Ind. Eng. Chem. Fundam. 1983, 22, 90-97
90
Morrison, T. J.: Johnstone, N. 6. 6. J. Chem. SOC. 1955, 3655-3659. Newman, J. S . “Electrochemical Systems”; Prentice-Hail: Englewood Cliffs, NJ, 1973. SeMell, A. “Solublities of Inorganic and Organic Substances”: Van Nostrand: New York, 1940. Shoor, S. K.; Walker, R. D.;Gubblns, K. E. J. Phys. Chem. 1969, 73,
3 12-3 17. Tiepel. E. W.; Gubbins, K. E. Ind. Eng. Chem. Fundam. 1973, 12, 18.
Received for review December 23, 1981 Accepted August 8, 1982
Prediction of Transport Properties. 2. Thermal Conductivity of Pure Fluids and Mixtures James F. Ely‘ and H. J.
M. Hanley
Thermophysical Properties Dlvlsion, National Engineering Laboratory, National Bureau of Standards, Boulder, Colorado 80303
A technique for the prediction of the thermal conductivity of nonpolar pure fluids and mixtures over the entire range of PVT states is presented. The model is analogous to the extended corresponding states viscosity model reported previously by Ely and Hanley in 1981. Calculations for the thermal conductivity require only critical constants, molecular weight, Pitzer’s acentric factor, and the ideal gas heat capacity as a function of temperature for each mixture component as input. Extensive comparisons with experimental data for pure fluids and nonpolar binary fluid mixtures including paraffins, alkenes, aromatics, and naphthenes with molecular weights to that of C24are presented. The average absolute deviation between experiment and prediction is less than 7 % for both pure species and mixtures.
Introduction We have recently discussed the prediction of the viscosity (7)of pure fluids and mixtures via the extended corresponding states one fluid model (Ely and Hanley, 1981 (hereafter denoted by part 1);Ely, 1981). It was stressed that the method is predictive and the number of mixture components is, in principle, unlimited. The method was applied to nonpolar fluids over a wide range of states from the dilute gas to the dense liquid. Here we extend the model to the thermal conductivity (A). It differs from other corresponding states thermal conductivity models (Mo and Gubbins, 1976; Hanley, 1976,1977;Haile et al., 1976; Murad and Gubbins, 1977; Christensen and Fredenslund, 1980; Teja and Rice, 1980) in the scope of application and in the fact that it requires no transport data as input. The basic idea is as before, namely that the configurational properties of a single-phase mixture can be equated to those of a hypothetical pure fluid. The properties of this fluid are then evaluated via corresponding states with respect to a given reference fluid (methane) at the appropriate corresponding pressure and temperature, or density and temperature. It is appreciated at the onset that there are both formal and practical difficulties with the thermal conductivity. For example, (1) a onefluid model must ignore the contribution of diffusion to the conductivity (Hanley, 1977a), and (2) any corresponding states argument cannot correctly take into account the effect of internal degrees of freedom on the thermal conductivity-an effect which may be large in the dilute gas (Hanley, 1977b). Also, the state of the art for measuring thermal conductivity is relatively poor; few data are accurate to within 10% and the range of data is usually limited to the dilute gas or to the saturated liquid. The scope of this work is the same as in part 1: we discuss results for the paraffins, alkenes, aromatics, and naphthenes to molecular weights of C2@ Input data are the critical parameters, molecular weight, and acentric factor of the pures or of the species in a mixture. In addition, the ideal gas heat capacity at constant pressure for
each species is required to account for the internal degrees of freedom of a polyatomic molecule.
Thermal Conductivity Model We postulate that the thermal conductivity of a pure substance or mixture may be divided into two contributions-one arising from the transfer of energy from purely collisional or translational effects, A’, and the other from the transfer of energy via the internal degrees of freedom, A”. We further assume that this latter contribution is independent of the density and may be calculated from the modified Eucken correlation for polyatomic gases, viz.
where A”, is the internal contribution for component a , M , is the molecular weight, q,* is the dilute gas viscosity of component N which is calculated by the method outlined in part 1, C$ is the ideal gas heat capacity, R is the gas constant and fint has a constant value of 1.32. For a mixis calculated via the empirical mixing rule (Li, ture, ,A,” 1976) A”,i,(T)
= CCX,XBA‘“B . B
(2)
where (A’”fl)-l
= 2[(x’”)-l
+ (X”O)-l]
(3)
We emphasize that in general the assumption that the internal contribution is independent of density must be incorrect. The translational contribution A’ is calculated via the corresponding states method outlined in part 1. Briefly, we postulate that the translational mixture thermal conductivity is identical with that of a hypothetical pure fluid, denoted by a subscript x , viz. A’mix(P,T) A’~(P,T) (4)
This article not subject to US. Copyright. Published 1983 by the American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 91
The corresponding states principle is then invoked on the hypothetical pure fluid X ’ x ( P , T ) = X’o(P0,To)Fx (5) where the subscript “0” refers to the reference fluid and
The equivalent temperature (To)and density (PO) for the reference fluid are defined by the relations Po = Phx,o; To = T / f x , o (7) The equivalent substance reducing ratios fx,o and hx,Oare defined by the mixing rules
fx,ohx,o=
CCw@a,ha,
(8)
tions, eq 1 and 5, respectively. The internal contribution presents no difficulty since it is based on dilute gas values which in principle are available to as low a temperature as may be encountered. For the collisional contribution, we have values of the total conductivity of methane &,(po,To)which includes a dilute gas contribution comprised of an internal degrees of freedom term and a translational term X’o(po,To).Thus,we require for methane X’O(P0,TO) = ~o(PotT0)- X”O(T0)
We have for the dilute gas
Thus
a ,
and where the asterisk denotes a dilute gas value. Since the methane thermal conductivity correlation of Hanley et al. is of the form XO(P0,TO) = Xo*(To)
where X0(l) is the first density correction to the thermal conductivity and A&, is the high density contribution, we find, on combining eq 16 and 17
and
h
+ hO(’)(TO)PO+ A&,(Po,To) (17)
1 = - (hall3 h81/3)3(1 -la,) ” 8
+
Finally
f, = ~~,(TR”,VR~,W,)T,~/T,O (12) h, = ~ , ( ~ R ~ , V R ~ , ~ , ) ~ ~ ~ / V(13) ,O
where 0, and $=I are the shape factors of Leach and Leland, whose detailed functional forms are given in part 1: w, is Pitzer’s acentric factor, k,, and I, are binary interaction constants, and the subscript “R” denotes a quantity reduced by its critical point value. As in our previous work with viscosity, we have set k,, and I,, equal to zero. The mixing rule for the mass was chosen to be analogous to that used for the viscosity in part 1 Mx-1/2fx,01/2hx,{4/3 = CCx~xgM,8-1/2f,X,1/2hag-4/3 (14) . B
where
Methods analogous to those used in part 1were used to extend this equation to the high reduced densities and low reduced temperatures required in this work. Briefly, pseudo-methane experimental data were defined via eq 1, 5, 6, and 18, viz.
where the superscript exptl refers to experimental data of a selected fluid or fluids and X xis a correction for noncorrespondencediscussed below. Since the first two terms of eq 18 make negligible contributions in the high-density region, only the AXo term requires modification. Thus AxO(P0,To) =
M
-1
=
!(&p + M8-1) 2
(15)
Other mass mixing rules such as those proposed by Mo and Gubbins (1976) were explored but offered no advantage over eq 14. Reference Fluid Equations As was mentioned in the Introduction, methane was chosen as the reference fluid in this work. This choice, which is entirely arbitrary, was based on the existence of experimental data for both the PVT and thermal conductivity of methane over the entire range of fluid states. The PVT relation is the 32-term Bender type BWR equation described in part 1 and the analytical form reported by Hanley et al. (1974) was used for the thermal conductivity. As we discussed at length in part 1, however, the range of states likely to be encountered for the fluids and mixtures of interest here are simply not covered by the methane fluid surface. For this reason, the reference fluid correlations were extended into a pseudo-liquid region with the extended equations for the equation of state and shape factors reported in part 1. For the thermal conductivity, we require extended equations for both the collisional and internal contribu-
Actual methane data along with pseudo data generated from methane and propane were then fit to the functional form given for this term by Hanley et al. (1974). The resulting coefficients and functional forms for the translational methane thermal conductivity are summarized in Table I. Comment on the Thermal Conductivity in the Critical Region The modern theory of transport phenomena (Sengers, 1971,1972) predicts an infinite thermal conductivity at the pure fluid critical point and a large enhancement in the vicinity of the critical point, shown schematically as a function of temperature and density in Figure 1. This enhancement has been observed experimentallyfor several fluids (for example, Sengers, 1971; Swinney and Henry, 1973; Roder, 1982). For a mixture, however, theory indicates that the enhancement is not present as has also been inferred experimentally by the light-scattering measurements of Ackerson and Hanley (1980). The absence of the en-
92
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Table I. Reference Fluid Translational Thermal Conductivity Correlationa h',(P0,To) = &*'(To) t h ( l ) ( T o ) P 0+ Ah,(P,,T,) 15R
h,*'(T,) =
-no* (To) 4M0
9
q o * ( T o )=
n=l
CnT0("-4)/3
h o ( ' ) ( T 0= ) b , e b,[b, - I n (TO/b,)l7 A h o = exp
[.,
t
a,/To]
exp[(a,
+ a,/T03'2W,0.'+
( P ~ / P ,-
t a , / T , t a,/ToZ)l- 1.0
l)~:.~(a,
_~
i
a1
b,
1
-7.1977082270Et0 8.5678222640Et 1 1.2471834689Et1 -9.8462522975Et 2 3.5946850007E-1 6.9798412538Et 1 -8.7288332851Et 2
-0.252762920E-t 0 0.3343285903+0 1.12 0.1680Et 3
2 3 4 5 6 7 8 9
Ci
2.907741307Et6 -3.312874033Et6 1.608101838Et6 -4.33 1904871Et 5 7.062481330Et4 -7.116620750Et3 4.325174400Et 2 -1.44591 1210Et 1 2.037119479E-1
4 DENSITY, p
Figure 1. Schematic representation of the pure fluid critical point thermal conductivity enhancement.
hancement in a mixture is due, in principle, to the different physical criteria for criticality in a pure fluid and mixture. A difficulty could arise, therefore, if one included the anomalous critical point contribution, Ax,, in the reference fluid thermal conductivity (eq 18) since it appears possible that the equivalent substance reducing ratios (eq 7) for a given px and T, could lead to poc and Toc.In practice this will not occur; the critical point of the hypothetical pure substance (Txc,pxc)is virtually inaccessible since the true critical temperature of a mixture T, is greater than T,' and the true critical density pc is larger than p; (Hanley, 1977b; Kreglewski, 1973; Rowlinson, 1969). The effect of the anomalous critical point contribution on the behavior of the conductivity of a mixture along the gas/liquid critical line is shown in Figure 2 for methane/propane mixtures. The effect is small which is consistent with the above argument. Because the effect is small, we have chosen not to include Axc in this work. Obviously, however, the neglect of this contribution will give rise to large errors in the predicted conductivity very close to the critical point of a pure fluid. Correction for Noncorrespondence It was found in part 1that a small correction factor was needed to account for possible failures of the corresponding states model. In that paper a correction factor was in-
0.5
0 X,
1 .o
CH,
-__
Figure 2. The critical excess thermal conductivity coefficient
AAc
along the plait point curve for a methane/propane mixture (Hanley, 1977a).
troduced based on Hanley's (1976) earlier work. That correction factor follows from the modified Enskog theory (Hanley et al., 1972) and depends only on PVT properties of the fluid mixture. For the thermal conductivity we have adopted a similar correction factor, viz.
xx =
([
1-
2-%)"*I$*
(19)
where for a mixture Zxc= ~ X , Z , ~ This . expression has a basis in the Enskog theory in that the density dependence of the thermal conductivity is dependent on the derivative (dP/dT),. The correction factor eq 19 is applied to the entire collisional contribution to the thermal conductivity, i.e. x'x(P,,Tx) = ~ ' o ( ~ o , T o ) F , X , (20) Using eq 2, 5 , 19, and 20, we have then for our final working equation xmixb,n
= X'O(PO,TO)F,X,+
A''miX(n
(21)
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 93
Table 11. Pure Fluid Thermal Conductivitv Data Sources fluid ethane propane n - bu t ane is0 bu tane n-pentane n-hexane 2-methylpentane 3-methylpentane 2,2-dimethylbutane n-heptane
2,4-dimethylpentane n-octane 2,2,4-trimethylpentane n-nonane 2,2,5-trimethylpentane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-hep tadecane n-octadecane n-nonadecane n-eicosane n-henicosane n-docosane n-tricosane n-tetracosane ethene propene 2-methylpropene 1 hexene 1-heptene 1-octene 1-nonene 1-decene benzene toluene ~
ethylbenzene o-xylene m-xylene p-xylene propylbenzene isopropyl benzene n-butylbenzene tert-bu tylbenzene biphenyl cy clopen tane meth ylc yclopentane cyclohexane methy lc yclohexane
sources Carmichael et al. (1963); Keyes (1954); Lenoir and Junk (1953); Le Neindre e t al. (1969) Brykov e t al. (1970); Carmichael e t al. (1968a); Leng and Comings (1957); Roder and de Castro (1981); Ryabtsev and Kazaryan (1969) Brykov e t al. (1970); Carmichael and Sage (1964) Kazaryan and Ryabtsev (1969) Brykov e t al. (1970); Bogatov ( 1 9 6 9 ) ; Carmichael e t al. (1969); Vilim (1960) Brykov e t al. (1970); Sakiadis and Coates (1955); Tsederberg (1965) Sakiadis and Coates (1955) Sakiadis and Coates (1955) Sakiadis and Coates (1955) Brykov e t al. (1970); Ogiwara e t al. (1980); Rastorguev e t al. (1968); Sakiadis and Coates (1955) Sakiadis and Coates (1957) Brykov e t al. (1970); Ogiwara et al. (1980); Tsederberg (1965) Sakiadis and Coates (1955) Brykov e t al. (1970); Ogiwara e t al. (1980) Sakiadis and Coates (1955) Brykov et al. (1970); Carmichael and Sage (1967); Naziev and Aliev (1973a); Ogiwara e t al. (1980); Rastorguev e t al. (1969);Sakiadis and Coates (1955) Naziev and Aliev (1973b) Sakiadis and Coates (1957) Mustafaev (1972a) Bogatov e t al. ( 1 9 6 9 ) ; Mustafaev (1972b) Naziev and Aliev (1973b); Bogatov e t al. (1969) Mustafaev (1973) Mustafaev (1973); Rastorguev and Bogatov (1972) Naziev and Aliev (1973b); Mustafaev (1973); Rastorguev and Bogatov (1972) Naziev and Aliev (1974) Rastorguev and Bogatov (1974) Rastorguev and Bogatov (1974) Rastorguev and Bogatov (1974) Rastorguev and Bogatov (1974) Rastorguev and Bogatov (1974) Kolomiets (1974) Vargaftik (1975) Ryabtsev and Kazargan (1970) Naziev and Abasov (1970); Brykov e t al. (1970) Brykov e t al. (1970); Naziev and Abasov (196913) Naziev and Abasov (1969a) Mustafaev (1974) Mustafaev (1974) Akhundov ( 1 9 7 4 ) ; Johnston e t al. (1965); Ogiwara e t al. (1980); Rastorguev and Pugach (1970) Akhundov and Gasanova (1969); Geller and Rastorguev (1968); Geller and Zaporazhan (1974); Rastorguev e t al. (1969); Rastorguev and Pugach (1970); Brykov e t al. (1970); Johnston e t al. (1965) Akhundov (1974); Rastorguev and Pugach (1970) Akhundov and Gasanova (1969b); Rastorguev and Pugach (1970) Akhundov and Gasanova (1969b); Rastorguev and Pugach (1970) Akhundov and Gasanova ( 1 9 6 9 ~ )Rastorguev ; and Pugach (1970); Ogiwara e t al. (1980) Guseinov e t al. (1976) Guseinov e t al. (1976); Rastorguev and Pugach (1971) Rastorguev and Pugach (1970) Rastorguev and Pugach (1971) Hedley e t al. (1970); Ziebland and Burton (1961) Sakiadis and Coates (1957) Sakiadis and Coates (1957) Briggs (1957); Naziev e t al. (1974); Sakiadis and Coates (1957) Briggs ( 1 9 5 7 ) ;Sakiadis and Coates (1957)
Summary of the Calculation Procedure A summary of the calculation procedure to evaluate thermal conductivity is as follows. Input parameters are the critical temperature, volume, and pressure, the acentric factor, ideal gas heat capacity, and molecular weight of each component of the mixture of interest. These parameters for the reference fluid are required with an equation of state and some functional form for the thermal conductivity for this reference fluid. Typical experimental input would be the pressure, temperature, and mixture composition. The density of the fluid or mixture is obtained by finding the equivalent pressure of the reference substance via the ratio p o = pxhx,o/ fx,o from the corresponding pressure in the mixture, p x . Initially, the shape factors in eq 1 2 and 13 are set to
unity. Given po = p(po,To),the density po follows. Thus, a first guess of the density is that obtained using classical 2-parameter corresponding states. Repeated iterations using eq 12 and 13 give the final density. Having, therefore, final values of p, fx,o, and hx,O,one can evaluate FA, po, and Toand hence X’,(po,T0) and X”(79, thereby obtaining a value for X,(p,n. Results for Pure Fluids Given the reference equation of state for methane and the equation for the thermal conductivity, Table I, it was straightforward to calculate the thermal conductivity of pure species a by eq 21. A considerable data base was assembled to provide a thorough test of the procedure and Table I1 lists the fluids and appropriate references. The data were evaluated as far as possible for accuracy and
94
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Table 111. Summary of Calculated Results for Pure Fluid Thermal Conductivity (AAD = Average Absolute Percent Deviation; BIAS = Percentage Bias) fluid
48
ADD 5.05 7.23 6.85 7.81 4.32 8.18 2.08 3.69 2.18 1.84 10.92 0.51 11.85 3.60 5.48 3.31 7.64 11.85 5.65 7.24 5.41 6.43 6.21 4.92 5.70 6.61 2.37
48
1.24
ili
ethane propane n-butane isobutane n -pent ane n-hexane 2-methylpentane 3-meth ylpentane 2,2-dimethylbu tane 2,3-dimethyl butane n-heptane 2,4-dimethylpentane n-octane 2,2,4-trimethylpentane n-nonane 2,2,5-trimethylpentane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pen tadecane n-hexadecane n-heptadecane n-octadecane n-nonadecane n-eicosane n-henicosane
133 23 8 38 114 90 162 :3 3 3 3 310 3 169 3 22 3 182 86 3 24 6 361 192 112 191 23 4 174
BIAS 2.94 7.07 3.85 -3.80 3.88 -7.58 -2.08 -3.69 -2.18 0.43 -1 0.47 0.51 -11.72 1.30 -4.23 -2.16 -7.09 -1 1.85 -5.65 -7.20 -5.28 -5.46 -6.11 -4.33 -5.01 -5.65 2.32 1.07
20,-
1
fluid n-docosane n-tricosane n-tetracosane ethene propene 2-methylpropene 1-hexene 1-heptene 1-octene 1-nonene 1-decene benzene toluene ethylbenzene o-xylene m-xylene p-xylene propyl benzene isopropyl benzene n-bu tyl benzene tert-butylbenzene biphenyl cyclopentane me thylcyclopentane cyclohexane me thylcyclohexane overall
N 48 42 42 114 42 83 147 148 107 219 223 166 54 7 183 187 167 209 70 143 70 70 14 3 3 136
ADD
BIAS
1.07 0.71 1.37 7.05 9.40 6.69 5.14 5.69 7.22 1.93 2.13 2.85 9.21 5.70 7.79 8.08 7.54 4.74 5.15 7.67 9.69 15.13 3.20 1.04 4.71
6388
6.56
0.92 0.17 -0.82 -5.56 -9.40 5.85 - 2.26 -4.01 -6.79 1.55 0.72 -1.50 7.06 4.59 7.26 6.29 2.46 -3.07 3.63 7.65 9.65 -15.13 -3.20 -1.04 -4.25 3.76 -1.30
9
2Or
4.13
I-hexane
ethane
-201
-2oL
'e
i-nonone
n-hexane
'Or
k
0
x
'
n
'Or
i
n-decane
. -2oL 201
cyclohexane
..
- 2 O L - A -
1 0
1
__
2 0
-
- I
-2oL
30
REDUCED DENSITY,
p,
Figure 3. Comparison of calculated and experimental thermal conductivity of ethane, n-hexane, n-decane, n-tetracosane, and cyclohexane. Literature references are given in Table 11.
internal consistency based on our experience of assessing data for simple fluids. It turns out that in general the data situation is not too satisfactory, especially in that the more recent accurate experimental techniques, such as the transient hot wire method, have been applied to only a few organic liquids [de Castro et al., 19771. A realistic guess as to the accuracy of the data quoted is 5 1 5 % and probably much worse for the very heavy species. The results are summarized in Table I11 and in Figures 3 and 4. Shown are the average absolute percent deviation, AAD, and the average percentage error, BIAS. The maximum error observed using the method was typically
-2o.---
0
-J-
10
2 0
30
REDUCED DENSITY,
p,
Figure 4. Comparison of calculated and experimental thermal conductivity of 1-hexene, 1-nonene, benzene, p-xylene, cumene, and carbon dioxide. Literature references are given in Table 11.
15% outside of the critical region and much larger in that region. This quantity, however, is not very instructive since it tends to be very sensitive to experimental uncertainties. The overall deviations between experiment and eq 21 are very satisfactory especially considering the limited input (T,, p o V,, M , CPo(7'), and a) needed to make the predictions. They become somewhat worse, and positive, as the freezing point of the fluid is approached, however.
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 95
Table IV. Summary of Calculated Results for Binary Mixture Thermal Conductivity" component 1 component 2 source(s)
N
15 Carmichael e t al. (1968) 6 Parkinson (1974) 14 Naziev a n d Nurberdyev (1974a); Parkinson (1974) 6 Ogiwara et al. (1980) 6 Parkinson (1974) 6 Parkinson (1974) 2 Mukahamedzyanov e t al. (1964) 9 Mukahamedzyanov e t al. (1964) 6 Mukahamedzyanov et al. (1964) Parkinson (1974) 6 6 Mukahamedzyanov e t al. (1974) 279 Naziev and Nurberdyev (1974a) 279 Naziev and Nurberdyev (197413) 8 Ogiwara et al. (1980) Jamieson et al. (1972); Vernart (1968); 32 Bashirov (1966); Fillipov and Novoselova (1955) 4 cyclohexane 5 Parkinson (1974) n-heptane 2,2,4-trimethylpentane Parkinson (1974) 6 o-xylene Parkinson (1974) 6 6 2,2,4-trimethylpentane Parkinson (1974) Parkinson (1974) 6 n-heptane methylcyclohexane Parkinson (1974) 6 4 toluene Mukhamedzyanov et al. (1974) 6 2,2,4-trimethylpentane Parkinson (1974) 167
n-bu tane 2,2,4-trimethylpentane n-octane n-octane 2,2,44rimethylpentane n-decane n-hexadecane 2,2,4-trimethylpentane n-octane n-heptadecane 2,2,4-trimethylpentane 2,2,5-trimethylpentane n-tetradecane n-heptane 1-hexene n-octane n-heptane benzene toluene
me thane 2,3-dimethylbutane n-hexane n-hep tane
toluene o-xylene cyclopentane cyclohexane methylc yclohexane overall
AAD
BIAS
12.20 4.70 6.17 2.01 1.96 8.99 3.44 4.97 4.77 4.71 5.06 8.11 8.84 3.55 6.15
12.09 4.10 -6.17 -2.01. 1.81 -8.99 -3.44 -4.97 -4.66 4.71 -5.06 -8.09 -8.83 -3.39 6.15
4.42 6.29 10.78 15.71 11.51 0.16 5.20 4.12 11.72 -11.72 1.42 6.83 4.42 6.29 10.78 15.71 11.51 0.66 5.20 4.12
" AAD = average absolute percent deviation; BIAS = average percent deviation. It is not clear why this should be the case, but one can speculate that the effect of density on the internal degrees of freedom become important in this region.
Results for Mixtures The main objective of this work was to develop a procedure to predict the transport properties of mixtures: there is a real need for such a procedure, yet none exists which does not require in some way transport data of the pure components. Our method, however, avoids this and the mixture evaluations are no more complicated than those for the pures, given the mixing rules, eq 8,9,and 14. It is also worth remarking that the procedure also gives the mixture density automatically. Table IV lists the mixtures, data sources, number of points considered, and the ADD and BIAS between experiment and our procedure. Figure 5 is typical of the deviations observed. In general, the results are excellent with an average absolute percent deviation of approximately 7%. An assessmentof the method should, however, bear in mind that the data situation for mixtures is not very good, an assignment of 5 1 5 % on the accuracy of the data is reasonable. Figure 6 compares the predicted and experimental thermal conductivities for the cyclopentaneln-heptane system at 273.15 K. Although the overall agreement is very good, the model seems to predict a linear composition dependence, whereas the data show some curvature. Unfortunately, the experimental data for mixture thermal conductivity as a function of composition are very sparse and do not allow more detailed evaluations. More experimental work is clearly needed. Conclusion We have presented a predictive procedure to estimate the thermal conductivity of nonpolar pure fluids and mixtures over the entire range of fluid states, from the dilute gas to the dense liquid. Extensive comparisonswith data show that the thermal conductivity for both pure fluids and binary mixtures-Cl to C, inclusive, aromatics and others-is predicted to within an absolute percent deviation of about 7%. Not shown in this paper are com-
::
methaneln-butane -20
=!
toluene/n- heptane
: : o
I
m
Tu
-20
t
I-hexeneln-heptane
-20 1.0
2.0
3.0
REDUCED DENSITY
'4
Cyclopentaneln-Heptane 0 Experimental
,
i
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
96
density is predicted to better than 1% and the dilute gas conductivity is predicted to within 5-10%. The basis of the method is the one fluid corresponding states concept with the extended corresponding states approach included. The method is predictioe and requires only the common characterization parameters of the pures as input; F,pc, Vc,w, Cpo,M. The number of components of a mixture which can be considered is, in principle, unrestricted. The application of the approach to polar fluids and to molecules with complex structure is under current investigation. Computer Program A computer program for predicting the transport properties (viscosity and thermal conductivity) and the density of pure fluids and their mixtures is available from the Gas Processors Association, Tulsa, Oklahoma, a t a nominal cost. The program is essentially that used to generate the results reported here. Acknowledgment This work was supported by the Office of Standard Reference Data and, in part, by the Gas Research Institute. We are grateful to R. P. Danner for making his thermal conductivity data base available to us and to Scott Starsky for help with the data reduction. Mrs. Karen Bowie helped substantially to prepare the paper. Nomenclature F = dimensional scaling factor for thermal conductivity M = molecular weight N = number of molecules in a system R = universal gas constant T = absolute temperature V = volume X = thermal conductivity non-correspondence correction Z = compressibility factor, p V / R T a , b, c = correlation parameters f = equivalent substance temperature reducing ratio h = equivalent substance volume reducing ratio k = energy binary interaction constant 1 = volume binary interaction constant p = absolute pressure x = mole fraction Greek Letters h = thermal conductivity 0 = energy shape factor p = density 4 = size shape factor w =
Pitzer's acentric factor
Subscripts a , @ = pure components or species in a mixture = binary pair in a mixture
X =
thermal conductivity
= viscosity
0 = reference fluid value
= fluid of interest, mixture or pure R = reduced value at the critical point
x
Superscripts c = critical value
exptl = experimental value * = dilute gas transport property value Literature Cited Ackerson, 8. J.; Hanley, H. J. M. J. Chem. Phys. 1980, 73, 3568. Akhundov. T. S. Izv. Vyssh. Ucheb. Zaved., Neff i Gaz 19748, 17, 78. Akhundov. T. S.: Gasanova. N. E. I z v . Vvssh. Ucheb. Zaved.. Weft i Gaz 1974b, 17, 24. Akhundov. T. S.; Gasanova. N. E. I z v . Vyssh. Ucheb. Zaved., Neffi Gaz 1969a. 12. 59. Akhundov. T. 'S,; Gasanova, N. E. I z v . Vyssh. Ucheb. Zaved., Neffi Gaz 1969b. 12, 67
Akhundov, T. S.; Gasanova, N. E. I z v . Vyssh. Ucheb. Zaved., Neff i Gaz I989c, 12, 73. Bashirov, M. Ya. Dokl. Akad. Nauk Azerb. 1966,2 2 , 20. Bogatov. G. F. I z v . Vyssh. Ucheb. laved., Neff i Gaz 1969, 12, 60. Bogatov, G. F.; Rastorguev, Yu. L.; Grigor'ev, B. A. Khim. i Tekhnol Topi. i Masel 1969, 14, 31. Briggs, D. K. H. Ind. Eng. Chem. 1957,49, 418. Brykov, V. P.; Mukhamedzyanov, G. Kh.; Usmanov, A. G. J . Eng. Phys. (USSR)1970, 18, 62. Carmichael, L. T.; Berry, V.; Sage, B. H. J . Chem. Eng. Data 1983,8 , 281. Carmichael, L. T.; Sage, B. H. J. Chem. Eng. Data 1964,9 , 511. Carmlchael, L. T.; Sage, B. H. J. Chem. Eng. Data 1987, 12, 210. Carmichael, L. T.; Jacobs, J.; Sage, 8. H. J . Chem. Eng. Data 19688, 13, 40. Carmichael, L. T.; Jacobs, J.; Sage, 8. H. J. Chem. Eng. Data I968b, 13, 489. Carmichael, L. T.; Jacobs, J.; Sage, B. H. J . Chem. Eng. Data 1989, 14. 31. Christensen, P. L.; Fredenslund, A. A. Chem. Eng. Sci. 1980,35, 871. de Castro, C. A. N.; Calado, J. C. G.; Wakeham, W. A. "Absolute Measurements of the Thermal Conductivity of Liqulds Using a Transient Hot-wire Technique"; Proceedings of the Seventh Symposium on Thermophysical Properties, A. Cezarliyan, Ed., American Society of Mechanical Engineers: New York, 1977. Ely, J. F.; Haniey, H. J. M. Ind. Eng. Chem. Fundam. 1981,2 0 , 323. Ely, J. F. J. Res. Natl. Bur. Stand. ( U . S . ) 1981,8 6 , 597. Filippov, L. P.; Novoselova, N. S. Vesfnik Mosk. G a s . Univ. Ser. F i . . Mat. i Estestven. Nauk 1955, 10, 37. Geller, V. 2.; Rastorguev, Yu. L. Tepioenergeika 1988, 75, 80. Geller, V. 2.; Zaporazhan, G. V. I z v . Vyssh. Ucheb. Zaved., Neffi Gas 1974, 17, 69. Guselnov, K. D.: Mirzoev, B. M.; Glymanov, A. A. Zh. Fiz. Khim. 1976,50, 1995. Haile, J. M.; Mo. K. C.; Gubbins, K. E. Adv. Cryog. Eng. 1976, 2 1 , 501. Hanley, J. H. M. Cryogenics 1976, 16, 643. Hanley, H. J. M. J. Res. Natl. Bur. Stand. ( U S . ) 1977a, A82, 181. Hanley, H. J. M. "Prediction of the Thermal Conductivity of Fluids Mixtures"; I n Proceedings of the Seventh Symposlum on Thermophysical Properties, p 668, American Soclety of Mechanical Englneers: New York, 1977b. Hanley, H. J. M.: McCarty, R. D.; Cohen, E. G. D. fhysica 1972, 6 0 , 322. Hanley, H. J. M.; McCarty, R. D.; Haynes, W.M. J. fhys. Chem. Ref. Data 1974,3 , 979. Hedley, W. H.; Milnes, M. V.; Yanko, W.H. J , Chem. Eng. Data 1970, 15, 122. Jamieson, D. T.; Irving, J. B.; Tudhope, J. S. "Llquld Thermal Conductivity-A Data Survey to 1973"; National Engineering Laboratory Report No. 601, Her MaJesty's Stationary Office, Edinburgh, 1975. Johnson, H. L.; Allman, H. W.; Rubin, T. J . Chem. Eng. Data 1985, 10, 240. Kazaryan, V. A.; Ryabtsev, N. I. Gazov. Delo. 1989, 10, 27. Keyes. F. G. Trans. Am. SOC.Mech. Eng. 1954, 75, 809. Kolomiets, A. Y. Heat Transfer Sov. Res. 1974,6 , 42. Kreglewski, A. "Mixtures of Fluids"; Thermodynamics Research Center, College Station, TX, 1973. Le Neindre, 8.: Tufeu, R.; Bury, P.; Johannln, P.; Vodar, B. "Experimental Study of Thermal Conductivity Coefficients of Methane and Ethane Between 25 C and 450 C at Pressures up to 1000 Bars"; I n Proceedings of the 8th Conference on Thermal Conductivity, Ho, C. Y.; Taylor, R. E., Ed., Plenum Press: New York, 1969; pp 229-246. Leng, D. E.; Comings, E. W. Ind. Eng. Chem. 1957,4 9 , 2042. Lenoir. J.; Junk, W.; Comings, E. Chem. Eng. frog. 1953,4 9 , 539. Li, C. C. AIChE J. 1976,2 2 , 927. Mo, K. C.; Gubbins, K. E. Chem. Eng. Commun. 1974,2 , 1281. Mo, K. C.; Gubbins, K. E. Mol. Phys. 1978,3 1 , 825. Mukhamedzyanov, G. Kh.; Usmanov, A. G.; Tarzimanov, A. A. Izv. Vyssh. Ucheb. Zaved., Neff i Gar 1984, 14, 689. Murad, S . ; Gubbins, K. E. Chem. Eng. Sci. 1977,3 2 , 499. Mustafaev, R. A. replofiz. VYS. remp. 19728, 10, 906. Mustafaev, R. A. I z v . Vyssh. Ucheb. Zaved., Neff i Gaz 1972b, 15, 20. Mustafaev, R. A. Inzh. Fiz. Zh. 1973,2 4 , 663. Mustafaev, R. A. Tepiofiz. Vys. Temp. 1974, 12, 883. Naziev, Y. M.; Abasov, A. A. Vyssh. Ucheb. Zaved., Neff i Gar 1989a, 12,
65. Naziev, Y. M.; Abasov, A. A. Vyssh. Ucheb. Zaved., Neff i Gaz 1969b, 12, 81. Naziev, Y. M.: Abasov, A. A. Khim. i Tekhnol. Topi. i Masel 1970, 75, 22. Naziev, Y. M.; Abasov, A. A.; Nurberdyev, A. A,; Shakhverdyev, A. N. Zh. Fiz. Khim. 1974,48, 436. Naziev, Y. M.; Aiiev, M. A. Inzh. Fiz. Zh. 19738,2 4 , 1033. Naziev, Y. M.; Aliev. M. A. I z v . Vyssh. Ucheb. Zaved.. Neft i Gaz 1973b, 16, 73. Naziev, Y . M.; Aliev, M. A. Izv. Vyssh. Ucheb. Zaved., Neff i Gaz 1974, 17, 71. Naziev, Y. M.; Nurberdyev, A. A. Chem. Tech. (Le/pz/g)1974,2 6 , 24. Ogiwara, K.; Arai. Y.; Saito, S. Ind. Eng. Chem. Fundam. 1980, 19. 295. Parkinson, W.Ph.D. Thesls. Unlverslty of Southern California, 1974. Rastorguev. Yu. L.; Grigor'ev, B. A,; Bogatov. G. F. Inzh. Fiz. Zh. 1969a, 17, 470. Rastorguev, Yu. L.;Grigor'ev, B. A.; Bogatov. G. F. Inzh. Fir. Zh. IQSQb, 17, 847. Rastorguev, Yu. L.; Bogatov, G. F.;Grigor'ev. B. A. Khim. i Tekhnol. Top/. i Masel 1974,9. 54. Rastorguev, Yu. L.;Bogatov, G.F. Khlm. i Tekhnol. Topi. i Masel 1972. 17, 13. Rastorguev, Yu. L.; Bogatov, G. F.; Grigor'ev, B. A. I z v . Vyssh. Ucheb. Zaved.. Neff i Gaz 1989, 12, 69. Rastorguev, Yu. L.; Bogatov, G. F.; Grlgor'ev, 8. A. Izv. Vyssh. Ucheb. Zaved., Neff i Gaz 1986, 1 7 , 59.
Ind. Eng. Chem. Fundam'. 1983, 22, 97-104 Rastorguev, Yu. L.: Pugach, V. V. 1970, 13. 69. Rastorguev, Yu. L.; Pugach, V. V.
Izv. Vyssh. Ucheb. Zaved., NeftiGaz Izv. Vyssh. Ucheb. Zaved., N e f t i G a z
1971,1 4 . a i . Roder, H.; de Castro, C. A. N. J. Chem. Eng. Data 1982,2 7 , 12. Roder, H. M. J. Res. Nstl. Bur. Stand. ( U . S . ) 1982,8 7 , 279.
Rowlinson, J. S. "LiquMs and LiquM Mixtures", 2nd ed.; Plenum Press: New York, 1969. Ryabtsev, N. I.; Kazaryan, V. A. Gazov. Prom. 1969, 1 4 , 46. Ryabtsev, N. I.; Kararyan. V. A. Gazov. &lo. 1970,2 , 36. Sakiadis, 8. C.: Coates, J. AIChE J. 1955, 1 , 275. Sakiadis. B. C.; Coates, J. AIChE J. 1957,3 , 121. Sengers, J. V. "Critical Phenomena": Proc. Int. School of Physics "Enrico Fermi", Course LI, Green, M. S.,Ed.; Academic Press: New York, 1971. Sengers, J. V. Ber. Bunsenges. Phys. Chem. 1972, 76, 234.
97
Swlnney, H. L.; Henry, D. L. Phys. Rev. 1973,AB, 2566. Teja, A. S.; Rice, P. Chem. Eng. Sci. 1981,3 6 , 417. Tsederberg. N. V. "Thermal Conductivlty of Gases and Liquids"; MIT Press: CambrMge, MA, 1965; Chapter 7. Vargaftik, N. B. "Tables on the Thermophysical Properties of Liquids and Gases", 2nd ed.: Wlley: New York, 1975. Vernart, J. E. S. "The Thermal Conductlvity of Binary Organic Liquid Mlxtures"; Proc. 4th Symposium on Thermophysical Properties, 1968. Villm, 0. Collect. Czech. Chem. Commun. 1960,25, 993. Ziebland, H.: Burton. J. T. A. J. Chem. Eng. Data 1961,6 , 579.
Received for reuiew F e b r u a r y 16, 1982 Accepted O c t o b e r 1, 1982
Film Models for Multicomponent Mass Transfer: A Statistical Comparison Lawrence W. Smith' and Ross Taylor' Department of Chemical Engineering, Ciarkson College of Technology, Potsdam, New York 13676
The rates of multicomponent mass transfer predicted from several approximate solutions of the Maxwell-Stefan equations for a film model of steady-state diffusion are compared with the fluxes calculated from an exact solution for many hundreds of thousands of example problems. It is found that the assumption that the matrix of multicomponent diffusion coefficients remains constant is an excellent one and that the solution of the linearized equations due to Toor and to Stewart and Prober always provides adequate estimates of the mass-transfer rates. The method of Taylor and Smith (a generalization of a method by Burghardt and Krupiczka) is the better of two explicit methods, bettering even the solution of the linearized equations in a number of cases. The explicit method of Krishna performs well if the rates of mass transfer are low. The simple effective diffusivity methods are woefully
inadequate.
Introduction Multicomponent mass transfer is encountered in many important processes such BS condensation, distillation, and gas absorption. The rigorous calculation of the rates of mass transfer in mixtures with three or more components is complicated by the coupling, or interaction, between individual concentration gradients. Possible consequences of this coupling are that a species may transfer in the direction opposite to that expected (reverse diffusion), may not diffuse at all even though a concentration gradient for that species exists (a diffusion barrier), or may diffuse in the absence of any driving force (osmotic diffusion) [see, e.g., Toor (1957), Krishna and Standart (1979),Reinhardt and Dialer (198l)l. It is now well established that diffusion in multicomponent ideal gas mixtures is accurately described by the Maxwell-Stefan equations. Exact analytical solutions of these equations for the general n-component case are known only for a film model of steady-state one-dimensional mass transfer [Krishna and Standart (1976, 1979), Taylor (1981a, 1982a)l. In addition to these exact solutions, a number of approximate solutions have also been published. The simplest and most widely known are those based on the concept of an "effective diffusivity" [Wilke (1950), Bird et al. (1960)l. The drawback of the effective diffusivity formulas as they are frequently used is that they do not accurately reflect the character of multicomponent diffusion. More rigorous methods that are able to predict the various interaction phenomena, based on the assumption that the matrix of multicompo'Indiana Institute
of
Technology,
Fort
Wayne,
IN 46803.
nent diffusion coefficients remains constant over the diffusion path, are due to Toor (1964) and to Stewart and Prober (1964). Notwithstanding the present availability of a generally applicable exact solution, approximate methods continue to be published [Burghardt and Krupiczka (1975), Krishna (l979,1981a), Taylor and Smith (1982)l and to find applications in design calculations [Bandrowski and Kubaczka (1981), Hegner and Molzahn (1979), Webb et al. (1981)l. Even as recently as 1975, Sherwood et al., in discussing three of the effective diffusivity methods, were able to write that "the procedures have not been tested by comparison with the rigorous equations over a wide enough range to provide an adequate judgement of their relative merits". Their remarks remain true today. Indeed, they could be applied with considerable justification to most of the other approximate methods cited above. The objective of the present paper is to provide a comprehensive statistical comparison of many of the approximate solutions of the Maxwell-Stefan equations for a film model of steady-state diffusion. All of the methods included in this comparison have the important property of being applicable to mixtures with any number of constituents. The Equations of Multicomponent Mass Transfer For steady-state unidirectional diffusion under isobaric, isothermal conditions the Maxwell-Stefan equations can be written as dy. ' Y i N k - Y k N i ' Y i J k - ykJi (1) do k = l C a i k / b k=l ca,k/h
-L=c k#i
0196-4313/83/1022-0097$01.50/00 1983 American Chemical Society
=c k#i
