2064
Ind. Eng. Chem. Res. 1998, 37, 2064-2068
CORRELATIONS A Correlation for the Prediction of Thermal Conductivity of Liquids Dana M. Klaas and Dabir S. Viswanath* Department of Chemical Engineering, University of MissourisColumbia, Columbia, Missouri 65211
Correlation and prediction of transport properties are important in the design of heat- and masstransfer equipment. Prediction of the thermal conductivity of liquids based only on theoretical grounds does not give good results because the theories describing the liquid state are far from satisfactory. In this paper, a semitheoretical method for the prediction of thermal conductivity is proposed and the results are compared with the recent methods by Arikol and Gurbuz and with the methods recommended by Reid et al. Introduction Thermal conductivity is an important property in the prediction of heat- and mass-transfer coefficients under both laminar and turbulent regimes. A number of correlations have been developed to predict thermal conductivity (Reid et al., 1987). Among them are the recent correlations due to Baroncini et al. (1979), Nagvekar (1984), and Arikol and Gurbuz (1992). Reid et al. in the current edition of their book, The Properties of Gases and Liquids, have included the Baroncini method and have compared this method with the methods of Sato and Reidel and of Missenard and Riedel. However, many of these methods are restricted to homologous series and require more than two or three parameters. In addition, these parameters have not been correlated with the readily available physical properties. Not only does the correlation presented in this paper have a theoretical basis but also the two parameters are correlated with readily available physical properties. Theory Horrocks and McLaughlin (1960, 1963) considered the energy transport occurring in a liquid as due to convective and vibrational contributions. The convective transport occurs due to molecules “hopping” from occupied sites to holes in the liquid quasi-lattice. This transfer is conditioned by both whether a molecule has sufficient energy to move and whether there is a hole close enough to accommodate it. Horrocks and McLaughlin show that the convective contribution is negligible and accounts for less than 5% of the thermal conductivity values for liquids. The vibrational effect is a function of the distance between the nearest neighbors and the difference in energy between layers of the quasi-lattice due to a temperature gradient. Vibrating molecules transfer energy whenever they collide. The higher the * Corresponding author. Telephone: (573) 884-0707. Fax: (573) 884-4940. E-mail:
[email protected].
temperature, the more the molecules will vibrate, sending the energy and heat down the gradient. The relation given by Horrocks and McLaughlin is
λ ) 2pvmlCv
(1)
where p is the probability of energy transfer on collision, v is the vibrational frequency, m is the number of molecules per unit area, l is the distance between adjacent planes, and Cv is the specific heat. The 2 accounts for the fact that a molecule crosses a plane perpendicular to its direction of motion twice in every complete vibration. A limitation of the Horrocks and McLaughlin theory is that the authors assumed p to be unity as it is an extremely difficult task to evaluate the probability of energy transfer on collision. Equation 1, in turn, leads to the temperature dependence of thermal conductivity as
( )
1 dλ λ dT
p
〈 (
) -R
δ ln(v) 1 3 d ln(V)
)〉
(2)
p
where R, the coefficient of thermal expansion, controls the temperature dependence. The Gruneisen constant, (δ ln v/δ ln V)p, is independent of temperature. Using the equation of state for a liquid in the form PVl ) ZlRT and taking the changes with respect to temperature gives
(
)
dZl dVl R ) Z +T dT P l dT
(3)
where P is the pressure, Vl is the liquid volume, Zl is the liquid compressibility factor, T is the temperature, and R is the gas constant. The partial derivative of Z with respect to temperature is very small for a liquid. Viswanath and Rao (1970) showed that combining eqs 2 and 3 yields
() ( ) λ T )A λ0 T0
S0888-5885(97)00683-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/01/1998
-b
(4)
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 2065 Table 1. Summary of Results with Parameters A and B Obtained from Figure 2 no. of substances no. of data points
alkanes
alkenes
alcohols
aromatic
halogen
misc.
glycols
nitrogen
total
21 284
3 20
13 93
8 53
10 51
13 66
3 34
2 10
73 611
no. of occurrences error range (%)
alkanes
alkenes
alcohols
aromatic
halogen
misc.
glycols
nitrogen
total
40
73 147 38 25 1 0 0
10 7 2 1 0 0 0
15 37 22 15 4 0 0
9 26 6 7 5 0 0
13 25 6 6 1 0 0
8 23 18 12 4 0 1
2 9 4 10 8 1 0
1 5 3 1 0 0 0
131 279 99 77 23 1 1
where λ0 is the value of thermal conductivity at T0 and A and b are constants for a given substance. The present method of estimating thermal conductivity is based on eq 4. The values of A and b were determined using statistical software package Systat and graphical analysis on Excel. New Correlation of λ0. Physical and chemical properties of a substance depend on the structure of the molecules and attractive and repulsive force fields. These factors along with the polar and nonpolar characteristics of the molecules play an important role in the behavior of the substances. In developing a correlation for A and b in eq 4, these factors were considered. Molar polarization was chosen as the parameter to characterize the behavior of molecules. Molar polarization is defined as (Viswanath and Prasad, 1974, 1981)
Pmc ) Rm + 4πNµ2/9kTc
(5)
where Pmc is molar polarization, Rm is molar refraction ) (M/F)[(n2 - 1)/(n2 + 2)], µ is dipole moment in Debye units, N is Avogadro’s number (6.023 × 1023 molecules/ mol), k is the Boltzmann constant, Tc is critical temperature in K, n is the refractive index, M is molecular weight in g/mol, and F is the density in g/cm3. The advantages to using molar polarization are as follows: (a) It has a sound theoretical basis as it is derived from the Clausius-Mossotti (Debye, 1929) equation. (b) It takes into consideration the structure of the molecules in the molar refraction term. (c) It accounts for the polar nature of the molecules in the dipole moment, µ, term. (d) It is temperature dependent. (e) It contains parameters which can be determined easily. (f) It does not contain parameters such as critical properties which are not easily determinable. Further, if a compound decomposes on heating, it will be difficult to determine the critical properties. No such thing is encountered in this parameter. (g) Dipole moments are available for more compounds compared to other properties such as the critical properties. (h) The value of molar polarization does not depend on the technique used to evaluate it. For example, different values of acentric factors are used by different authors based on the vapor pressure-temperature data used and the method of evaluating the slope at Tr ) 0.7. Present Correlation. Experimental thermal conductivity values for a variety of substances at different temperatures were gathered mainly from two sources,
Figure 1. Temperature vs thermal conductivity for a typical substance (n-decane).
Figure 2. Correlation for A and b vs Pmc.
Jamieson and Tudhope (1963) and Vargaftik (1975). Jamieson and Tudhope compiled thermal conductivity data from a number of sources and then analyzed and ranked the data based on accuracy. Wherever possible, the most accurate data as determined by Jamieson and Tudhope were used in this work; however, the scatter of the values is apparent as shown in Figure 1 for n-decane. The scatter in the “best” experimental data is close to (5-10%. The data tabulated by Vargaftik, on the other hand, appear to be smoothed-out, as is also shown in Figure 1. Vargaftik’s alkane data were used to develop the new correlation. The data were plotted using the lowest temperature and the corresponding
2066 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Table 2. Comparison of the New Method with the Arikol and Gurbuz Methoda new method
a
Arikol and Gurbuz
substance
T range (K)
AAD (%)
MAD (%)
AAD (%)
MAD (%)
methane propane n-pentane isopentane n-hexane n-heptane n-octane isooctane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane n-octadecane n-nonadecane n-eicosane
99.2-112.2 213.2-223.2 273.2-303.2 273.2-293.2 273.2-333.2 273.2-353.2 233.2-393.2 290.0-370.0 233.2-413.2 253.2-433.2 253.2-453.2 273.2-473.2 273.2-501.2 293.2-513.2 293.2-533.2 303.2-553.2 303.2-573.2 305.2-573.2 313.2-593.2 313.2-613.2
2.7 0.2 1.7 0.3 2.8 1.7 2.1 2.7 3.0 2.0 2.3 3.2 2.3 2.0 2.1 2.3 2.3 2.5 2.0 1.6
7.2 0.3 9.5 0.5 9.3 7.4 4.9 6.5 8.8 6.2 6.1 9.5 5.7 7.9 11.6 8.5 8.4 9.1 7.7 3.6
6.4 4.6 2.7 3.2 4.9 6.2 5.6 8.1 3.6 4.0 3.8 8.7 4.3 11.4 6.4 4.7 4.6 8.9 5.9 13.2
11.8 5.5 11.4 4.5 8.4 14.1 8.0 9.6 8.8 8.0 7.7 10.7 5.6 12.7 7.6 5.9 5.9 11.2 7.1 13.8
AAD ) average absolute deviation (%). MAD ) maximum absolute deviation (%).
Table 3. Comparison between Calculated and Experimental Values of Liquid Thermal Conductivity (Taken from Reid et al., Table 10-7) percent errorb compound propane n-pentane n-decane cyclohexane methylcyclopentane benzene ethylbenzene ethanol n-octanol tert-butyl alcohol m-cresol aniline propionic acid methylene chloride carbon tetrachloride ethyl bromide chlorobenzene iodobenzene ethyl acetate butyl acetate acetone diethyl ether acetaldehyde a
T, K 323 293 303 314 349 293 293 311 293 323 389 293 353 293 313 347 293 311 293 353 290 285 253 293 253 293 293 233 353 253 353 293 333 293 273 313 293 293
λL,a
exptl
0.0783 0.114 0.111 0.127 0.119 0.124 0.121 0.115 0.148 0.137 0.114 0.132 0.118 0.165 0.152 0.135 0.166 0.116 0.150 0.145 0.178 0.173 0.159 0.148 0.110 0.103 0.103 0.141 0.111 0.106 0.0938 0.147 0.141 0.137 0.171 0.151 0.129 0.190
Latini et al.
Sato and Riedel
Missenard and Riedel
new method
-19 -5.7 -5.9 -3.2 -2.9 -1.2 -3.2 -2.2 0 1.9 5.1 2.0 2.9 -3.3 0 3.5 -11 4.5 10 3.8
27 20 20 -2.0 -1.8 11 13 14 -3.4 -2.1 0 2.2 3.2 15 19 22 -19 26 -3.6 -8.6 -15 -3.4 -13 -15 -0.8 -1.6 7.7 0.0 4.1 -0.4 -0.9 -7.1 -12 -4.9 -2.2 0.5 4.5 -12
18 17 17 9.5 9.8 3.7 3.8 4.7 -5.1 -4.0 -1.8 4.4 5.3 24 28 32 5.6 77 28 21 10 15 -6.3 -7.9 15 14 -6.9 2.6 7.1 5.1 4.5 3.1 -2.7 9.2 3.7 6.6 22 -11
36.1 1.3 1.7 0.2 -0.4 -2.1 0.3 1.4 -7.0 -6.0 -0.3 0.0 -1.2 2.7 6.7 12.2 -3.3 10.3 2.1 -6.6 -5.1 -4.4 2.0 -0.6 0.2 -3.1 0.5 5.0 1.3 7.5 -2.5 1.0 -3.2 1.4 2.8 6.7 5.6 -0.2
-8.9 -17 -19 -6.4 -7.3 2.0 -0.5 2.4 -15 -17 2.9 2.4 2.5 -9.8 -6.9 3.9
All values of λL are in W/(m‚K). b Percent error ) [(calcd - exptl)/exptl] × 100.
thermal conductivity data as T0 and λ0, respectively, and the values for A and b were evaluated for individual substances. These A and b values for different substances were then plotted against molar polarization, as shown in Figure 2. The trendline equations derived
from this plot were then used to calculate thermal conductivity of the substances at various temperatures using eq 4. As can be seen, the correlation of A and b with molar polarization is weak, but the correlation was developed using only alkane data. It is likely that the
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 2067 Table 4. Summary of Results with Constants A ) 1.0 and b ) 2/3 no. of substances no. of data points
alkanes
alkenes
alcohols
aromatic
halogen
misc.
glycols
nitrogen
total
21 284
3 20
13 93
8 53
10 51
13 66
3 34
2 10
73 611
no. of occurrences error range (%)
alkanes
alkenes
alcohols
aromatic
halogen
misc.
glycols
nitrogen
total
40
73 75 86 49 1 0 0
11 7 1 1 0 0 0
22 20 20 29 2 0 0
18 11 11 8 5 0 0
18 16 8 9 0 0 0
24 18 10 10 2 1 1
6 4 4 8 8 4 0
3 1 1 5 0 0 0
175 152 141 119 18 5 1
values of A and b will differ slightly depending on the homologous series used. The present correlation gives good results for the temperature range between the normal melting point and the normal boiling point of a substance. Beyond this region the pressure contribution would be greater and would likely have to be factored into the correlation. Results and Discussion Table 1 contains a summary of the results. The analysis shows that over 83% of the 611 data points tested have a deviation less than 5%, with the average deviation being approximately 2%. Since the correlation was developed using only the alkane data, it gives excellent results for broad types of C-H compounds including alkanes, alkenes, aromatic hydrocarbons, and cyclic hydrocarbons, whereas the predicted thermal conductivities of C-OH compounds such as alcohols show slightly higher errors. This could, in part, be due to the occurrence of hydrogen bonding. The data set also includes other hydrogen-bonding substances, highly polar compounds, and halogen-substituted compounds besides a variety of aromatics and alcohols. The results from this new correlation were compared to the values calculated using the Arikol and Gurbuz method (1992). This method was chosen in part because it is a recent correlation and in part because the authors report fairly good results. As can be seen in Table 2, in most cases the average and maximum deviations are lower for the new correlation compared to the results calculated using the Arikol and Gurbuz method. Overall, the two methods show similar results. However, the correlation of Arikol and Gurbuz is based on homologous series and, in addition, it has more adjustable parameters. Table 3 shows a comparison of the present method with the correlations of Latini et al., Sato and Riedel, and Missenard and Riedel. These methods were compared by Reid et al. (1987), in their monograph The Properties of Gases and Liquids, as acceptable correlations to predict the thermal conductivity of liquids. The results in Table 3 show that the new method gives better results except for propane at 323 K. This temperature is above the boiling point of propane, and prediction methods should be restricted to temperatures at or below the boiling point unless satisfactory pressure corrections are developed particularly for low-boiling substances. The present method has other advantages. As pointed out by Reid et al., the predicted data for certain compounds, such as cresols, depend on whether they are treated as aromatics or alcohols. The present
correlation does not depend on this type of judgment as the input data for a particular substance is unique and the method, at present, does not depend on the homologous series. Another advantage is that several methods depend on parameters such as critical properties, boiling point, and acentric factor, and these properties do not show appreciable differences for isomers. On the other hand, molar polarization changes appreciably among isomers as both molecular structure and dipole moment are involved in its evaluation. This allows for better characterization of the substances involved in the correlation and hence more accurate prediction of thermal conductivity data. The present correlation has not taken advantage of the temperature function in the definition of molar polarization. This will be incorporated in future work, and the results will be published at a later date. As molar polarization is characteristic of the structure of the molecules, a study of different transport properties of liquids and liquid mixtures using this parameter would increase the understanding of the equilibrium and nonequilibrium operative in liquids and liquid mixtures. Figure 2 shows that the value of b is fairly constant at approximately 2/3 and that A increases slightly but is approximately unity. These values for A and b were used in the correlation with good results (see Table 4). This means that the thermal conductivity of a substance can be found at various temperatures just by knowing the thermal conductivity at a reference temperature. This makes predicting thermal conductivity especially easy. In conclusion, the proposed correlation for thermal conductivity not only has a strong theoretical basis but also predicts thermal conductivity values with better accuracy. The method will be tested more exhaustively and applied to liquid mixtures. Literature Cited Arikol, M.; Gurbuz, H. A New Method for Predicting Thermal Conductivity of Pure Organic Liquids and Their Mixtures. Can. J. Chem. Eng. 1992, 70, 1157. Baroncini, C.; Di Filippo, P.; Latini, G.; Pacetti, M. Thermal Conductivity of Liquids: Comparison of predicted values with experimental results at different temperatures. High Temp.High Pressures 1979, 11, 581. Debye, P. Polar Molecules; The Chemical Catalog Co., Inc.: New York, 1929. Horrocks, J. K.; McLaughlin, E. Thermal Conductivity of Simple Molecules in the Condensed State. Trans. Faraday Soc. 1960, 56, 206. Horrocks, J. K.; McLaughlin, E. Temperature Dependence of the Thermal Conductivity of Liquids. Trans. Faraday Soc. 1963, 59, 1709.
2068 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Jamieson, D. T.; Tudhope, J. S. A Simple Device for Measuring the Thermal Conductivity of Liquids with Moderate Accuracy; NEL Report No. 81; National Engineering Laboratory: East Kilbride, Glasgow, U.K., 1963. Miller, J.; Joseph, W.; McGinley, J. J.; Yaws, C. L. Thermal Conductivity of Liquids. Chem. Eng. 1976, 133. Nagvekar, M. A Group Contribution Method for Liquid Thermal Conductivity. M.S. Thesis, The Pennsylvania State University, State College, PA, 1984. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill Book Co.: New York, 1987. Vargaftik, N. B. Tables on Thermophysical Properties of Gases and Liquids; Hemisphere Publishing Co.: Washington, DC, 1975. Viswanath, D. S.; Rao, M. B. Thermal conductivity of liquids and its temperature dependence. J. Phys. D: Appl. Phys. 1970, 1444.
Viswanath, D. S.; Prasad, D. H. Generalized Thermodynamic Properties of Real Fluids Using Molar Polarization at the Critical Temperature as the Third Parameter; Department of Chemical Engineering, Indian Institute of Science: Bangalore, India, 1974. Viswanath, D. S.; Prasad, D. H. A New Three Parameter Law of Corresponding States. Presented at the AIChE National Meeting, Houston, TX, 1981.
Received for review September 22, 1997 Revised manuscript received January 29, 1998 Accepted February 17, 1998 IE9706830
