J . Phys. Chem. 1994,98, 1306-1310
1306
A Method for the Prediction of the Transport Properties of Dense Fluids: Application to Liquid *Alkanes T. F. Sun, J. Bleazard, and A. S. Teja' School of Chemical Engineering and Fluid Properties Research Institute, Georgia Institute of Technology, Atlanta, Georgia 30332-0100 Received: July 9, 1993; In Final Form: October 13, 1993'
A new method is presented for the prediction of the transport properties (viscosity, thermal conductivity, and diffusion coefficient) of liquids. The method is based on the rough hard sphere theory of Chandler and incorporates the recent molecular dynamics results for Lennard-Jones fluids reported by Heyes. The Lennard-Jones parameters and the effective hard sphere diameter required in the calculations were determined from a knowledge only of the density-temperature behavior of the fluid a t atmospheric pressure using the Ross variational perturbation method. Analytical expressions are presented for the transport properties of the n-alkanes, and comparisons are shown between calculations and experiment. The results generally agree with published results within experimental error. The analytical expressions also allow the transport properties of the alkanes to be calculated a t conditions of temperature and pressure where direct measurements would be difficult.
Introduction The transport properties of dense fluid mixtures are required in many engineering calculations involving fluid flow, heat transfer, and mass transfer. They are also of interest because they provide a framework for an understanding of intermolecular forces. Since it is unlikely that experimental measurements for these properties at all conditions of interest can be found in the literature, reliable methods for their estimation are of considerable importance in process calculations. The transport properties of gases may be predicted from kinetic theory. However, no satisfactory theory exists for liquids, although some success has been achieved by the application of the rough hard sphere theory of Chandler. The objective of the present work was to build on the success of the rough hard sphere theory and combine it with some recent results obtained by molecular dynamics simulations in order to obtain a method for the correlation and prediction of the diffusivity, thermal conductivity, and viscosity of fluids over a wide range of conditions.
Development of the Method According to Chandler,' the experimental diffusion coefficient D and viscosity g of a real liquid can be approximated by a rough hard sphere diffusion coefficient DRHSand viscosity ~ R H S . Furthermore, DRHSand ~ R H Sare proportional to DSHSand ~ S H S for a smooth hard sphere fluid as follows:
between 0.9 and 0.7 for slightly acentric molecules such as C4H8, CClF3, C6H6, and C6H12, and it varies from 0.6 to0.3 for molecules such as CCl3F, CHF3, Si(CH3)4, and C7H14 (methylcyclohexane) which exhibit significant departure from sphericity.* The decrease in CDwith increasing complexity in molecular shape suggests that coupling produces an additional mechanism for molecular relaxation. From fits of high-pressure NMR results for D, it has been found that CDis nearly independent of density, although it is a function of temperature. The rough hard sphere theory has also been successful in interpreting high-pressuredata of shear viscosity for many organic liquids.2 Once again, the coupling parameter C, was found to be almost independent of density but showed strong dependence on temperature. Moreover, core-size parameters of fluids derived from experimental data of high-pressure self-diffusivity were found to be within 1-2% of the values obtained from experimental data of high-pressure viscosity. One limitation of the rough hard sphere theory is that it is valid in a narrow range of densities. In particular, it fails at reduced densities pa3 > 0.97, as the hard sphere system is metastable under these condition^.^ The theory is also not useful when the shape of the molecules deviates significantly from spherical. To overcomesome of these limitations, we propose the following:
and in addition where CD and C, reflect the degree of coupling between translational and rotational motions and are unity for a smooth hard sphere fluid. The diffusion coefficients of real liquids are therefore smaller than those calculated by assuming a smooth hard spherical core, whereas the viscosities are larger. The rough hard sphere model has proved successful in interpreting several sets of measurements of self-diffusion coefficients of polyatomic fluids made by high-pressure NMR spin echo techniques, indicating the physical significance of the coupling parameter CD. At normal liquid states, CD is 1 for nearly spherical molecules such as CH4, CF4, and SF6. It lies Abstract published in Aduance ACS Abstracts, January 1, 1994.
0022-365419412098-1306%04.50/0
In eqs 3-5, DLJ,~ L J and , XLJ are the diffusivity, viscosity, and thermal conductivity of a Lennard-Jones (L-J) fluid. The L-J potential was chosen as a model for real liquids because it is a realistic potential for molecular liquids even at dense fluid condition^.^ In addition, exact results obtained by the method of molecular dynamics (MD) for the diffusivity, viscosity, and thermal conductivity of an L-J fluid over a wide range of density and temperature were recently p ~ b l i s h e d .Finally, ~~~~~ the use of different types of experimental data (pVT,gas solubility,viscosity, diffusivity, surface tension, high-pressure NMR, and Raman spectroscopy) has been explored for the evaluation of the , ~ a consequence, L-J parameters of the L-J p ~ t e n t i a l . ~As 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1307
Transport Properties of Dense Fluids parameters for a wide variety of organic liquids are available using either equilibrium or transport property data. The L-J (12-6) potential relates the intermolecular potential energy u(r) for a pair of molecules to the separation distance r as follows:
where e represents the depth of the potential well and a is the separation distance a t zero energy. Using this potential function, H e y e ~performed ~ ~ ~ , ~equilibrium and nonequilibrium molecular dynamic (MD) computer simulations to obtain the diffusivity DLJ,viscosity ~ L J and , thermal conductivity XLJ of an L-J fluid over almost the entire fluid region. As expected, he found that the repulsive part of the L-J potential determines the liquid structure and therefore has the greatest effect on the thermodynamic and transport properties of the fluid. The attractive part of the potential was found to play only a minor role. Therefore, as a reference model, Heyes chose the hard sphere potential to characterize the transport properties of the L-J fluid. By modifying the Enskog hard sphere theory and introducing a temperature-dependent hard sphere diameter, he summarized his MD results in the simple analytical form:
qLJ/qo = [1 - exp{-0.2195(V/Vo - 1.384))]’.075
X,,/X
= 1 -exp{-0.1611(V/Vo-
1.217))
(8) (9)
where DO,70, and XO are the diffusivity, viscosity, and thermal conductivity for a dilute gas and are given in classical kinetic theory8 as
In eqs 10-12, k is Boltzmann’s constant, T i s the temperature, m is the mass of the molecule, p is the number density (= N/V), and WJ)and Q(2,2) are collision integrals.* The ratio V / Vo of the molecular volume to the volume at close packing is given by
V/Vo= 2’ ’/pDHS3
(13)
in which DHSis an effective hard sphere diameter. Equations 7-9 are similar to the equations for the transport properties of hard sphere fluids derived by D y m ~ n dexcept , ~ that the coefficients of eqs 7-9 were obtained by fitting the MD results of an L-J fluid. In order to apply these equations to a liquid, one J ULJ as well as DHS. The needs to have the L-J parameters ~ L and latter is not specified for an L-J fluid and must therefore be defined. A method for obtaining these parameters for any fluid based on the Ross perturbation approachlo is outlined below.
Determination of e, u, and & Using the Ross Variational Perturbation Theory In this section, we describe a method for the calculation of the parameters e and a and DHSfor liquids from experimental density data a t ambient pressures using the Ross perturbation approach. The Ross perturbation approach was chosen since it is well suited for liquids and has the advantages of accuracy and simplicity when compared with other approaches, such as those of Barker and Henderson” and Weeks, Chandler, and Andersen.12 Also, our experience with these theories suggests that the Ross variational approach works well in many cases where the other
theories fail. The three parameters u, e, and &s for a liquid were obtained as follows: 1. Initial values of c and uof the fluid were chosen. (Literature values for similar substances may be selected, since the method is not sensitive to the values of e and u.) 2. An experimental P, V, T data point is selected, and the volume Vis converted to the number density p (= N A / V ,where NAis Avogadro’s number). 3. An initial value of DHS at the given T and p is chosen. 4. The Helmholtz energy A is calculated at the given T and p as follows:
whereAideal,Acs, A,,,, and Am are the ideal, Carnahan-Starling, Ross, and perturbation contributions to the Helmholtz energy. These contributions are obtained from
A,, = N k T ( 4 r - 3?)/(1 - {)’
(15)
A , = 0.5iVp somgHs(r) u(r) dr
(17)
with the packing traction
p given by
r= .lrpDH,3/6
(18) In eq 17, ~ H is S the-hard sphere radial distribution function and u(r) is the Lennard-Jones potential function given in eq 6. A computational algorithm for g ~ s ( ris) given in ref 8 and is based on a closed-form analytical expressionI3derived from the PercusS a complicated function Yevick e q ~ a t i 0 n . lThe ~ resulting ~ H is of the reduced density and the molecular separation distance r. Integration of eq 17 was therefore performed numerically using the Gauss-Legendre method. 5 . Varying DHSand repeating the above calculations allows the Helmholtz energy A to be obtained as a function of DHS.The value of &s selected corresponds to a minimum in the Helmholtz energy A . 6. The pressure may now be calculated at the given T and p by differentiating eq 14 as follows:
( 8 A I d V ) T = -P (19) 7. Steps 2-6 are repeated for other P, V, T data points. 8. The deviations between calculated and experimental pressures a t all data points (Le. at all values of T and p) are minimized using a nonlinear least-squares method to yield the final values of e and a (and the values of DHSas a function of temperature and density). It should be mentioned that perturbation theory has, in many cases, been applied to model systems, for which the properties are “exactly” known from computer simulations. The central goal of this type of computation is a check of the accuracy of the theory. In recent years, perturbation theory has also been compared with “real experimental data”. For instance, the Ross theory was applied to dense fluid regions of neon to obtain an effective exponential -6 potential by fitting high-pressure p V T andspeedofsounddata.15 InthisandotherI6studieqexperimental data at high pressures were employed to calculate potential parameters. However, high-pressure experimental pVTdata are very scarce for the liquid n-alkanes studied in this work. Therefore, the density-temperature relations at atmospheric pressure were used in this work. From our calculations on various liquids, we have found that the difference between L-J parameters derived from high-pressure p V T data using the Ross method and those from density data at atmospheric pressure agree with each other within a few percent.
Sun et al.
1308 The Journal of Physical Chemistry, Vol. 98, No. 4, 1994
TABLE 1: Comparison of 4 1 s Calculated in the Present Work with Values Obtained Using High-Pressure Transport Prooertv Data ~
-
~~
DHS
substance 2-ethvlhexvl benzoate
temp (K) this work 313 7.26 373 7.22 2-meth ylcyclohexane 223 5.73 298 5.68 pyridine 303 4.96 423 4.91 benzene 303 5.00 333 4.98 cyclohexane 313 5.38 383 5.36 fluorotrichloromethane 341 4.88 460 4.82 carbon tetrachloride 283 5.16 328 5.14 I
-
(4
lit
7.34 7.18 5.78 5.74 4.94 4.90 5.12 5.11 5.52 5.50 5.03 4.90 5.27 5.22
7:
G
t 6 1
5 m
.-
: .
L
.
5
5 -
:
:
+
ref diff (W) 2
2
-1
L
2
-1
7
2
+0.5
2
-2.4
2
-2.8
2
-2
.
I
I
I
-
4 -
.
3
3
22
-2
5 10 15 n - Alkane Carbon Number
0
20
Figure 1. L-J potential diametersof the n-alkanes as a function of carbon
number. 600
In order to employ the potential parameters calculated using
-
the above procedure for predicting transport properties, it must first be demonstrated that the method yields essentially the same values using either equilibrium or transport property data. In Table 1, we compare our values obtained using ambient pressure density data with those obtained from high-pressure D and 7 data. The values of the core size obtained by us are generally slightly lower ( 0.7 ( T > 120 K) for C2H6 or T* > 0.6 ( T > 300 K) for C16H34, C, was found to decrease with increasing
Transport Properties of Dense Fluids
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1309
1 c
1.5
0"
9
-5
oc
:
1
LL m
.-5 a
0.5 -
s
.,. e
c 2 -.
rl
0 ' 0.5
0
F' 0.6
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I
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0.7
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1
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0.8
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1
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0.9
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, , ,
1.0
1.1
Temperature ( T' )
Figure 4. Ln C, as a function of were calculated using eq 23.
P for the n-alkanes.
The solid lines
Temperature ( T')
Figure 6. CDof the n-alkanes as a function of reduced temperature P. The solid lines were calculated using eq 25.
the diffusivity as a function of T* and n was obtained by fitting 70 experimental data points for the n-alkanes reported in ref 21, including data for C7, C9, Clo, C12, c14,and c16. The correlation is shown in Figure 6, and an analytic form is given below. The AAD of the fit was
