is adequate for calculating latent heats of vaporization for coal-liquid model compounds, hydrocarbons and normal fluids with small deviation. The correlation works well over the entire domain of liquid-vapor coexistence from the freezing point to the critical point. This work constitutes the first comprehensive model incorporating renormalization group theory and phenomenological scaling theory, and predicts precisely the latent heat of vaporization for complicated coal-liquid model compounds and other hydrocarbons from the freezing point to the critical point using only two estimated quantities, the critical temperature and the acentric factor. One of the merits of the model is that it covers the entire range of vapor-liquid coexistence, including the critical region. I
I
I
I
I
I
NAPHTHALENE
I
c
i 8 *I 0 i
-I- orno0 a0 ,
“00
oooo~~ 0~~0
q
n-OCTANE
..DD
8 g +I-
a &A
-I 300
Fig. 4. Percentage
THIANAPHTHENE
,%&A“AllAa
lJJ 05 1 400
1 500 T/K
Afi
Ita
*,Ad* 600
error in latent heat of vaporization
& 700
L as a function
of temperature.
10 ACKNOWLEDGEMENTS The authors wish to thank Mr. Mamoru Omiya for useful discussions concerning programming. The authors also thank the United States Department of Energy and Phillips Petroleum Company for their continued financial support during this project.We also acknowledge initial funding provided by the Electric Power Research Institute to assemble the experimental apparatus used to obtain a major portion of the experimental data utilized in developing the proposed correlation. LIST OF SYMBOLS
constants in eqn. (5) A: constants in eqn. (6) B, constants in eqn. (5) Bzconstants in eqn. (6) D, constants in eqn. (8) E? constants in eqn. (8) latent heat of vaporization calculated latent heat of vaporization experimental latent heat of vaporization reduced latent heat of vaporization vapor pressure (MPa) critical pressure (MPa) PC reduced vapor pressure (P/P,) P, gas constant R T temperature (K) critical temperature (K) T, reduced temperature (T/T,) T, freezing point (K) T, scaling exponents in eqn. (5) a, P, A w acentric factor of substance under investigation acentric factor of reference compound (n-decane) *I
A,,
A,,
A,
A:, A:, B,, B,, B:, Bf, D,, 02, E,, E2, L L talc. L exp. L* P
APPENDIX
Example K Data:
A
1. Calculate the latent heat of vaporization
T, = 751.35 K, w = 0.302.
of naphthalene
at 553. I5
11
From eqn. (7), c = (T, - T)/T, = (751.35 - 553.15)/751.35 From eqns. (5) and (6), L& = 4.44847, LG, = 3.13215 From eqn. (3), o* = w/O.490 = 0.6163
= 0.2638
L* = L*(0)+ w*L*(1)= 6 *3789 L cdc.= L*RT, = 6.3789 x 8.3145 X 751.35 = 39.85 kJ mol-’ Reported Lexp,= 39.82 kJ mol-‘; error = 0.08%.
Lcalc,- L,_=
0.03 kJ mol-‘;
percent
Example 2. Calculate the latent heat of vaporization of n-octane at 533.15 K Data: T, = 569.35 K, o = 0.394. E = 0.0636, L;, = 2.63153, L1;, = 1.71058 w* = 0.8041 L* = 4.00697 L ca,c,= 18.97 kJ mol-’ Reported Le.+ = 18.80 kJ mol-‘; error = 0.89%.
Lcdc,- Lexp,= 0.17 kJ mol-‘;
Example 3. Calculate the latent heat K
ofvaporization
Data:
percentage
oj thianaphthene at 635.15
T, = 752 K, w = 0.294.
c = 0.1554, L,$ = 3.66504, L;, = 2.55305 w* = 0.600 L* = 5.19687 L ca,c,= 32.49 kJ mol-’ Reported Lexp,= 32.90 kJ mol-‘; centage error = - 1.2%.
Lca,c,- Lexp,= -0.41
kJ mol-‘;
per-
REFERENCES Chart, N.H., 1965. J. Chem. Eng. Data, 10: 207-210. Goodwin, R.D. and Haynes, W.M., 1982. Thermal Properties of Propane. Natl. Bur. Stand. (U.S.), Monogr. 170: 249. Goodwin, R.D., Roder, H.M. and Straty, G.C., 1976. Thermal Properties of Ethane. Natl. Bur. Stand. (U.S.), Techn. Note 684: 326. Griffiths, R.B., 1967. Phys Rev., 158: 176-187. Hestermans, P. and White, D., 1961. J. Phys. Chem., 65: 362-365. Piker, KS., Lippmann, D.L., Curl, R.F., Huggins, C.M. and Petersen, D.M., 1955. J. Am. Chem. Sot., 77: 3433-3440.
12
Riedel, L., 1954. Chem. Ing. Tech., 26: 679-683. Sivaraman, A. and Kobayashi, R., 1982. J. Chem. Eng. Data, 27: 264-269. Sivaraman, A. and Kobayashi, R., 1983. J. Chem. Thermodyn. (accepted for publication). Sivaraman, A., Martin, R.J. and Kobayashi, R., 1983. Fluid Phase equilibria, 12: 175-188. Tamir, A., Tamir, E. and Stephan, K., 1983. Heats of Phase Change of Pure Components and Mixtures. Elsevier. Torquato, S. and Stell, G.R., 1982. Ind. Eng. Chem. Fundam., 21: 202-205. Vargaftic, N.B., 1975. Tables of Thermophysical Properties of Gases and Liquids. 2nd edn. Hemisphere, New York. Widom, B., 1965. J. Chem. Phys., 43: 3898-3905. Wieczorek, S.A. and Kobayashi, R., 1981. J. Chem. Eng. Data, 26: 11-13. Wilson, K.G. and Fisher, M.E., 1972. Phys. Rev. Lett., 28: 240-243.
B.V., Amsterdam
-
Printed
in The Netherlands
CORRELATION FOR PREDICTION OF LATENT HEAT OF PURE COMPONENTS INCORPORATING RENORMALIZATION GROUP FORMULATIONS WITH CORRESPONDING-STATES PRINCIPLE A. SIVARAMAN,
J. MAGEE
and RIKI
KOBAYASHI
George R. Brown School of Engineering, Department of Chemical Engineering, Rice University,
P.O. Box 1892, Houston, TX 77251 (U.S.A.) Received
August
8, 1983; accepted
in final form September
23, 1983)
ABSTRACT Sivaraman, A., Magee, J. and Kobayashi, R., 1984. Correlation for prediction of latent heat of pure components incorporating renormalization group formulations with correspondingstates principle. Fluid Phase Equilibria, 16: 1-12. A corresponding-states correlation has been developed using renormalization group theory, phenomenological scaling theory and Pitzer et al’s (1955) three-parameter corresponding-states principle to predict latent heats of vaporization for simple fluids (aliphatic and atomatic hydrocarbons, including complex coal-liquid model compounds) from the freezing point to the critical point. The resulting correlation consists of an expansion in the acentric factor which is truncated after the second term, in accordance with the linear dependence observed for experimental data. The dimensionless latent heat of vaporization defined as L* = L/RT, is given by L* = LX + w*L,+,), where L’ is a nonanalytic function (Torquato and Stell, 1982). Subsequent tests of the ability of this correlation to predict latent heats of vaporization over a board domain of reduced temperatures for a large number of different types of compounds have confirmed the validity of the present approach. The results show root-mean-square deviations between reported and predicted latent heats of vaporization in the range 0.46-6.9358 for aliphatic and aromatic hydrocarbons including one-, two- and three-ring coal-liquid model compounds in the range of reduced temperatures 0.02 i L = (T,
- T)/T,
< 0.69.
INTRODUCTION
In the design of coal-conversion processes it is necessary to know the physical properties of the products, including their latent heats of vaporization. These are difficult to measure at elevated temperatures, since thermal degradation occurs for most coal-liquid model compounds_ Hence it is necessary to have a general correlation to predict latent heats with the least possible uncertainty over a wide temperature range. Although several meth037%3812/84/$03.00
0 1984 Elsevier Science Publishers
B.V.
2
ods have been proposed for estimating latent heats of vaporization (Riedel, 1954; Tamir et al., 1983; Chen, 1965) none of them is capable of predicting latent heats over the broad domain of values of interest here. Utilization of the renormalization group-theoretical approach (Wilson and Fisher, 1972) and the phenomenological scaling hypothesis (Widom, 1965; Griffiths, 1967) together to predict pure-component properties in the vicinity of the critical point is a new development. This work presents a simple correlation developed to predict latent heats of vaporization of coal-liquid model compounds (one-, two- and three-ring aromatics), and extended to other hydrocarbons and normal fluids, including, of course, simple fluids. The corresponding-states principle is applied to develop a generalized correlation. A theoretically derived expression (Torquato and Stell, 1982) is used in the present model to represent the latent heat of vaporization. However, the model parameters are treated as compound-independent by using the corresponding-states principle. As stated above, although the correlation was developed and tested for predicting latent heats of vaporization of coal-liquid model compounds, it has been extended to accommodate all other hydrocarbons and normal fluids in general. The principle of corresponding states (Pitzer et al., 1955) is frequently useful for predicting the properties of a large class of compounds from a knowledge of the properties of a few of its members. CORRELATION
FORMULATION
The proposed corresponding-states correlation heat consists of a two-term expansion:
for the dimensionless
L* = L*(0) + i&FL* (1) where
latent
(1)
L* = L/RT,,
(2) w* = w/o, R is the gas constant (8.3145 J mol-’ K-l); and w the acentric factor of the substance defined by Pitzer (1955) as w=
-logP,-1.0
(3)
T,(K) is the critical temperature under
investigation,
the latter
(4)
with P,= P/P,,where P is the vapor pressure at T,= 0.7; w, = 0.490; and PC is the critical pressure. The acentric factor can easily be obtained accurately
3
from a minimum amount of available data; it is an important parameter which correlates all the coal-liquid model compounds studied in this work. L& and L.$, are the reduced latent heats of vaporization taking into consideration the singular (near-critical) and analytic background contributions (Torquato and Stell, 1982): L$,, = A#
+ A@+*
+ Agc’-a+a + B,c + B,c* + B3c3
+ B;r’ L1;, = A;@ + A;@+* + A;c’-~‘+~ + B*16 + B*c* 2
(5) (6)
where c = (T, - T)/T,
(7)
The critical exponents (Y, p and A are given by (Y= l/8, j3 = l/3 and A = l/2 (Wegner’s first “gap” exponent), obtained from renormalization group-theoretical calculations. In eqns. (5) and (6), A,, A,, A,, B,, B,, B, and A:, AT, AT, B:, Bz, B: are system-independent constants. RESULTS AND DISCUSSION
The reduced latent heats of vaporization model compounds as well as the hydrocarbon from a fit of the model
L* of each of the coal-liquid compounds were interpolated
L = D,rO.“” + D2~o.8333+ D3<1.2083 + E,E + E,c* + E,c’
(8)
at various values of E, namely, 0.02, 0.06, 0.08, 0.10, 0.20, 0.23, 0.26, 0.35, 0.42, 0.45, 0.50, 0.525, 0.555, 0.60 and 0.69, and plotted against w* (varying from 0 to 1) as shown in Fig. 1. The curves obtained are linear. Their slope tends towards zero as the temperature tends to T,. Results for some thirteen different compounds, including various hydrocarbons and coal-liquid aromatic compounds for which the relevant data are known precisely, are plotted in Fig. 1. Experimental latent heat of vaporization data for benzene, toluene, n-hexane, ethylbenzene, m-xylene, n-heptane, n-butylbenzene, n-decane and isobutane were taken from Vargaftic’s tables (1975). Data for fluorene and 1-methyhraphthalene were obtained in the authors’ laboratory (Sivaraman and Kobayashi, 1982; Wieczorek and Kobayashi, 1981). Data for methane and ethane were obtained from Hestermans and White (1961) and Goodwin et al. (1976). Table 1 presents the physical properties of the simple fluids considered. Table 2 presents the coefficients of the curves fitted to the latent heats of vaporization. Comparison of interpolated and experimental latent heats of vaporization gave the average deviations also reported in Table 2. Good fits are obtained using eqn. (8) as a model. As an example,
o
0.2
0.4 W”:
0.6 0.8 w/w,
Fig. 1. Dependence of reduced latent several reduced temperatures C.
TABLE
1
Physical
constants
heat of vaporization
L* upon
acentric
factor
o* at
of simple fluids studied
Substance
Mol. wt.
w
w*
T. WI
Methane Ethane Isobutane Benzene Toluene n-Hexane Ethylbenzene m-Xylene Fluorene l-Methylnaphthalene n-Heptane n-Butylbenzene n-Decane
16.043 30.070 58.124 78.108 92.134 86.178 106.160 106.160 166.230 142.200 100.205 134.212 142.286
0.008 0.098 0.176 0.212 0.257 0.296 0.301 0.331 0.334 0.340 0.351 0.392 0.490
0.016 0.200 0.359 0.433 0.524 0.604 0.614 0.675 0.682 0.694 0.716 0.800 1.000
190.55 305.33 408.13 562.60 593.95 507.85 619.55 619.15 870.0 772.00 540.16 660.95 619.15
P, (MPal 90.70 89.90 113.55 278.65 178.15 177.80 178.20 225.28 389.65 251.15 182.60 185.15 243.50
4.599 4.884 3.647 4.894 4.114 2.969 3.607 3.516 4.702 3.567 2.736 2.888 2.117
D1 - 3.1728 8.1906 19.1006 69.9844 15.9982 -6.0044 41.7044 37.1990 - 4044.9956 2979.0877 37.3461 - 0.4860 40.1308
Compound
Methane Ethane Isobutane Benzene Toluene n-Hexane Ethylbenzene m-Xylene Fluorene l-Methylnaphthalene n-Heptane n-Butylbenzene n-Decane
489.9883 336.9943 - 93.8664 - 1248.6766 616.7414 2151.9841 - 604.4192 - 129.8145 116916.2900 - 88601.5530 - 219.2860 1184.2508 49.0697
4
716.6062 452.1371 - 352.0698 - 2121.6613 612.1521 3970.0380 - 1500.6937 - 598.3529 141688.2726 - 110129.8878 - 641.4468 1040.8836 - 355.5899
4
Coefficients of curves fitted to latent heats of vaporization (eqn. (8))
TABLE 2
E2
- 156.6397 - 74.0431 92.2219 587.4095 - 50.5704 1188.4977 455.1850 228.1847 -20199.7137 15189.7441 213.8806 - 16.4311 173.0617
4
- 1082.3986 - 716.2352 387.0218 2963.2079 - 1144.5860 - 5297.2873 1803.2369 598.9994 - 238269.3142 18315.6775 732.1476 -2115.4310 217.0293
50.9335 9.9884 - 23.0364 - 248.2035 - 6.9103 463.0202 - 153.3696 - 96.8468 4113.1624 - 2342.1463 - 85.8480 - 35.7952 - 58.9371
E,
0.150 0.197 0.103 0.050 0.067 0.882 0.077 0.041 0.024 0.012 0.245 0.034 0.144
Average deviation (%)
6
METHANE -
Predicted o Experimental
T/K
Fig. 2. Latent heats of vaporization reported for methane, as a function
predicted using eqn. (8) compared of temperature.
to experimental
data
the predicted and experimental values of the latent heat of vaporization of methane are shown for comparison in Fig. 2. Least-squares linear regression of L* against w* for various e values yield TABLE
3
Terms of correlation c
function
for latent heat of vaporization
at various
c
4)
=?I,
0 0.02 0.06 0.08 0.10 0.20 0.23 0.26 0.35 0.42 0.45 0.50 0.525 0.555 0.600 0.690
0 1.6206 2.5195 2.8744 3.1007 4.0280 4.2369 4.4247 4.8970 5.1668 5.2666 5.4525 5.4916 5.5688 5.7382 6.0267
0 1.0646 1.6512 1.9034 2.1351 2.8080 2.9676 3.1180 3.5206 3.8568 4.0091 4.1946 4.3330 4.4625 4.5964 4.7937
reduced
temperatures
1
TABLE 4 Coefficients of least-squares analyses Coefficient
Value
Lb @P. (511 Al A, A, Bl B, B,
- 0.932980 275.553255 416.646872 - 617.167986 - 94.438858 29.557315
LX, (ew. (6)) A: At A; Bf Bt B;
10.494541 - 351.097613 -617.139173 854.731448 155.934841 - 50.592504
8 I
Fig. 3. Dependence of correlation functions L,,* for latent heat of vaporization upon reduced temperature c
a
values of L& as the intercepts and of L$, as the slopes. The terms of the correlation functions L& and L1;, obtained are tabulated for various E values ranging from 0 to 0.69 in Table 3, and the coefficients of the respective least-squares analyses in Table 4. Figure 3 presents the dependence of L& and L$, upon E. The newly predicted latent heats of vaporization were evaluated by substituting into eqn. (1) L&, L$, and w* at temperatures approaching the vicinity of the critical point T,. The percentage root-mean-square deviations in the latent heat of vaporization given by L rms =
(m L
talc.
-
2/n)“2
Lexp. )/Lexp. x 1001
for methane, ethane, isobutane, benzene, toluene, n-hexane, ethylbenzene, m-xylene, fluorene, 1-methylnaphthalene, n-heptane, n-butylbenzene and n-decane were 0.99, 1.11, 1.39, 2.83, 0.54, 2.11, 0.53, 1.12, 4.17, 0.81, 0.88, 0.46 and 0.55, respectively. The present correlation was also applied to other compounds to test its validity. Fourteen different compounds including various hydrocarbons and coal liquids were tested, namely, propane, n-pentane, n-octane, n-nonane, naphthalene, o-xylene, p-xylene, diphenylmethane, a&dine, thianaphthene, bicyclohexyl, 2_methylnaphthalene, 9-fluorenone and carbazole. A close comparison of available data for latent heats of vaporization (Goodwin and
TABLE 5 Application
of correlation
for latent heat of vaporization
Compound
Number Temperature (K) L,, of points
Propane n-Pentane n-Octane n-Nonane Naphthalene o-Xylene p-Xylene Diphenylmethane Acridine Tbianaphthene Bicyclohexyl 2-Methylnaphthalene 9-Fluorenone Carbazole
22 17 32 34 35 35 29 22 20 22 17 20 20 12
110-346 153-313 223-533 223-553 363-703 253-593 293-573 425-635 465-625 425-635 425-585 425-605 505-635 525-635
1.03 0.84 0.86 1.77 0.99 0.57 1.16 0.80 5.29 1.20 1.58 2.12 6.93 2.42
to other compounds
(W) Data sources Goodwin and Hayes (1982) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Vargaftic (1975) Wieaorek and Kobayashi (1981) Sivaraman and Kobayashi (1983) Wieczorek and Kobayashi (1981) Wieczorek and Kobayashi (1981) Wieczorek and Kobayashi (1981) Sivaraman et al. (1983) Sivaraman et al. (1983)
Haynes, 1982; Sivaraman and Kobayashi, 1983; Sivaraman et al., 1983) with the values predicted using this model gave L,,, values of 1.03, 0.84, 0.86, 1.77, 0.99, 0.57, 1.16, 0.80, 5.29, 1.20 1.58, 2.12, 6.93 and 2.42, respectively, as listed in Table 5. Appendix A illustrates the procedure for calculating the latent heat of vaporization for three compounds, namely, naphthalene, n-octane and thianaphthene. The variation of the percentage error in the latent heat of vaporization with temperature is shown for these three compounds in Fig. 4.
CONCLUSIONS
From a comparison of experimental latent heats of vaporization and the values predicted by the present model, it can be seen that the proposed generalized correlation is adequate for calculating latent heats of vaporization for coal-liquid model compounds, hydrocarbons and normal fluids with small deviation. The correlation works well over the entire domain of liquid-vapor coexistence from the freezing point to the critical point. This work constitutes the first comprehensive model incorporating renormalization group theory and phenomenological scaling theory, and predicts precisely the latent heat of vaporization for complicated coal-liquid model compounds and other hydrocarbons from the freezing point to the critical point using only two estimated quantities, the critical temperature and the acentric factor. One of the merits of the model is that it covers the entire range of vapor-liquid coexistence, including the critical region. I
I
I
I
I
I
NAPHTHALENE
I
c
i 8 *I 0 i
-I- orno0 a0 ,
“00
oooo~~ 0~~0
q
n-OCTANE
..DD
8 g +I-
a &A
-I 300
Fig. 4. Percentage
THIANAPHTHENE
,%&A“AllAa
lJJ 05 1 400
1 500 T/K
Afi
Ita
*,Ad* 600
error in latent heat of vaporization
& 700
L as a function
of temperature.
10 ACKNOWLEDGEMENTS The authors wish to thank Mr. Mamoru Omiya for useful discussions concerning programming. The authors also thank the United States Department of Energy and Phillips Petroleum Company for their continued financial support during this project.We also acknowledge initial funding provided by the Electric Power Research Institute to assemble the experimental apparatus used to obtain a major portion of the experimental data utilized in developing the proposed correlation. LIST OF SYMBOLS
constants in eqn. (5) A: constants in eqn. (6) B, constants in eqn. (5) Bzconstants in eqn. (6) D, constants in eqn. (8) E? constants in eqn. (8) latent heat of vaporization calculated latent heat of vaporization experimental latent heat of vaporization reduced latent heat of vaporization vapor pressure (MPa) critical pressure (MPa) PC reduced vapor pressure (P/P,) P, gas constant R T temperature (K) critical temperature (K) T, reduced temperature (T/T,) T, freezing point (K) T, scaling exponents in eqn. (5) a, P, A w acentric factor of substance under investigation acentric factor of reference compound (n-decane) *I
A,,
A,,
A,
A:, A:, B,, B,, B:, Bf, D,, 02, E,, E2, L L talc. L exp. L* P
APPENDIX
Example K Data:
A
1. Calculate the latent heat of vaporization
T, = 751.35 K, w = 0.302.
of naphthalene
at 553. I5
11
From eqn. (7), c = (T, - T)/T, = (751.35 - 553.15)/751.35 From eqns. (5) and (6), L& = 4.44847, LG, = 3.13215 From eqn. (3), o* = w/O.490 = 0.6163
= 0.2638
L* = L*(0)+ w*L*(1)= 6 *3789 L cdc.= L*RT, = 6.3789 x 8.3145 X 751.35 = 39.85 kJ mol-’ Reported Lexp,= 39.82 kJ mol-‘; error = 0.08%.
Lcalc,- L,_=
0.03 kJ mol-‘;
percent
Example 2. Calculate the latent heat of vaporization of n-octane at 533.15 K Data: T, = 569.35 K, o = 0.394. E = 0.0636, L;, = 2.63153, L1;, = 1.71058 w* = 0.8041 L* = 4.00697 L ca,c,= 18.97 kJ mol-’ Reported Le.+ = 18.80 kJ mol-‘; error = 0.89%.
Lcdc,- Lexp,= 0.17 kJ mol-‘;
Example 3. Calculate the latent heat K
ofvaporization
Data:
percentage
oj thianaphthene at 635.15
T, = 752 K, w = 0.294.
c = 0.1554, L,$ = 3.66504, L;, = 2.55305 w* = 0.600 L* = 5.19687 L ca,c,= 32.49 kJ mol-’ Reported Lexp,= 32.90 kJ mol-‘; centage error = - 1.2%.
Lca,c,- Lexp,= -0.41
kJ mol-‘;
per-
REFERENCES Chart, N.H., 1965. J. Chem. Eng. Data, 10: 207-210. Goodwin, R.D. and Haynes, W.M., 1982. Thermal Properties of Propane. Natl. Bur. Stand. (U.S.), Monogr. 170: 249. Goodwin, R.D., Roder, H.M. and Straty, G.C., 1976. Thermal Properties of Ethane. Natl. Bur. Stand. (U.S.), Techn. Note 684: 326. Griffiths, R.B., 1967. Phys Rev., 158: 176-187. Hestermans, P. and White, D., 1961. J. Phys. Chem., 65: 362-365. Piker, KS., Lippmann, D.L., Curl, R.F., Huggins, C.M. and Petersen, D.M., 1955. J. Am. Chem. Sot., 77: 3433-3440.
12
Riedel, L., 1954. Chem. Ing. Tech., 26: 679-683. Sivaraman, A. and Kobayashi, R., 1982. J. Chem. Eng. Data, 27: 264-269. Sivaraman, A. and Kobayashi, R., 1983. J. Chem. Thermodyn. (accepted for publication). Sivaraman, A., Martin, R.J. and Kobayashi, R., 1983. Fluid Phase equilibria, 12: 175-188. Tamir, A., Tamir, E. and Stephan, K., 1983. Heats of Phase Change of Pure Components and Mixtures. Elsevier. Torquato, S. and Stell, G.R., 1982. Ind. Eng. Chem. Fundam., 21: 202-205. Vargaftic, N.B., 1975. Tables of Thermophysical Properties of Gases and Liquids. 2nd edn. Hemisphere, New York. Widom, B., 1965. J. Chem. Phys., 43: 3898-3905. Wieczorek, S.A. and Kobayashi, R., 1981. J. Chem. Eng. Data, 26: 11-13. Wilson, K.G. and Fisher, M.E., 1972. Phys. Rev. Lett., 28: 240-243.