Self- and Mutual Diffusion in Dense Supercritical Gas Mixtures Harry 1. Frisch Department of Chemistry, State Cniversity of N e w York at Albany, Albany, N.Y. 12203
Edward McLaughlin*' Department of Chemical Engineering and Chemical Technology, Imperial College, London, S.W. 7 , England
The theory of diffusion in a dense fluid of mixed hard spheres i s used as a basis to calculate the self- and mutual diffusion coefficients in a mixture of methane and perdeuteriopropane for which experimental self-diffusion coefficients are available as a function of composition and pressure. Agreement between theory and experiment for the self-diffusion of perdeuteriopropane is, with the exception of one or two points, close to the accuracy of the experimental results ( =t5%).
k m i d a b l e experimental difficulties have created a distinct absence of experiniental values of the diffusion coefficients of dense supercritical gases as functions of pressure aud composition. This absence, even for simple chemical species, in turn means that the usual corresponding states types of correlation cannot be set up to estimate these properties. It is therefore relevant t o see what can be done by alternative means. I n recent' developments (A-Chalabi and Alclaughlin, 1970; D o u g l a s and Frisch, 1969) of the theory of diffusion in dense fluid mixtures expressions have been obtained for the self-diffusion coefficients D1, DZ and the mutual diffusion coefficient D for binary mixtures of hard spheres. From the equations it is possible to examine the pressure, temperature, and composition dependence of the various diffusion coefficients. T o date for dense supercritical gases only D a t infinite dilution of species 1, viz. , D2-1, has been compared with experiment and the purpose of the present paper is t o use the most recent development of the theory t o examine reported measurements (Woessner, et al., 1969) as functions of pressure and composition. Theory of Diffusion
The limiting values of the mutual, D , and self-diffusion coefficients, Dt, are lim D 21-
=
D2"'
1
lim Dl
=
DIO
x1+1
lirn D1 XI-0
lim D
=
D1"2
x1+0
lim D2 = D2"1
theory can be written in reduced form (Al-Chalabi and RlcLaughlin, 1970)
.
+
D1"2
(4) where bo = 2/JIu113. 922, gll, and 912 are corresponding terms in the equation of state of a hard-sphere fluid mixture
(5) where ni is the number density of species i. Now the equation of state of a pure or mixed fluid of hard spheres is not known exactly but the Percus-Yevick approximation has given results which lie close to computer calculations and as it provides convenient analytical expressions, it can be used t o evaluate eq 2 and 3. I n this approximation (Lebowitz, 1964)
(1)
lirn DZ = D20
(2)
R]"2;}-1
where v10 is the volume per molecule of pure species 1 and v is the volume of t h e mixture per molecule. The ratio of t h e diameters u of the particles is given by r = u22/u1, and their mass ratio by R = mZ/ml. The various g's are contact pair distribution functions. 91' is a contact pair distribution function of pure species 1and determines the equation of state
(1
Xl+l
=
2R
+
r)g12
= v11
+
(6)
Q22
where
x1+0
where Dl0 and D20 are the self-diffusion coefficients of the respective pure fluids of species 1 and 2 and x1 is the mole fraction of species 1. The various coefficients are illustrated in Figure 1 for a particular system. The equations for Dl and D 2 on the basis of hard sphere
with a corresponding expression for g22 with the species number interchanged. E , which is the ratio of the volume of the molecules to the volume of the system, is given b y
t Present address, Department of Chemical Engineering, Louisiana State University, Baton Rouge, La. 70803.
where vl*
=
=
ol*(zl
+ x#)/v
nu113/6 = bo/4 and o
=
l / n with n
(8) =
nl
+ nz.
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
35
XI
Figure 1. Variation with composition of methane ( x , ) of the various diffusion coefficients in the system CH4-C3D8 at 57.8"C and 121.7 bars. The curve for D1 runs through the experimental points X. Curves D' and D, are calculated, with 0 indicating the experimental points on the Dz curve (diffusion coefficients 1 OB/(mZ sec-I)).
Comparison of Theory with Experiment
From the above system of equations it can be seen t h a t DI and D2 can be calculated from the density of the fluid of known XI, u1*, r , and R. I n order to compare theory with experiment it is convenient to take the ratio of DJ/Dz given b y
eter. With this fixed value of ul* and the various experimental v values for the relevant pressures and compositions (Olds, et al., 1943; Reamer, et al., 1949, 1950) the remaining Dl/Dz ratioswerecalculated. Table I shows the results obtained by using these calculated
eq 9 after eliminating gI2in terms of gI1 and 922 with Table I. Comparison of Calculated and Experimental Diffusion Coefficients D, Obtained from D1 for Various Pressures for the System CH,-C3D8 at 57.8"C p = 121.7 bars
Di
D?
(exptl)
The theory as developed is for hard spheres and hence t o get any realistic agreement with experiment it is conventional to obtain parameters b y fitting the equations a t a particular experimental point. To do this the hard-sphere diameters were taken as the corresponding Lennard-Jones dilute gas eq 9 viscosity diameters using the equation UE&2
=
.2Q2(2!2)*
Ind. Eng. Chem. Fundam., Val. 1 1 , No. 1, 1972
0 0.3 0.5 0.7 1.0
(11)
rvhere Q ( 2 , z ) * is the collision integral. The diameter of C3D8 was taken (Hirschfelder, et al., 1954) as that of C3Hs. M'ith R known, eq 9 was fitted a t the experimental pressure of 121.7 bars a t a composition of methane x1 = 0.5 a t 573°C using the experimental D1/D2 value. This yields g22/g11 so that with the known volume v a t this p , T,and x1 eq 10 is used t o obtain u1* which was treated as the disposable param36
x1
x
1.78 3.60 6.17 10.9 19.1 DI
0 0.3 0.5 0.7 1.0
1.49 2.52 3.88 6.46 ...
1.49 2.57 3.88 6.11 10.6
p = 156.2 bars Dz D?
(exptl)
xi
02 (exptl) (calcd) IO8 m 2 sec-l
x
1.66 3.25 4.86 7.85 14.40
(exptl) (calcd) IO8 m2 sec-l
1.37 2.29 3.94 5.07 .,.
1.37 2.40 3.21 4.60 8.16
p =
Di
138.3 bars D2
(erptl)
(exptl)
X
Dz (calcd)
lo8 m z sec-l 1.38 2.41 3.82 5.60 . _ .
1.67 3.38 5.30 9.19 16.48
1.38 2.45 3.43 5.28 9.26
p = 173.4 bars
Di
D2
(exptl)
(exptl)
X
1.55 3.03 4.50 7.03 13.18
Dz
(calcd) 108 m z sec-'
1.30 2.13 3.51 4.34 ,..
1.30 2.26 3.03 4.19 7.53
ratios and the experimental D I values t o give D2. At' x1 = 0 the rm-erse ol)cra,tion is carried out and the experimental Dso x d u c is used t o give tlie D1"* result which is in italics in t,he table aiid which was not measured. On tlie whole the agreemciit, n.it,li experiment where the D? values have a reported accwracy of ==I 5% is remarkably good with the exception of the value at p = 156.2 bars and :cl = 0.5. If t'lie mutual diffusioii coefficient is requircd it call be calculated by similar methods from the corresponding equation
IT
here
D'
=
D / ( b hi al i) 111x l ) T , n
(13)
~ i t l al i the activity of species 1. Alteriiativel~with the g2? gil values available aiid tlic experimental D1 T d u e s then it i b wnpler to m e tlie equation
Figure 1 gives a relxesentative set of curves for 5" = 57.8"C aiid p = 121.7 bars, but no mutual diffusion Iiieasurenieiits are available for comparisoii. I t is possible of course to fit the ratios D 1 ~ D 1and o D?lDf separately at a fixed pressure, temperat~ure,anti composition and use the zil* values ohtaiiied to calculate the diffmioii coefficients for other pressures aiid compositions rather tliaii fitting tlie ratio as aboi-e. Other procedures involving the ratio of dense to dilute gas diffusion coefficients may be used (AIcConalogue and AIcLauglilin. 1969) and have also been shon-ii to be satisfactory. It should be pointed out, however, that while the parameters obtained a t one temperat'ure can be used for generating t'he pressure and composition dependence of the diffusion coefficients they are unlikely to be useful when used a t a different temperature, as wlieii hard sphere t'lieory is applied t o real systems temperature dependent diameters are necessary. Conclusion
Provided volumetric data are available, it appears t h a t the theory of diffusion in systems of mixed hard spheres may be used to estimate diffusion coefficients 111 dense supercritical gases as fuiictionb of pressure and compositioii. A limited amount of diffusion data is necessary to enable this to be done. Once D1, D2>and D have been obtained experimentally for any simple mixture over a range of pressures and as
functions of composition it will then be possible to check more fully the domain of validity and limitations of the method. Nomenclature
a i = activity of species i bo = hard-sphere second virial coefficient Di0 = self-diffusion coefficient of pure species i D i = self-diffusion coefficient of species i in the mixture Di"j = self-diffusion coefficient of species i in the mixture infinitely diluted by species j D' = activity-corrected mut'ual diffusion coefficient of species i and ,j D = mutual diffusion coefficient of species i a n d j gio = contact pair distribution fuiict'ion of pure species i g L t = contact pair distribution function of species i in the mixture = contact pair distribution function of species i and j qij iii the mixture k = 13oltzinann constant = mass of part.icle i = number density of species i in the mixture ni 7~ = number density of the mixture p = pressure r = ratio of diameters u j j / u j j of particles in the mixture R = ratio of masses mj/mi of particles ill the mixture r = volume of the inisture per molecule vi* = volume of a inolecule of species i = volume per molecules of pure fluid of species i x i = inole fraction of species i ziio
GRI:ISRLI.:TTl:RS
E
=
uti
=
Q2(*,2)* =
ratio of volume of the molecules to the volume of the em ciiaiiieter of a particle of species i v iscosity collision integral for the Lennard-Jones 12-6 poteiit#ial
literature Cited
AI-Chalabi, H. A., McLaughliii, E., 31ol. Phys. 19, 703 (1970). I)ouglaca, 1). C., Frisrh, H I,., J . Phys. Chem. 73, 3039 (1969). Hirsrhfelder, J. O., Curtis*, C. F., Bird. 13. B., "llolecular Theorv of Gases and Liqiiids," p 1110, \%ley, S e w York. N. Y.. 1954. Lebowitz. J. L.. I'hus. RPL,. 133. ,4885 11964'1. lIcConalogue, 11. J . ) lIrLaughlin, E., J f d . Phys. 16, 501 (1969). Olde, 13. H., Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Cheni. 35, 922 (1043). Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chena. 41, 482 (1949). Reamer, H. H., Sage, B. H., Lacey, W.X., I n d . Eng. Chern. 42, 534 11950). W'oe&r, 11. E., Snowden, B. S.,Jr., George, R . A., Melrose, J. C., I X D . ESG.CHl:M., FITDAM. 8, 779 (1969). 13 i,ri;iv~:~) for review September 29, 1970 ACCIKPTKD September 7, 1971 This work was supported in part, by the Petroleum Research Fund, administered by the American Chemical Soc,iety, Grant No. 3519C-5,6, and the National Science Foundation, Grant No. GP-1988 1.
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
37
