194
Ind. Eng. Chem. Res. 1993,32, 194-199
Thermal Conductivity of Hydrocarbon Mixtures: A Perturbation Approach Alexander Vdsquez and Julio G. Briano* Department of Chemical Engineering, University of Puerto Rico, Mayaguez, Puerto Rico 00680-5000
A model to predict the thermal conductivity of liquid hydrocarbons and their mixtures is presented. The reference fluid is a pure liquid paraffin at zero pressure. Three different factors are added to the model as perturbation from the property of the reference fluid. The first correction accounts for the aromatic character of the liquid, and the second takes into account the effect of pressure. The property of the mixture is expressed as a function of the thermal conductivity of the pure fluid, plus a term accounting for mixing. The last correction term was theoretically derive6 based on a continuous thermodynamic approach. The normal boiling temperature and the Watson parameter are used as characterizing variables. The thermal conductivity of pure substances may be predicted with average deviations below 2% with respect to experimental data. Also the average deviations for data of liquid mixtures were around 3%. Several applications of the model to petroleum fractions and coal-derived liquids show deviations comparable to previous models. Introduction The continuous mixture model (DeDonder, 1931; Aria and Gavalas, 1966; Briano and Glandt, 1983; Cotterman et al., 1985; Kehlen et al., 19851, which considers the composition of a mixture as a continuous distribution function, has been used by several authors to describe the liquid-vapor equilibrium problem in petroleum fractions and related fluids (see, for example, Bowman and Edmister, 1951; Briano, 1983; Cotterman and Prausnitz, 1985). However, there are only a few studies using this model to calculate bulk properties (Melo et al., 1987; Lug0 and Briano, 1991). The calculation of bulk properties generally involves integrations over the whole composition domain; thus the results are mainly affected by the momenta of the distribution and are independent of ita shape. Lug0 and Briano (1991) used these ideas to predict the viscosity of hydrocarbon mixtures and found that the continuous formulation could be applied to discrete mixtures as well as to continuous ones. Several reports are available on the modeling of transport properties for petroleum fractions and coal-derived liquids (Baltatu et al., 1985; Gray et al., 1983; Mallan et al., 1972; Hwang et al., 1981; Twu, 1985; Aasberg-Petersen et al., 1991). Most of these models adopt a single equivalent pure substance to represent the mixture, and the predictions are based on the application of the corresponding states principle. The Model A typical example of a continuous mixture is found in a petroleum fraction. Such liquids can be modeled as a discrete mixture of continuous mixtures of homologous compounds (Briano, 1983). In this case the composition is represented by a bivariate density function p(f,Tb)where one of the distributed variables, f , takes only discrete values representing the families; the other variable, Tb, is a continuous random variable which, for practical applications, is selected to be the normal boiling temperature of the pure substances. For a mixture of F families, p(f,Tb) must be such that
From a distillation curve such as a true boiling point (TBP) analysis, one gets information only on the marginal distribution p(Tb) F
p(Tb)
f= 1
(2)
The other marginal distribution, Xf,which gives the fraction of each family is defined as (3) Thus, the potential information available to us is in the form of the marginal distributions. One Family of Compounds. For a single family of compounds, a bulk property of a continuous mixture can be expressed as a perturbation series expansion in terms of the dispersion of the distribution function. Such series must converge. Since the expansion k wried out in terms of the width of the distribution function, the model is appropriate only for statistically narrow distributions. Briano and Glandt (1983) derived the perturbation using a statistical thermodynamic framework and it can be expressed as M(TP;P(Tb)) = W T P )
+ tlM1(TP) + ...
(4)
In eq 4, M is the property of the mixture, Mo is the property of the pure fluid represented by the mean of the composition function, at the same temperature and pressure of the mixture, 7 is a “small parameter” associated to the variance of the distribution function, and W(T,P) is the first correction term due to mixing. It should be observed that the majority of the models for bulk properties of mixtures represent the property of the system as that of a pure equivalent substance through the use of some mixing rules. Equation 4 can be interpreted as a generalization of such case, where the first term on the right side is the property of an equivalent substance and the second term is a correction. Following the above arguments, the thermal conductivity of a mixture can be written as X(TP;p(Tb))
It is almost impossible to obtain such a composition function because the bivariate character of the distribution is generally “hidden” in the characterization of the mixture.
cPcf,Tb)
Xo(T,OP,O;Tbo)4- ?X’(T,OP,O;Tbo) ( 5 )
where Xo is the thermal conductivity of a pure reference fluid with normal boiling temperature given by Tb0
lTTdJ(Tb)dTt,
0888-588519312632-0194$04.00/00 1993 American Chemical Society
(6)
Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 198 which coincides with the first moment of the distribution. A' is the correction term which only depends on properties of the pure reference fluid, and v is the variance of the composition density function,
v = l,,Tb2P(Tb)dTb - (Tho)'
(7)
Note that in eq 5 the reduced properties of the pure reference fluid are used rather than the temperature and pressure themselves, with T, = T/T: and P,= P/P:. It should be emphasized that the reference fluid is a hypothetical one and does not necessarily represent a real compound. Mixture of Families. To extend the model to the case of mixture of families, a perturbation "across families" is proposed. A continuous variable which gives an ideal of the paraffinic character of the mixture will be used. The classical Watson parameter K , (Watson and Nelson, 1933), which is still used in the characterization of petroleum fractions, was selected: (8) Here, the boiling temperature is expressed in degress Rankine and SGmImis the specific gravity of the liquid. A single mixture of paraffins will be the reference fluid. A second-order perturbation in terms of Kw- K,P is performed to give X(TQ&,;P(Tb)) = Xp(TQ℘P(Tb)) + ai(Kw - Kwp)+ a,(Kw- KWpY(9) In this equation, K,P is the value of the parameter for a pure paraffii with Tb= Tbo.A correlation for this quantity in terms of Tbis given in the Appendix. XP is the property of the paraffinic mixture used as reference and a1and a2 are the correction terms which could depend on pressure, temperature, and normal boiling temperature. Additional cross terms, involving both small parameters ( K , - K,P) and 9, are neglected.
Pure Fluids Low Pressure. To produce a practical model for mixtures according to eq 5, it is essential to develop an equation for predicting the thermal conductivity of pure liquid hydrocarbons. The following equation is proposed for representing the property of liquid paraffins at low pressures: X = A(Tb)[(1 - T,)o.38/T:/6] (10) The form of eq 10 is based on that proposed by Baroncini et al. (1981). Here A(Tb) is characteristic of each compound and must be determined by the use of appropriate data. Several existing models (Kandiyoty et al., 1972; Teja and Rice, 1981; Ogiwara et al., 1980) were used to generate uniformly weighed data. More than 2000 values for about 40 compounds in the whole domain of normal boiling temperature and reduced temperature were used. The fitting procedure showed that a quadratic dependence of A on Tb was a very accurate representation for paraffins. A(Tb) = 00 + Q l T b + U2Tb2 (11) ai)s values are given in the Appendix. When eq 10 was compared against experimental data from the literature, an average absolute deviation of 1.8% was found for about 600 data points. In Figure 1, the experimental and the calculated values for the thermal conductivity of paraffm at low pressures are displayed.
CcicU o l e a
~ o r r n o ic o n d u i t
Y t i
fw,m-~'
Figure 1. Comparison between experimental and calculated thermal conductivities of liquid paraffins at low pressure. Table I. Mean and Maximum Absolute Deviations in the Predicted Thermal Conductivity of Liquid Hydrocarbons at Low Preasure mean max dev dev component N" --(%) (%) rep n-pentane 17 3.47 6.82 6, 7, 11, 12, 18 n-hexane 21 1.30 3.26 6, 7, 11, 18, 20 2-methylpentane 16 1.39 3.03 6, 7, 11, 18 n-heptane 43 1.83 4.89 6, 7, 11, 12, 17, 18, 20 1-octene 9 3.54 5.10 8, 11 2-methylheptane 23 2.96 6.36 11 n-octane 48 2.06 4.63 6, 7, 11, 12, 17, 18, 20 n-nonane 38 1.70 4.24 6, 7, 11, 17, 18 n-decane 36 2.13 4.97 6. 7. 11. 17, 18 n-undecane 27 1.35 3.64 7; ii n-dodecane 32 1.60 3.46 7, 11 n-tridecane 26 1.42 4.48 7, 11 n-tetradecane 40 1.94 4.35 6, 7, 11 n-pentadecane 34 1.68 4.77 7 n-hexadecane 39 1.72 4.51 6, 7 n-heptadecane 34 1.34 4.79 7, 11 n-octadecane 34 1.33 4.76 7, 18 n-nonadecane 27 1.09 3.23 7 n-docosane 34 2.15 5.22 7 n-tricosane 34 2.64 5.10 7 n-tetracosane 27 1.01 2.65 7 cyclohexane 25 1.66 4.59 2, 6, 11 methylcyclohexane 24 4.32 5.93 11 benzene 114 1.98 6.65 2, 6, 10, 13, 17-1 toluene 134 1.42 6.64 2,6, 10, 13, 17-1 l rn-xylene 15 1.43 4.55 11,18 o-xylene 16 2.12 4.65 11,18 p-xylene 11 1.50 3.17 11, 17 ethylbenzene 13 1.83 3.43 11,lS isopropylbenzene 14 2.91 5.18 11,18 n-butylbenzene 7 1.46 3.23 11 tert-butylbenzene 6 2.42 3.88 11
-
total a
1018
1.85
6.82
N is the number of experimental points. bSee Table V.
For nonparaffmic compounds, the perturbation scheme represented by eq 9 is adopted. Thus, the thermal conductivity of pure hydrocarbon liquids is given by
(12)
where T,P = T/TcPand P,P = PIP$. The critical parameters are those of a paraffin with the same normal boiling temperature as the real compound. The critical constants for paraffins are correlated against the normal boiling temperature as described in the Appendix. The correction factors aland a2were found to be constant and obtained by fitting a very large number of experimental points. In Table I, the average and the maximum absolute deviations
196 Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 Table 11. Mean and Maximum Absolute Deviations in the Predicted Thermal Conductivity of Liquid Hydrocarbons at High Pressure mean dev max dev comDonent N (%) (%) ref" 3.11 9.97 3, 11 123 n-butane 3.54 10.90 4, 11 115 n-pentane 1.88 8.97 11 89 l-hexene 11 3.57 8.57 n-hexane 134 11 2.08 9.11 l-heptene 99 11 4.11 9.99 304 n-heptane 11 2.21 9.46 l-octene 100 4.11 10.14 11 n-nonane 125 1.15 3.51 5, 11 154 n-decane 11 1.68 4.08 51 n-dodecane 11 2.68 6.93 66 n-tridecane 11 3.02 6.74 n-tetradecane 30 11 3.76 8.60 n-hexadecane 30 11 1.00 2.80 48 n-heptadecane 14 1.96 5.27 cyclohexane 48 11, 14 1.20 6.32 121 benzene 11 2.01 10.17 306 toluene 11 70 ethylbenzene 4.58 8.70 11 5.96 10.56 69 ieopropylbenzene 11 6.89 10.75 68 1,2,4-trimethylbenzene 11 6.22 10.11 70 1,3,5-trimethylbenzene 11 6.57 10.55 n-butylbenzene 67 11 tert-butylbenzene 67 5.42 10.86 m-xylene 11 70 2.75 5.48 11 2.51 4.64 o-xylene 70 p-xylene 11 58 1.84 3.85 2552
total
3.17
0
2
4
6
8 10 12 Reduced pressure
14
16
18
Figure 2. Thermal conductivity of liquid n-decane. The data are from Carmichael and Sage (1967).
1
10.90
See Table V.
of the model with respect to experimental data are presented. The absolute average deviation of the model was found to be less than 2% when compared to more than lo00 data points from different sources. High Pressure. The effect of pressure on liquid properties is noticeable only at very high pressures. There are several methods described in the literature that take into account this effect in the estimation of thermal conductivity (Missenard, 1970; Andersson, 1978; Ely and Hanley, 1983; Rodher and Nieto de Castro, 1982). The following empirical equation is proposed: A=
XPr=0[1
+ (P,P)'g(T,P)I
(13)
with g(TrP) = bo + blT,P + b2(TrP)2 (14) where all constants (b/s and y) are given in the Appendix. Notice that the correction on pressure has been formulated using the equivalent paraffin as a basis. Thus, if eq 13 is viewed as a perturbation from a fluid at P, = 0, the cross terms involving reduced pressure and ( K , - K,P) have been neglected. The complete equation for a pure hydrocarbon is given by the expression
1
aJKW- KwPl2 [1 + (PrP)'g(Trp)](15)
The equation only requires the normal boiling temperature and the Watson parameter for the specific compound to allow computation of the thermal conductivity of the pure liquid hydrocarbons at T and P. In Table 11, the results obtained with eq 15 for liquids at high pressures are tabulated as average and maximum absolute deviations from
Figure 3. Thermal conductivity of liquid benzene. The data are from Li et al. (1984).
experimental data. As an example in Figure 2, the thermal conductivity of n-decane is plotted against the reduced pressure for several levels of reduced temperature. A similar plot for benzene is displayed in Figure 3. In both cases the model of eq 15 follows remarkably well the experimental data.
Mist ures The model for mixtures as expressed in eqs 5 and 6 may be written as follows:
(16) Here Xo is the thermal conductivity of a pure fluid, with normal boiling temperature coinciding with the mean of the distribution, and it is given by eq 15. The only unknown in this equation is the function h(Tr),which is a simpiified form for the correction term described by eq 5. This term was chosen to be of a form similar to that previously obtained for viscoeitieS (Lug0 and Briano, 1991). Using experimental information for several discrete mixtures of hydorcarbons, it was found that a linear correlation with reduced temperature was enough to give excellent results. Thus w - 1 ) = 8 1 + 82Tr (17) Figure 4 shows a comparison between experimental and calculated values of thermal conductivity of hydrocarbon mixtures. Table I11 shows the mean and maximum absolute deviations of the model when compared to 163 data points. If should be pointed out that the correction term given in eq 17 is only of importance for mirtureS diSpl&lg
Ind. Eng. Cham. Res., Vol. 32, No. 1, 1993 197
i a l c u l a t e d .-er-3
CS13L:t
i
Dressure (MPa)
t i ' A +-*)
Figure 4. Comparison between experimental and calculated thermal conductivities of hydrocarbon mixtures.
Figure 6. Thermal conductivity of a coal-liquid fraction. The solid line was computed with eq 16. The dashed lines and the experimental points are from Baltatu et al. (1985). 1
0 '5,
0 13
i9C
33C
33C
3.2
342
-enae-3+vre3?K'1
Figure 5. Thermal conductivity of a liquid mixture of n-hexanel n-decane. The solid line represent eq 16. The dashed line is the same model excluding the correction for mixing (h(T,)= 0). The data are from Mukhamedzyanov et al. (1964). Table 111. Mean and Maximum Absolute Deviations of the Thermal Conductivitv of Hydrocarbon Mixtures mean dev max dev N (Ti) (Ti) ref" mixture n-heptadecaneln-octane 5 2.67 4.67 11 2 1.87 3.31 11 n-heptaneln-hexadecane n-octanelisooctane 3 5.22 9.93 11 8.54 11 n-octanelisooctane 6 4.74 n-octaneln-tetradecane 4 7.41 11.98 16 2.29 16 n-hexaneln-decane 24 0.61 1.76 16 n-decaneln-tetradecane 24 0.92 18 6.31 7.66 21 n-heptane/cyclohexane 39 1.70 3.67 11 benzeneltoluene benzene/cyclohexane 10 3.11 4.77 11 10.51 11 cyclohexane/ toluene 5 5.27 6.81 9.61 21 toluenelbenzeneln-pentane 13 ~~
total a
153
2.91
11.98
See Table V.
a dimensionless variance, 7/(Tbo)*,larger than 0.001. As an example, in Figure 5 the thermal conductivity of a binary liquid mixture containing n-hexane and n-decane is plotted as a function of temperature. In this figure, the dashed lines are the resulta of the model without the correction term for mixing (i.e., h(T,) = 0), and the solid lines are computed using eq 16.
Petroleum Fractions and Coal-Derived Liquids As an application of the method, we computed the thermal conductivity of petroleum fractions and of liquids derived from coal using as parameters the mean molar boiling temperature and the variance of the boiling temperature distribution. To obtain these basic parameters, it is first necessary to obtain the molar equivalent of the
~
_250
350
300
400
450
Temperature ( K )
Figure 7. Thermal conductivity of petroleum fractions. The experimental data are from Mallan et al. (1972).
volumetric distillation temperatures. This may be done using the correlation proposed by Zhou (1984):
Tb0 = Tv- exp(-1.15158 - 0.01181T,2/3 + 3.70684m1/3) (18) where Thois the degrees Celsius, and T,is the mean volumetric boiling temperature obtained from an ASTM distillation in degrees Celsius, which is defined by Tv = (7'10 + T30 + Tw + T,o + 7 ' ~ ) / 5 (19) where Tlo, T30, TM,etc. represent the temperature in an ASTM distillation for which lo%, 30%, etc. of the volume has been distilled. In eq 18, m is the slope of the ASTM distillation curve in degrees Celsius per percent (OC/%): m=
Tw - TlO
90 - 10 Finally, the variance of the molar boiling temperature is obtained from the correlation reported by Melo et al. (1987) as a function of m. 7
= 863.67m
+ 914.72m2
(21)
One of the most relevant studies on this subject is that of Baltatu et al. (1985). They extended the method proposed by Ely and Hanley (1981, 1983) to compute the thermal conductivities of petroleum and coal-liquid fractions. Baltatu et al. (1985) adopted different approaches to determine the basic input parameters for the corresponding states model. In Figure 6, the thermal conductivity for a coal-liquid fraction identified as SRC-I naphtha (Baltatu et al., 1985) is plotted as a function of pressure. In the same figure, the prediction made by Baltatu et al. (1985), using the Kesler and Lee (1976) equations for es-
198 Ind. Eng. Chem. Res.,Vol. 32, No. 1, 1993 Table IV. Thermal Cdnductivity of Petroleum and Coal-Liquid Fraction6 mean dev max dev fraction N Tb SG K, (%) (70) ref" 6.3 11.6 15 oil no. 2 7 421.1 0.806 11.31 21.6 15 20.8 6 566.2 0.875 11.50 oil no. 4 10.6 15 9.7 oil no. 6 5 492.6 0.807 11.90 18.8 21.9 15 oil no. 8 11 616.4 0.877 11.80 3.4 7.7 15 oil no. 9 7 388.7 0.779 11.40 16.9 21.4 15 oil no. 10 9 557.1 0.856 11.70 17.0 15 15.7 oil no. 11 6 498.4 0.955 10.10 7.1 15 6.2 oil no. 21 21 524.4 0.817 12.00 17.9 15 14.0 oil no. 22 6 603.3 0.857 12.00 0.3 0.8 15 oil no. 23 15 535.7 0.803 12.30 4.7 8.8 9 CLD no. 2 4 376.0 0.770 12.60 4.1 7.3 9 CLD no. 4 5 413.0 0.812 12.60 18.7 9 8.1 CLD no. 6 10 470.0 0.954 12.60 8.5 9 3.8 CLD no. 8 12 532.0 0.976 12.60 9.7 9 5.3 CLD no. 10 12 571.0 0.997 12.60 CLD no. 12 12 652.5 1.079 12.60 8.31 8.8 9 total
148
8.4
21.9
See Table V. Table V. Sources for Experimental Data on Thermal Conductivity of Pure Hydrocarbons Used in Tables I-IV ref eren ce no. 1 Baltatu e t al. (1985) 2 Baroncini et al. (1984) 3 Carmichael and Sage (1964) 4 Carmichael et al. (1969) 5 Carmichael and Sage (1967) Chhabra et al. (1980) 6 7 Engineering Sciences Data Unit (1975a) 8 Engineering Sciences Data Unit (1975b) 9 Gray e t al. (1983) 10 Jaimeson and Hastings (1969) 11 Jaimeson et al. (1975) 12 Kandiyoti et al. (1972) 13 Li (1976) Li e t al. (1984) 14 Malian e t al. (1972) 15 Mukhamedzyanov et al. (1964) 16 17 Ogiwara e t al. (1980) 18 Rahalkar et al. (1969) 19 Rowley and White (1987) 20 Rowley and Gubler (1988) 21 Rowley (1986)
timating pseudoparameters, is included. Figure 7 shows our prediction for the thermal conductivity of two different petroleum fractions, oil no. 6 and oil no. 23 as reported by Mallan et al. (1972). In Table lV, data for both petroleum and coal-liquid fractions are compared to the present model. Average deviations in the range of 10% were found. It should be mentioned that, for most of the fractions treated, the complete characterization needed in the model was not available. Finally, Table V summarizes the different literature sources of experimental data used in this work.
SG = specific gravity T = temperature Tb = normal boiling temperature T, = critical temperature T, = mean volumetric temperature XI = fraction of family ao,a l , a2 = constants in eq 11 bo, b,, b2 = constants in eq 14 c = number of components in a discrete mixture f = discrete random variable g = empirical function defined in eq 14 h = empirical function defined in eq 17 m = slope of ASTM distillation curve p = density distribution function x i = molar fraction of component i Greek Letters a2 = correction terms 01, O2 = constants in eq 17 y = constant in eq 13 cy1,
7 =
X =
variance thermal conductivity
Subscripts i = component indices r = reduced property Superscripts 0 = reference fluid 1 = correction term p = paraffin
Appendix Paraffin Basic Properties. T,P = 9.292 + 1.756Tb0- 8.66 x 10-4(Tb0)2 P,P
+
= 6228.4 - 10.8Tb0 3.80
K,P = 14.25 - 7.739
X
X
10-3(Tb0)2
10-3Tb0+ 9.426
X
10-'j(Tbo)2
Characterization of Mixture. Tb0
= ixiTb, Or lT@(Tb) dTb i=l
Thermal Conductivity of P u r e Fluids.
-
Acknowledgment Support for this work was received from the NSFEPSCoR Program through Grant RII-8610677. Nomenclature A = characteristic function F = number of different families of homologous compounds K, = Watson parameter M = property of a mixture Mo = property of a reference fluid M' = correction term due to mixing P = pressure P, = critical pressure
h(T,) = 0.7789 - 1.2971:
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