Ind. Eng. Chem. Res. 1999, 38, 4513-4519
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A Group Contribution Method for the Prediction of Thermal Conductivity of Liquids and Its Application to the Prandtl Number for Vegetable Oils Christopher M. Rodenbush, Dabir S. Viswanath,* and Fu-hung Hsieh† Department of Chemical Engineering, University of MissourisColumbia, Columbia, Missouri 65211
Data on thermal conductivity of liquids, as a function of temperature, are essential in the design of heat- and mass-transfer equipment. A number of correlations have been developed to predict thermal conductivity of liquids with limited success. Among the correlations proposed so far, only the correlation due to Nagvekar and Daubert (Ind. Eng. Chem. Res. 1987, 26, 1362) is based on group contributions. In this paper, a new group contribution method is developed based on the Klaas and Viswanath (Ind. Eng. Chem. Res. 1998, 37, 2064) method for prediction of thermal conductivity of liquids and the results are compared to the method of Nagvekar and Daubert and other existing correlations. The present method predicts thermal conductivity of some 228 liquids that encompass 1487 experimental data points with an average absolute deviation of 2.5%. The group contribution method is used to examine the temperature dependence of Prandtl number for vegetable oils. Introduction Thermal conductivity of liquids is an important transport property in engineering design. It is required to evaluate dimensionless numbers, such as Prandtl, Lewis, Fourier, and others, that are used in heat- and mass-transfer calculations. Design engineers may tend to feel that the temperature dependence of the different properties, such as thermal conductivity, that enter into such dimensionless numbers tend to cancel, and the effect of temperature on these groups is not very significant. However, as will be shown later with Prandtl number as an example, some of these dimensionless numbers could change severalfold over a range of temperatures. Prandtl number changes approximately 16-fold over a temperature range of 300-550 K for vegetable oils with the corresponding overall heattransfer coefficient changing 4-fold. Experimental determination of thermal conductivity is difficult, time-consuming, and expensive. Further, in order to find the feasibility of any model, process, or procedure, where thermal conductivity enters as a physical property, it is necessary to estimate thermal conductivity quickly with some amount of accuracy. Apart from pure substances, engineers almost always encounter mixtures, and experimental determination of mixture properties adds an additional factor of difficulty and error. The primary focus of the following discussion is on pure substances, though the results could possibly extend to mixtures, with vegetable oils used as an example in this paper. A number of correlations have been developed for estimating thermal conductivity of liquids. Some of the most recent and accurate methods are due to Robbins and Kingrea,1 Missenard,2 Viswanath and co-workers,3-5 * Corresponding author: (telephone) (573) 884-0707; (fax) (573) 884-4940; (e-mail)
[email protected]. † Department of Biological and Agricultural Engineering, University of MissourisColumbia, Columbia, MO 65211.
Baroncini et al.,6-8 Nagvekar and Daubert,9 and Klaas and Viswanath.10 Of these methods, Viswanath and coworkers based their methods on the hole theory of Eyring11 and the vibrational theory of Horrocks and McLaughlin.12,13 Baroncini et al. developed very complicated temperature-dependent functions to correlate thermal conductivity. Nagvekar and Daubert correlated the thermal conductivity of liquids to a temperaturedependent function and presented a group contribution method to evaluate the two constants in the function. Klaas and Viswanath10 proposed a semitheoretical method to predict the thermal conductivity of liquids. They reported results of better accuracy when compared to other notable methods, such as recommended by Reid et al.,14 for the temperature range between the normal melting and boiling points of each substance. Klaas15 showed that the method developed has a sound theoretical basis and only needs two parameters. Klaas further demonstrated that it is possible to use the method to predict thermal conductivity of a liquid, at various temperatures, with the only input parameter required being a reference point for the liquid. The group contribution method developed in this paper uses the method of Klaas and Viswanath10 and the theory behind it. Although there are group contribution methods for other properties such as viscosity, there have been no group contribution methods for thermal conductivity except for the method of Nagvekar and Daubert.9 Nagvekar and Daubert’s method is fairly complicated and has separate contributions for C-H in alkanes, C-H in alkenes, etc., and at the same time does not differentiate among some compounds such as, for example, glycols and phenols. The results of the new proposed method are compared with the group contribution method developed by Nagvekar and Daubert and other existing correlations. The group contribution method developed in this paper is used to predict thermal conductivity of vegetable oils, which are essentially mixtures of fatty acids. The impact of the thermal conductivity property on
10.1021/ie990320v CCC: $18.00 © 1999 American Chemical Society Published on Web 09/25/1999
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Ind. Eng. Chem. Res., Vol. 38, No. 11, 1999
dimensionless numbers was considered by examining Prandtl number as a function of temperature for vegetable oils. Group Contribution Method Development The relation given by Horrocks and McLaughlin12,13 for thermal conductivity of a liquid based on vibrational theory 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. Klaas and Viswanath10 used eq 1 to develop the following correlation for the prediction of thermal conductivity of liquids:
(λ/λo) ) A(T/To)-b
(2)
where λo is the value of thermal conductivity at To while A and b are constants for a given liquid substance. Theory predicted the value of A to be unity and b to be 2/ . Klaas15 by fitting experimental data found that the 3 value of A ranged from 0.98 to 1.10 whereas the value of b ranged from 0.60 to 0.68. In an attempt to generalize this correlation, Klaas correlated A and b with molar polarization which is defined as
Pmc ) Rm + 4πNµ2/9kTc
(3)
where Pmc is molar polarization at the critical temperature, Rm is molar refraction ) (M/F)[(n2 - 1)/(n2 + 2)], µ is dipole moment (in debye units), N is Avogadro’s number, k is Boltzmann constant, Tc is critical temperature (in K), n is refractive index, M is molecular weight (in g/mol), and F is density (in g/cm3). In the present method, the values of A and b were taken as 1 and 2/3 respectively, and eq 2 is written as
(λ/λo) ) (T/To)-2/3
(4)
Equation 4 is rearranged to give
λ ) DT-2/3
(5)
where D ) λoTo2/3 and the value of D is evaluated by the group contribution technique, which is described in the next section. Group contribution methods assume that a particular property of an element or a group in a molecule is constant, regardless of the nature of the rest of the molecule. Thus, when the structure of the molecule is known, an additive property is estimated by summing up the contributions corresponding to the elements or groups of the molecule. In eq 5, thermal conductivity is a function of temperature and group contribution values are determined for the constant D, which is independent of temperature. This approach is particularly appealing for a property such as liquid thermal conductivity, where the effect of temperature is relatively small and the effect of pressure is negligible. For each substance, the value of D is the summation of the contributions of the various groups that are necessary to construct (comprising) the molecule. If a constant value for D can be determined for each substance, then the thermal conductivity of the liquid at various temperatures can
Figure 1. Total group contribution values for each substance in the n-alkane series.
be predicted from eq 5 by just knowing the structure of the substance. Group Contribution Method Values For group contribution methods, the simplest additivity law is known as the “zeroth”-order law, which is the law of additivity of atomic properties.16 In this law, one assigns partial values for the property in question to each atom in the molecule of which the molecular property is the sum of all the atomic contributions. One of the limitations of this method is that with the conservation of atoms in a chemical reaction this law would predict that any molecular property would also be conserved.16 The next higher approximation in additivity schemes is the additivity of bond properties. The group contribution method in this paper uses this scheme as a basis. Experimental data were gathered for a wide variety of substances from many sources, such as Jamieson and Tudhope17 and Vargaftik.18 For each experimental data point for a particular substance, the value for D in eq 5 was calculated and an average value of D was then determined. For this section, this calculated D is referred to as the value of D for the particular substance. The most basic homologous series of organic liquids is the normal alkanes, which are made of C-H and C-C bonds. Starting with methane, each successive group is constructed with the addition of two C-H bonds and one C-C bond. The value of D is determined from the sum of the contributions of each of these bonds. Figure 1 shows that the value of D is a linear function of the number of carbon atoms in the n-alkane series. This means that a constant value can be obtained for the C-H bond and the C-C bond. The value for the C-H bond was determined by dividing the value of D for methane by 4. The value for the C-C bond was determined by taking the value of D for propane, subtracting 8 times the value of the C-H bond from it, and then dividing the result by 2. Propane was used for this purpose because experimental thermal conductivity data for ethane are not available. Thus, for the n-alkane series
D(n-alkane) ) n(C-C)y(C-C) + n(C-H)y(C-H) (6) where n is the number of bonds and y is the contribution of the bond. The scheme of subtracting known values of bonds from the value of D, of a substance, is the common method used throughout to obtain a value for a previously unknown bond value.
Ind. Eng. Chem. Res., Vol. 38, No. 11, 1999 4515 Table 1. Values of the Constants in Eq 8
Figure 2. Total group contribution values for each substance in the n-alcohol series.
The values of D determined for octane and isooctane are 5778 and 4326, respectively. This large difference in D values shows that it would not be accurate to construct branched alkanes with the bonds and corresponding bond values used for normal alkanes. For branched alkanes, the bonds C-CH< and C-C< were used along with the C-H and C-C bonds of normal alkanes. The value for C-CH< was obtained by subtracting the C-C and C-H contributions from the value of D for isopentane. The value for C-C< was obtained in a similar manner using the value of D for isooctane. For normal alcohols, the contribution of the hydroxyl group bond, C-OH, was determined by subtracting the C-H and C-C contributions from the value of D of the alcohol. The same approach was applied to the following homologous groups: acids, halogen-substituted alkanes, esters, ketones, ethers, and aldehydes. Figure 2 shows the total group contribution of normal alcohols based on the number of carbon atoms in the substance. Ideally, Figure 2 would have displayed a linear trend that showed an increase of the value of D equal to the group contribution value of two C-H bonds and one C-C bond for each additional carbon atom added to the series. This would have allowed for the evaluation of the hydroxyl group contribution and would have given a constant value. However, as shown by Figure 2, the total group contribution of the n-alcohol series can be broken down into linear-type relations as a function of the number of carbon atoms. This allows for the hydroxyl group to have a group contribution value that is a linear function of the number of carbons to which it is attached. This observation was applied to each substituted group of the different homologous series examined. These substituted groups were considered end groups of their bonds, and the number of carbon atoms attached to a particular bond was determined by counting only the number of carbon atoms leading up to that bond. For example, ketones have two C-CdO bonds and the number of carbon atoms to which the two bonds are attached can differ. In the case of alcohols, acids, esters, and aldehydes, it was found that there was an aforementioned linear relationship for the first few carbon atoms and a different linear relationship for the remaining carbon atoms. Therefore different contributions were assigned for C < 3 and C g 3. Branched alcohols and branched acids were treated as their own separate linear functions of the number of carbon atoms attached due to the presence of C-CH< and C-C< bonds For alkenes, the value for the contribution of the Cd C bond was determined by subtracting the values of the C-C and C-H bonds from the value of D of 1-hexene.
oil
A
B
oil
A
B
corn cottonseed linseed olive
3070 3261 2768 3328
-6.40 -6.78 -5.62 -6.96
peanut rapeseed soybean sunflower
3630 3192 3052 3174
-8.03 -6.45 -6.31 -6.63
This substance was chosen for this task because unlike ethylene it follows the expected thermal conductivity trend of the other alkenes examined. For nitrogen-based compounds, the value of the bond of a carbon attached to a particular nitrogen group was determined in most cases by subtracting C-H, C-C, and N-H bond values from the value of D of the particular substance. The bond of N-H was assigned the same value as the C-H bond contribution. The construction of rings and ring attachments was treated separately from their nonring counterparts with the exception of the values for C-H and N-H bonds. As discussed by Benson,16 the principle of additivity schemes rests on the assumption that there is no significant interaction with more distant units. The introduction of a ring brings more distant units into proximity, thus, an expected departure from straight additivity laws. This is the reason for the different treatment of rings. For monocyclic hydrocarbons, at the same temperature the thermal conductivity of cyclopentane is higher than the value for cyclohexane. As a result, the contribution values of the C-C bonds of monocyclic hydrocarbons were treated as a whole and assigned as a function of the number of carbon atoms in the ring, which leads to a decreasing bond value the greater the number of carbon atoms. On the basis of the common method previously discussed, values for bonds attached to the monocyclic hydrocarbons were determined. Aromatic rings were constructed from C-H, C-C, and CdC bonds. The benzene ring was broken down into these bonds to allow for the construction of other unsaturated rings. As a result, one of the values of the contribution of the C-C and CdC bonds was set while the other one was then calculated. The value of the C-C bond was assigned the same value as the C-C bond of alkanes. The value of the CdC bond was then determined by subtracting the C-C and C-H bonds from the value of D of benzene. The values of the bonds attached to the aromatic ring were then determined. Bonds of side chains directly attached to a ring were treated differently compared to bonds not directly or indirectly attached to a ring. A few substances with multisubstituted functional groups were examined. These groups were treated on a different basis than previous treatment of these groups. This was because of the possible increase in the interactions of the molecules with the introduction of the second group. For multisubstituted halogens, the C-F, C-Cl, C-Br, and C-I bonds were assigned constant values. If these substances contained only one carbon atom, a correction was assigned for each of the following situations: CH2, >CH, and >C
