Ind. Eng. Chem. Res. 1996, 35, 2453-2459
2453
Extension of the Rough Hard-Sphere Theory for Transport Properties to Polar Liquids J. G. Bleazard and A. S. Teja* Fluid Properties Research Laboratory, School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
A simple extension of the rough hard-sphere theory is proposed to describe the thermal conductivity and viscosity of a wide variety of polar compounds. The method is based on the known transport properties of a smooth hard-sphere system together with a temperaturedependent hard-core volume and coupling parameters to account for deviations from true smooth hard-sphere behavior. A key advantage of the method is the ability to simultaneously correlate self-diffusion, viscosity, and thermal conductivity using a common characteristic volume V0 for each compound. Thus, information about V0 obtained from one transport property can be applied to calculate the other properties. The model is also well suited for extension to temperature and pressure conditions outside those at which experimental data are available. Results for 58 polar liquids in the temperature range from 293 to 493 K are presented with a maximum error of 5.0% for viscosity and 2.9% for thermal conductivity. Model parameters for a homologous series of compounds can be expressed as smooth functions of the size, allowing the prediction of the transport properties for compounds for which data are presently not available. Introduction Recently, several researchers (Assael et al., 1989; DiGuilio and Teja, 1992; Dymond and Awan, 1989) have adapted the hard-sphere theory to correlate and predict the transport properties of real fluids. These semiempirical schemes have provided a means of accurately representing the thermal conductivity λ, viscosity η, and self-diffusion D as a function of the liquid density. Also, if the pressure dependence of the liquid density is known, then these methods are able to represent the pressure dependence of the transport properties (Wakeham, 1989; Wakeham et al., 1990; Assael et al., 1990, 1994a). This is especially useful in industrial applications in which the transport properties may be required at pressures other than those at which experimental data are available. However, methods based on the rough hard-sphere theory are currently limited to nonpolar liquids at temperatures where experimental data are available (temperatures typically less than 373 K). The purpose of the present work was to extend the rough hard-sphere approach to polar liquids over an extended temperature range. The model was tested on 58 polar liquids, including hydrogen-bonded liquids, over a wide range of conditions. The parameters of the model were correlated with the size of the molecules in the case of a homologous series of compounds including diols, carboxylic acids, ethanoates, and poly(ethylene glycols). This facilitates the prediction of the thermal conductivity and viscosity of members of the homologous series when these properties are not available. Rough Hard-Sphere Model The origins of the hard-sphere approach lie in the van der Waals model in which the true intermolecular pair potential is replaced with a weak long-range attraction and a rigid-core repulsion. Dymond (1985) showed that for the transport properties, the van der Waals theory is equivalent to the hard-sphere theory, provided that the hard-sphere diameter is allowed to decrease with increasing temperature to account for the soft repulsive energy of real systems. To represent the transport properties of real liquids, a rough hard-sphere adaptation of the smooth hardS0888-5885(95)00758-5 CCC: $12.00
Table 1. Coefficients aDi, aηi, and aλi for Equations 13-15 i
aDi
aηi
aλi
0 1 2 3 4 5 6 7
3.330 76 -31.742 61 133.047 2 -285.191 4 298.141 3 -125.247 2
1.094 5 -9.263 24 71.038 5 -301.901 2 797.690 0 -1221.977 0 987.557 4 -319.463 6
1.0655 -3.538 12.120 -12.469 4.562
sphere model was proposed by Chandler (1975), who showed that, at sufficiently high densities, the transport coefficients of real fluids can be approximated by those of rough hard spheres (RHS). The rough hard-sphere coefficients, in turn, are proportional to the transport coefficients of smooth hard spheres (SHS) as follows:
D ≈ DRHS ≈ ADSHS
(1)
η ≈ ηRHS ≈ CηSHS
(2)
Here A and C are factors to account for the coupling between translational and rotational motions. The coupling factors are unity for smooth hard spheres. Chandler concluded that the coupling between translational and rotational motions has the effect of reducing the value of the self-diffusion of a rough hard sphere (A e 1) relative to that of a smooth hard sphere and increasing the value of the viscosity (C g 1). The above ideas of the rough hard-sphere model were extended to all the transport properties for real dense fluids (Li et al., 1985, 1986) as follows:
D* ) RDD*SHS
(3)
η* ) Rηη*SHS
(4)
λ* ) Rλλ*SHS
(5)
where RD, Rη, and Rλ account for the nonspherical shape and translational-rotational coupling. The superscript * represents the property in reduced form as determined © 1996 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996
Table 2. Constants of the Modified Rackett Equation (Equation 18) substance
Mw
Tc, K Diols 626 658 621.8 643 667 612.6 673 670 638.2 725.6 723.6 746.6 665.6 769.8
A, kg/m3
B
C
82.942 57 82.527 2 13.285 60.657 16.382 199.916 4.193 95 2.987 68 1.977 51 103.948 0 24.790 52 0.064 238 197.148 8 0.069 604
0.261 53 0.262 60 0.106 73 0.227 90 0.119 10 0.407 51 0.061 148 0.052 025 0.042 025 0.303 72 0.150 68 0.007 702 0.423 21 0.008 024
0.196 43 0.171 88 0.109 42 0.173 91 0.106 49 0.370 89 0.080 609 0.076 457 0.080 332 0.337 02 0.153 83 0.061 598 0.417 78 0.067 274
180.22 250.905
0.365 47 0.442 12
0.417 21 0.616 58
93.299 19 69.699 22 435.755 9 211.768 6 179.747 0 189.802 5 159.918 2
0.285 57 0.251 31 0.630 65 0.455 79 0.410 42 0.436 36 0.395 24
0.276 08 0.238 76 0.846 08 0.694 18 0.527 89 0.564 39 0.488 41
1,2-propanediol 1,3-propanediol 1,2-butanediol 1,3-butanediol 1,4-butanediol 2,3-butanediol 1,5-pentanediol 1,6-hexanediol 2,5-hexanediol 1,2-octanediol 1,8-octanediol 1,9-nonanediol 1,2-decanediol 1,10-decanediol
76.095 76.095 90.122 90.122 90.122 90.122 104.14 118.18 118.18 146.23 146.23 160.26 174.28 174.28
dimethyl disulfide diethyl disulfide
94.19 122.24
Disulfides 605 642.1
diethylenetriamine ethylenediamine N,N-dimethylethylenediamine triethylamine N,N-diethylethylenediamine N,N-diisopropylethylamine N,N,N′,N′-tetraethylenediamine
103.17 60.10 88.15 101.19 116.21 129.25 116.21
Amines 677 593 409.6 530.4 600.7 572.6 570.4
acetic acid propionic acid heptanoic acid octanoic acid nonanoic acid
60.05 74.08 130.19 144.22 158.24
Carboxylic Acids 592.9 601.3 663.1 677.1 689.8
238.810 209.304 238.723 4 210.243 2 219.214
0.417 91 0.402 55 0.453 10 0.429 70 0.440 31
0.526 86 0.508 11 0.607 33 0.543 14 0.571 03
2-methoxyethanol 2-(2-methoxyethoxy)ethanol 2-[2-(2-methoxyethoxy)ethoxy]ethanol 1-methoxy-2-propanol dipropylene glycol methyl ether dipropylene glycol ether ethylene glycol dimethyl ether 2-methoxyethyl ether triethylene glycol dimethyl ether tetraethylene glycol dimethyl ether 2-ethoxyethanol 2-(2-ethoxyethoxy)ethanol
76.10 120.15 164.20 90.12 148.20 134.18 90.12 134.18 178.23 222.28 90.12 134.18
Alcohol-Ethers 565.5 630.8 677.2 551 658 700 536 613.8 659.3 713.7 571.7 637
81.594 46 363.532 2 168.832 7 181.205 4 185.456 0 82.897 03 93.010 57 156.878 8 197.917 8 215.936 118.071 5 162.752 7
0.263 99 0.526 27 0.365 21 0.394 02 0.391 19 0.260 86 0.291 10 0.358 80 0.396 91 0.410 01 0.319 81 0.364 49
0.214 85 0.787 32 0.367 04 0.378 23 0.505 03 0.260 60 0.265 97 0.439 95 0.516 88 0.589 27 0.292 54 0.389 10
2-cyanopyridine 3,5-dichloropyridine
104.11 147.99
Pyridines 720 721.5
285.388 4 43.423 67
0.456 41 0.161 20
0.658 39 0.212 97
butyl ethanoate pentyl ethanoate hexyl ethanoate octyl ethanoate
116.16 130.19 144.21 172.27
Ethanoates 570.9 593.3 613 649.6
137.842 7 149.22 136.867 0 146.832 9
0.345 01 0.361 27 0.349 75 0.364 05
0.409 49 0.440 00 0.413 62 0.456 77
spheres given by
by the following:
[ ][ ] [ ][ ] [ ][ ]
nD V D* ) n0D0 V0
2/3
η V η* ) η0 V0
2/3
λ* )
λ V λ0 V0
[ ] [ ] [ ]
M ) 5.030 × 10 RT 8
1/2
1 ) 6.035 × 10 MRT
2/3
8
) 1.936 × 107
M RT
DV
-1/3
1/2
1/2
2/3
(6)
ηV
(7)
λV2/3
(8)
where n is the number density, V is the molar volume, M is the molecular weight, R is the gas constant, and T is the temperature (all properties in SI units). D0, η0, and λ0 refer to the transport properties of a dilute gas obtained from classical kinetic theory (McQuarrie, 1975). V0 is the volume of close packing of
V0 )
NAdHS3
x2
(9)
where NA is Avogadro’s number and dHS is the hardsphere diameter. It can be shown (Dymond, 1985) that in the case of smooth hard spheres, the reduced transport properties are functions only of the ratio of the molar volume of the system to the close-packed volume (V/V0) as follows:
D*SHS ) fD(V/V0)
(10)
η*SHS ) fη(V/V0)
(11)
λ*SHS ) fλ(V/V0)
(12)
For smooth hard spheres, expressions for the coef-
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2455
Figure 1. Correlation of viscosity with RHS model (solid line). Diols (0), disulfides (2), alcohol-ethers (*), amines (]), carboxylic acids (b), pyridines (9), ethanoates (×), ethanolamines (O), and poly(ethylene glycols) (+).
Figure 3. V0 of the diols with terminal OH groups. 1,3Propanediol (]), 1,4-butanediol (9), 1,5-pentanediol (4), 1,6hexanediol (×), 1,8-octanediol (O), 1,9-nonanediol (0), and 1,10decanediol (b).
fusion of each individual molecule and averaging the results. As a result of this limitation, Dymond (1987) determined accurate corrections to the Enskog theory for viscosity and thermal conductivity from experimental data of methane and argon, both considered to be smooth hard-sphere systems. Assael, Dymond, and their collaborators (1992a-e, 1994a,b) used the results from the corrected Enskog theory to determine the universal functions fD(V/V0), fη(V/V0), and fλ(V/V0). Experimental data for the n-alkane transport coefficients were used to extend the curves below V/V0 ) 1.5. The resulting universal curves were then expressed as follows: 5
log(fD(V/V0)) ) Figure 2. Correlation of thermal conductivity with RHS model (solid line). Diols (0), disulfides (2), alcohol-ethers (*), amines (]), carboxylic acids (b), pyridines (9), ethanoates (×), ethanolamines (O), and poly(ethylene glycols) (+).
7
log(fη(V/V0)) )
Table 3. References of Experimentally Measured Density G, Viscosity η, and Thermal Conductivity λ substance diols carboxylic acids ethanolamines poly(ethylene glycols)
property
ref
F, η F, η, λ F λ η, λ F, η λ F η λ
Sun et al., 1992 Bleazard et al., 1995 Sun et al., 1995 Bleazard and Teja, 1995 Bleazard et al., 1996 DiGuilio et al., 1992a DiGuilio et al., 1992b Tawfik and Teja, 1989 Lee and Teja, 1990 DiGuilio and Teja, 1990
ficients of thermal conductivity, viscosity, and selfdiffusion are given by the Enskog theory, with corrections determined by molecular dynamics studies (Alder et al., 1970; Easteal et al., 1983). However, the molecular dynamics corrections for thermal conductivity and viscosity involve greater uncertainties than corrections for self-diffusion because of the significantly greater computing time required for the evaluation of these properties for the whole system. Self-diffusion, on the other hand, can be determined by computing the diff-
∑ i)0
aηi
4
log(fλ(V/V0)) )
() () ()
aDi ∑ i)0
∑ i)0
aλi
V0
i
V
V0
i
V
V0 V
(13)
(14)
i
(15)
with the values of the coefficients aDi, aηi, and aλi given in Table 1. The self-diffusion, viscosity, and thermal conductivity of the n-alkanes were correlated using eqs 3-8 and the properties of smooth hard spheres (eqs 13-15). The characteristic volume, V0, and the coupling parameters, RD, Rη, and Rλ, for each n-alkane were determined at any temperature by superimposing plots of the reduced transport properties vs V/V0 onto the universal curves of eqs 13-15. V0 was found from a horizontal shift of the plot, while the R parameter was determined by a vertical shift. Slight adjustments were made until each R was constant and V0 was property independent and decreased smoothly with increasing temperature. Values for V0 and the R parameters were tabulated for each compound at the experimental temperatures and then expressed in equation form. The correlation was very sensitive to changes in V0, so the n-alkanes were split
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Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996
Proposed Scheme for Polar Liquids
Figure 4. Deviation of the model from experimental viscosity for the poly(ethylene glycol). Diethylene glycol (]), triethylene glycol (9), tetrakis(ethylene glycol) (4), pentakis(ethylene glycol) (b), and hexakis(ethylene glycol) (0).
The correlation developed by Assael et al. (1992a-e, 1994a,b) has been tested on several classes of compounds with good success. The purpose of this work was to extend the method to the transport properties of polar compounds which are liquids at room temperature and atmospheric pressure. Another important consideration of this work has been to better determine the temperature dependence of V0. Theory suggests that V0 should decrease with increasing temperature, and in the past, V0 was determined for each compound over a range of temperatures and then fit as a complex function of temperature. One of the goals of this work was to propose a simple equation that would adequately describe V0 over a large range of temperatures and for a wide variety of compounds. Density, viscosity, and thermal conductivity data for 58 polar compounds at atmospheric pressure and over a temperature range of 293-493 K were used to test the method. A linear temperature dependence was proposed for V0 and Rλ, and the parameters were determined directly from the experimental data. The viscosity and thermal conductivity of all 58 compounds could be correctly correlated by allowing Rη to be constant, Rλ to be a linear function of temperature (T), and V0 to be a linear function of 1/T. Thus,
V0 ) a0 +
a1 T
(16)
Rλ ) b0 + b1T
(17)
where T is expressed in kelvin. With a positive value for a1, V0 decreases with increasing temperature, as expected from theory. The density of each compound was measured in our laboratories and was correlated using a modified Rackett-type equation as follows:
F ) A/B[1+(1-Tr)
Figure 5. Deviation of the model from experimental thermal conductivity for the poly(ethylene glycols). Diethylene glycol (]), triethylene glycol (9), tetrakis(ethylene glycol) (4), pentakis(ethylene glycol) (b), and hexakis(ethylene glycol) (0).
into two groups (CH4-C4H10 and C5H12-C16H34) in fitting V0 as a function of temperature and n-alkane carbon number. RD, Rη, and Rλ were each fit as a function of carbon number only. It should be noted that pressure effects of the transport properties were accounted for by incorporating the pressure dependence of the molar volume into the plots of V/V0. Thus, V0 and the R parameters are independent of pressure. The experimental data for all the transport properties could be correlated with a mean-square percentage deviation of better than 3.0%. Assael et al. later used this technique to correlate the transport properties of several simple molecular liquids (1992b), n-alkane mixtures (1992c), aromatic hydrocarbons (1992e), n-alcohols (1994a), and several refrigerants (1994b). The coupling parameters were typically independent of temperature except in the case of some polar liquids where they were allowed to be temperature dependent. For all compounds, V0 was consistent for all transport properties and decreased smoothly with temperature.
C]
(18)
where A, B, and C are constants and Tr (T/Tc) is the reduced temperature. Table 2 contains the constants for eq 18, the critical tempertatures (Tc) used in the correlation, and the molecular weights of the compounds. The critical temperatures were estimated using Joback’s modification of Lyderson’s method as outlined by Reid et al. (1987). Results The 58 compounds measured in our laboratory were divided into 9 groups: diols, disulfides, alcohol-ethers, amines, carboxylic acids, pyridines, ethanoates, ethanolamines, and poly(ethylene glycols). The references for previously measured values of density, viscosity, and thermal conductivity for several compounds are listed in Table 3. The coefficients of the model were determined by simultaneously plotting reduced viscosity and reduced thermal conductivity vs V/V0. Values for a0 and a1 were determined by sliding the plots along the horizontal V/V0 axis to superimpose them onto the functions given for fη(V/V0) and fλ(V/V0), while values for Rη, b0, and b1 were determined by sliding the plots along the vertical axis. All coefficients were adjusted simultaneously to give the least sum of squares errors for both properties. For all compounds studied, a1 was positive, resulting in V0
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2457 Table 4. RHS Parameters and Comparison of Model substance 1,2-propanediol 1,3-propanediol 1,2-butanediol 1,3-butanediol 1,4-butanediol 2,3-butanediol 1,5-pentanediol 1,6-hexanediol 2,5-hexanediol 1,2-octanediol 1,8-octanediol 1,9-nonanediol 1,2-decanediol 1,10-decanediol dimethyl disulfide diethyl disulfide
105a0, 103a1, m3 mol-1 m3 K mol-1
Diols 1.394 9 1.043 6 1.848 5 1.491 8 1.229 0 1.787 6 1.163 2 1.204 2 2.546 0 1.413 2 0.944 95 1.328 9 1.685 3 1.389 1
4.9204 7.1660
2.720 6 3.931 5
Disulfides 1.156 6 1.116 9
diethylenetriamine ethylenediamine N,N-dimethylethylenediamine triethylamine N,N-diethylethylenediamine N,N-diisopropylethylamine N,N,N′,N′-tetramethylethylenediamine
4.3195 2.6581 6.2366 2.4125 7.8300 8.9248 7.7839
butyl ethanoate pentyl ethanoate hexyl ethanoate octyl ethanoate monoethanolamine diethanolamine triethanolamine N,N-dimethylethanolamine N,N-diethylethanolamine N-methyldiethanolamine N-ethyldiethanolamine diethylene glycol triethylene glycol tetrakis(ethylene glycol) pentakis(ethylene glycol) hexakis(ethylene glycol)
-0.223 34 0.224 49 -0.560 52 -0.220 80 0.127 41 -0.661 93 0.240 23 0.361 78 -0.925 66 0.283 40 0.922 93 0.722 47 0.717 62 1.047 9
8.186 2 6.207 8 12.313 11.021 8.542 7 12.572 10.351 12.029 21.435 15.832 13.598 17.484 19.444 18.889
4.6070 8.0562 8.7401 6.1452 8.4117 4.9296 4.4094 8.8955 10.380 13.619 5.2641 7.8791
2-cyanopyridine 3,5-dichloropyridine
b0
3.5094 4.1408 3.5390 4.1627 4.8808 3.4702 5.8673 6.8481 4.0811 8.3923 9.4658 9.6742 10.252 10.730
2-methoxyethanol 2-(2-methoxyethoxy)ethanol 2-[2-(2-methoxyethoxy)ethoxy]ethanol 1-methoxy-2-propanol dipropylene glycol methyl ether dipropylene glycol ether ethylene glycol dimethyl ether 2-methoxyethyl ether triethylene glycol dimethyl ether tetraethylene glycol dimethyl ether 2-ethoxyethanol 2-(2-ethoxyethoxy)ethanol
acetic acid propionic acid heptanoic acid octanoic acid nonanoic acid
Rη
4.0255 4.9475 9.0030 10.120 11.153
1.783 0 1.924 1
Alcohol-Ethers 3.513 8 0.966 84 0.859 15 3.451 6 0.779 73 1.444 6 10.261 1.676 6 1.389 1 5.964 8 2.539 0 0.370 08 10.272 0.424 11 0.830 13 18.974 2.318 9 -0.976 77 5.096 5 1.786 0 2.193 3 3.596 4 0.976 79 2.365 9 8.027 2 1.587 8 2.418 3 8.812 8 1.536 8 3.259 8 5.368 2 1.327 7 0.865 11 7.387 5 1.422 0 1.175 2 11.770 5.920 9 3.868 2 14.798 6.421 8 7.431 8 6.063 8
Amines 2.123 7 1.818 4 1.042 0 2.4782 1.171 1 1.345 9 1.344 8
0.015 709 0.232 74 1.182 6 0.522 71 1.485 2 1.742 8 1.583 8
Carboxylic Acids 0.321 83 0.761 74 0.590 26 1.026 4 0.977 54 0.804 94 5.526 9 1.444 8 1.490 2 6.577 3 1.493 1 1.699 0 7.881 9 1.562 4 1.827 8
viscosity thermal cond 103b1, K-1 AAD % MAD % AAD % MAD % 4.325 0 3.143 9 5.943 3 4.615 5 3.973 8 6.030 5 4.155 8 4.482 0 7.445 2 5.753 9 4.186 1 5.692 9 5.966 4 5.543 2
1.35 1.42 1.78 1.20 0.83 1.70 0.80 0.27 2.57 0.44 1.12 0.22 0.37 0.35
2.53 4.02 4.51 2.27 1.84 4.34 1.91 0.61 5.01 1.10 1.93 0.48 0.70 0.89
0.12 0.21 0.39 0.46 0.14 0.64 0.20 0.18 0.95 0.21 0.54 0.51 0.41 0.38
0.35 0.71 0.98 0.98 0.44 1.17 0.39 0.28 2.92 0.41 1.30 1.22 0.76 0.79
0.031 951 0.519 05
0.14 0.41
0.49 1.00
0.16 0.16
0.32 0.28
1.934 1 1.660 2 4.785 1 2.552 2 5.014 8 8.784 3 1.031 0 0.843 06 3.909 0 3.011 7 2.620 2 3.720 4
0.06 0.95 0.40 1.27 0.26 2.26 0.25 0.38 0.33 0.62 0.77 0.24
0.12 2.91 1.03 3.79 0.55 3.69 0.58 1.00 0.66 1.59 1.57 0.63
0.29 1.09 0.52 0.43 0.10 0.14 0.11 0.15 0.21 0.27 0.25 0.29
0.71 1.68 0.67 1.08 0.27 0.32 0.29 0.45 0.31 0.46 0.78 0.46
7.740 7 5.487 4 1.876 7 8.417 8 2.615 3 1.991 8 2.194 3
0.65 0.34 0.08 0.60 0.21 0.83 0.35
1.07 0.87 0.34 1.00 0.47 1.32 0.80
0.84 0.27 0.18 0.37 0.13 0.22 0.16
1.52 0.55 0.36 0.55 0.21 0.46 0.33
1.122 0 1.268 6 1.868 4 1.994 2 2.319 2
1.53 0.45 0.58 0.59 0.80
3.32 1.58 1.20 1.11 1.35
0.66 0.45 0.32 0.22 0.14
1.36 0.74 1.22 0.43 0.23
5.6965 6.2612
3.745 0 4.746 9
Pyridines 1.322 4 1.074 1
1.442 4 1.550 6
0.977 87 0.578 92
0.51 0.62
1.20 1.31
0.20 0.18
0.30 0.33
7.9584 9.1140 9.9551 12.011
3.045 8 3.695 1 5.247 3 6.981 1
Ethanoates 1.104 4 1.158 9 1.223 9 1.518 8
1.903 4 2.206 8 2.187 0 2.630 3
0.986 83 0.628 08 1.440 5 1.763 0
0.13 0.63 0.25 0.43
0.33 1.17 0.62 1.02
0.16 0.25 0.27 0.36
0.35 0.38 0.70 0.73
3.2106 3.8785 4.8376 5.9300 7.0374 6.0171 5.3479
Ethanolamines 5.290 4 1.179 2 0.107 22 14.632 2.379 8 -0.612 66 21.171 4.496 8 -1.402 0 5.508 0 0.977 84 0.441 21 9.843 8 1.136 5 0.527 20 12.375 1.809 0 -0.201 08 12.620 1.865 6 -0.100 76
3.322 3 6.669 8 10.963 2.635 6 4.036 2 5.767 8 5.163 4
0.83 0.84 2.40 1.53 0.70 1.78 1.72
1.17 1.92 3.34 2.87 1.80 2.95 2.92
0.51 0.21 0.27 0.59 0.61 0.08 0.13
1.11 0.39 0.70 1.51 1.43 0.15 0.21
4.7800 6.6584 8.0775 9.7329 10.606
Poly(ethylene glycols) 9.033 0 1.828 6 -0.048 89 13.318 2.201 9 0.216 76 18.905 2.836 0 0.232 77 23.805 3.469 0 0.244 84 30.860 4.585 0 0.269 87
5.690 3 7.125 0 9.570 8 12.172 15.084
1.28 1.29 1.51 1.47 1.82
2.08 2.21 2.48 2.37 3.12
0.78 0.61 0.61 0.53 0.92
1.94 1.25 1.30 0.88 2.21
decreasing with increasing temperature. The correlation was capable of correctly describing the experimental viscosity data (565 data points total) with an AAD of 0.76% and MAD of 5.0%, while thermal conductivity
(425 data points total) could be correlated with an AAD of 0.32% and MAD of 2.9%. Plots of the reduced viscosity vs reduced volume and reduced thermal conductivity vs reduced volume are shown in Figures 1 and
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Table 5. Coefficients of Equation 19 for Homologous Groups of Compounds homologous group
F(Cn)
f0
f1
diols with terminal OH groups
a0, m3 mol-1 a1, m3 K mol-1 Rη b0 b1, K-1 a0, m3 mol-1 a1, m3 K mol-1 Rη b0 b1, K-1 a0, m3 mol-1 a1, m3 K mol-1 Rη b0 b1, K-1 a0, m3 mol-1 a1, m3 K mol-1 Rη b0 b1, K-1
1.168 97 × 10-5 1.182 71 × 10-3 1.088 60 0.117 177 2.431 42 × 10-3 2.169 06 × 10-5 -1.434 31 × 10-3 0.336 078 0.199 373 1.005 01 × 10-3 4.008 70 × 10-5 -1.117 93 × 10-3 1.050 01 1.257 59 -1.902 76 × 10-4 3.003 58 × 10-6 2.380 61 × 10-3 1.756 05 -0.547 653 3.454 77 × 10-3
9.493 87 × 10-6 1.794 82 × 10-3 -6.111 57 × 10-4 -0.026 7247 3.317 97 × 10-4 8.984 62 × 10-6 7.766 05 × 10-4 0.246 837 0.205 101 4.796 21 × 10-5 1.000 18 × 10-5 1.019 18 × 10-3 -0.036 4169 0.169 387 2.426 7 × 10-4 2.490 36 × 10-5 2.641 20 × 10-3 -0.168 659 0.329 959 6.413 50 × 10-4
carboxylic acids
ethanoates
poly(ethylene glycols)
Table 6. Comparison of the Correlation Using Equation 19 with Experimental Data viscosity substance
thermal conductivity
2.38 0.93 1.90 1.71 3.36 0.58 1.03 5.08 6.09 1.74 1.13 1.47 0.59 1.44 1.23 0.50 1.29 1.37 1.51 1.53 1.78
5.28 3.35 5.49 2.70 9.33 1.17 1.59 10.40 13.74 3.51 2.66 3.69 0.955 2.53 2.23 1.27 2.29 2.81 3.52 2.84 3.24
0.53 0.45 0.30 0.64 1.06 0.71 1.15 0.74 0.69 0.34 0.47 0.24 1.12 0.96 0.59 0.52 0.82 0.67 0.60 0.57 0.94
1.72 1.33 0.60 0.83 1.51 1.17 2.04 1.65 1.25 1.42 0.86 0.46 1.55 1.68 1.27 1.15 1.38 1.81 1.81 0.96 2.46
2. Table 4 contains the values of the coefficients obtained for each of the compounds as well as the AAD and MAD. During the data analysis, it was observed that the slope of the reduced viscosity vs reduced volume V/V0 obtained from eq 14 decreases below a V/V0 of 1.1 and finally goes through a maximum near a reduced volume of 1. Thus, the equation was found to underpredict viscosity in this region of reduced volume. Three compounds (diethanolamine, triethanolamine, and 2,5hexanediol) each had one viscosity data point at a reduced volume below 1.1 and were therefore omitted from the data analysis. More experimental data below a reduced volume of 1.1 would be required to obtain an improved set of constants for eq 14. The parameters of the model were studied to determine if trends could be observed for groups of compounds. It was determined that the parameters for the homologous series of compounds could be expressed as a function of carbon chain length. The viscosity and thermal conductivity of four groups of homologous compounds were correlated: (1) diols with OH groups on the carbon chain ends; (2) carboxylic acids; (3) ethanoates; and (4) poly(ethylene glycols). V0 (a0 and
3.008 71 × 10-3 0.011 5494 1.134 34 × 10-7 2.890 93 × 10-5 -0.012 4858 -2.577 48 × 10-3 1.043 19 × 10-5 0.011 6387 -1.273 14 × 10-6 3.466 08 × 10-4 0.105 694 -0.032 5106 2.163 46 × 10-4
a1), Rη, and Rλ (b0 and b1) for each homologous group of compounds were expressed as functions of the number of carbon units Cn as follows:
F(Cn) ) f0 + f1Cn + f2Cn2
Cn AAD % MAD % AAD % MAD %
1,3-propanediol 3 1,4-butanediol 4 1,5-pentanediol 5 1,6-hexanediol 6 1,8-octanediol 8 1,9-nonanediol 9 1,10-decanediol 10 acetic acid 2 propionic acid 3 heptanoic acid 7 octanoic acid 8 nonanoic acid 9 butyl ethanoate 4 pentyl ethanoate 5 hexyl ethanoate 6 octyl ethanoate 8 diethylene glycol 2 triethylene glycol 3 tetrakis(ethylene glycol) 4 pentakis(ethylene glycol) 5 hexakis(ethylene glycol) 6
f2
(19)
Table 5 contains the values of fi for each of the four homologous groups, and Table 6 contains the comparison with experimental data for all compounds. Figure 3 is a plot of V0 as a function of temperature and carbon number for the diols with terminal OH groups. The V0’s of all homologous groups displayed similar trends with respect to temperature and molecular size. The method was capable of describing the viscosity of the four groups with an AAD of 1.8%, while the AAD for the thermal conductivity was 0.7%. Examples of deviation plots comparing the model with experimental viscosity and thermal conductivity are shown in Figures 4 and 5, respectively, for the poly(ethylene glycols). Conclusions The rough hard-sphere model has been shown to be capable of simultaneously correlating the transport properties of a wide variety of compounds. The scheme allows the correlation of the transport properties as a function of temperature and pressure by incorporating an equation of state to describe the molar volume. Simple equations for V0 and Rλ have been suggested to describe the temperature dependence of these parameters. Rη was assumed to be constant. The viscosity and thermal conductivity of 58 polar liquids were correlated with average and maximum absolute errors of 2.9% and 5.0%, respectively. It was also shown that for homologous groups of compounds, the parameters can be represented as a function of carbon chain length. This suggests further that the scheme could be extended to a group contribution method. Literature Cited Alder, B. J.; Gass, D. M.; Wainwright, T. E. Studies in molecular dynamics. VIII. The transport coefficients for a hard-sphere fluid. J. Chem. Phys. 1970, 53, 3813-3826. Assael, M. J.; Charitidou, E.; Wakeham, W. A. Thermal conductivity of liquids: Prediction based on a group-contribution scheme. Int. J. Thermophys. 1989, 10, 779-791. Assael, M. J.; Dymond, J. H.; Tselekidou, V. Correlation of highpressure thermal conductivity, viscosity, and diffusion coefficients for n-alkanes. Int. J. Thermophys. 1990, 11, 863-873.
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Received for review December 15, 1995 Accepted February 20, 1996X IE9507585
X Abstract published in Advance ACS Abstracts, June 1, 1996.
