DIMENSIONLESS NUMBERS applied stress A = elastic modulus = pgR/G time scale of flow D e (Deborah number) = relaxation time = RG/Up Le (Levich number)
:=
viscous stress total stress = pU/pgR2
inertia R e (Reynolds number) = viscous force =
PWP
I Y s (Weissenberg number) = (relaxation time) (shear rate) = pU/GR For tensorial quantities, the units of the physical components are given. literature Cited
(5) Bagley, E. B., Birks, A. M., Warren, G. G., “Polyethylene Flow Studies,” movie filmed at Central Research Laboratory C.I.L., McMasterville. (6) Caswell, B., Schwartz, W.H., J . Fluid Mech. 13, 417 (1962). (7) Chao, B. T., Phys. Fluids 5 , 69 (1962). (8) Hadamard, J., Corn@. Rend. 152, 1735 (1911). (9) Levich, V. G., Z h . Eksp. Teor. Fiz. 19, 18 (1949). (10) Lochiel, C., Ph.D. thesis, University of Edinburgh, 1964. (11) Maxwell, J. C., Trans. Roy. Soc. 49, 157 (1867). (12) Metzner, A. B., personal communication, 1965. (13) Metzner, A. B., White, J. L., Denn, M. M., “Constitutive Equations for Viscoelastic Fluids for Short Deformation Periods and for Rapidly Changing Flows. Significance of the Deborah Number,’’ to be published. (14) Moore, D. W.? J . F l u i d M e c h . 6, 113 (1959). (15) Zbid., 16, 161 (1963). (16) Rybzcynsky, W., Bull. Acad. Cracouia A-40 (1911). (17) Stokes, G. G., “Mathematical and Physical Papers,” Cambridge University Press, Cambridge, 1880. (18) \Yhite, J. L., J . Appl. Polymer Sci. 8, 2339 (1964). (19) LYhite, J. L., Metzner, A. B., Zbid., 7, 1867 (1963). (20) White, J. L., Metzner, A. B., Denn, M. M., Am. Inst. Chem. Engrs. Meeting, Philadelphia, 1965.
(1 ) Xstarita, G., “Sugli aspetti pratici dei problemi di meccanica dei fluidi viscoelastici,” Zng. Chzm. Ztal., in press. (2) Astarita, G., Apuzzo, G., A . Z. Ch. E. J . 11,815 (1965). RECEIVED for review December 15, 1965 (3) Astarita, G., Marrucci, G., Rend. Accad. Ltncei ( R o m e ) 8-36, 836 ACCEPTED May 5, 1966 (1964). (4) Astarita, G., Nicodemo, L., IND.EX. CHEM.FUNDAMENTALS Based on a paper given at the 1965 h m u a l Meeting, American Institute of Chemical Engineers, Philadelphia, Pa. 5 , 237 (1966).
DIFFUSIVITIES IN THE SYSTEM: CARBON D IOX I D El-N IT ROG EN-A RG0 N T H O M A S A. P A K U R A R ’ A N D J O H N R. F E R R O N Department of Chemical Engineering, University of Delaware, A’euark, Del.
’
Binary diffusivities are presented for the systems C02-Ar and Cop-N2 a t 1 atm. and 1081 to 1810’ K. Tracer diffu:jivities of 14C02through ternary mixtures are also given. The binary data, together with prior, low temperciture data, are correlated empirically (mean deviations of 2 to 3%) with the Lennard-Jones ( 1 2-6) poteintial. The resulting potential is related to that obtained from prior measurements of viscosity b y means of a rigid, spherocylinder model, thus providing means for prediction of one transport property from measurements of the other. Ternary measurements agree satisfactorily with values computed from binary diffusivities.
Concentration effects on binary diffusivities are small and just beyond the observable
range.
of radioactivity downstream from a point source of W 0 2 in a n isothermal, laminar stream of combustion products from a flat, diluted, carbon monoxide-oxygen flame have provided high temperature diffusivities in several previous studies. T h e self-diffusion coefficient of carbon dioxide has been determined a t 1 atm. and 1103’ to 1944’ K. (70, 20). Measuremmts of binary diffusivities of the systems C02-argon and Con-nitrogen a t 1156’ to 1676’ K. (27) and of COL-water a t 975‘ to 1616’ K. (70) have been reported. I n addition, by use of tritiated water vapor as tracer, the binary diffusivity of COZ-water a t 1100’ to 1570’ K. and the selfdiffusivity ofwater a t 950’ to 1400’ K. (8)have been studied. In this paper, binary diffusivities of C02-Ar and c 0 2 - N ~ are reported for the widest temperature ranges (1132’ to 1798’ K. and 1081’ to 1810’ K., respectively) which can be EASUREMENTS
Present address, E. I. du Pont de Nemours and Co., Inc., Richmond, Va.
attained with the present apparatus. I n addition, tracer diffusivities of 14C02in ternary mixtures of carbon dioxide, nitrogen, and argon are given for the temperature range 1176’ to 1525’ K . T h e binary data are correlated in terms of spherical and of spherocylindrical models. Tracer diffusivities of 1dCOz are compared with those computed from binary data using mixture rules derived for the effective diffusivity from the StefanMaxwell equations (6, 25). Experimental Methods
The experimental apparatus and procedure and the methods of data analysis have been described in detail ( 9 ) . A brief summary of these and description of one improvement that has been made for the current measurements are provided here. Figure 1 illustrates schematically the major parts of the apparatus. Premixed carbon monoxide, oxygen, and diluents are fed to the water-cooled, porous-bronze burner. Combustion VOL. 5
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occurs in a uniform flame about '/4 inch high. Combustion gases rise in laminar flow through the measuring section, which is maintained a t the temperature of the experiment by electrical heating coils, then through the cooling section to the chimney, where the combustion gases are diluted with air and discharged outside the building. Tracer gas, 1 4 C 0 2in ordinary carbon dioxide, enters through a capillary of 0.5-mm. i.d. T h e gas sample is withdrawn continuously, through a capillary tube of the same size, and passed to a radioactivity counting cell. Samples are taken a t several heights along the axis of the measuring section. The radioactivity relative to that of the incoming tracer gas is inversely proportional to the height of the measurement. The tracer diffusivity of 1 4 C 0 2is obtained from the slope of the linear plot of the reciprocal of relative radioactivity us. height. If the concentration of 1 4 C 0 2is small (about 0.5% or less), one can relate tracer diffusivity (U1*) to the self-diffusivity of carbon dioxide (Oil), the binary diffusivity ( D ~ z )and , the mole fractions (XIand XZ) of bulk-gas components by
Dll is known from similar measurements in pure CO2 (9, ZO), and x1 and x2 have been fixed a t known values. Determinations of D1* a t various temperatures thus provide, through Equation 1, values of 0 1 2 a t the same temperatures. The slope of the line relating radioactivity and reciprocal distance also depends upon the tracer flow rate, usually 0.3 to 1.7 ml. per minute a t room temperature, which must be measured independently. I n previous work, this measurement has been the source of the largest experimental errors. The technique used involved injecting a soap bubble into the gas and observing the time of its passage through a small pipet. While this method is convenient, its reproducibility depends to some extent upon the amount of soap used. Moreover, the liquid soap employed tends slowly to contaminate the radioactivity measuring system, increasing the background count and the error of the measurement. To alleviate these difficulties, the soap-bubble technique has been replaced in the current experiments by the metering system shown schematically in Figure 2. Two pistons, with different diameters, are connected to a common piston rod and mounted in precision-bore glass cylinders. Water is metered accurately into the lo\\er cylinder a t a relatively large rate (5 to 20 ml. per minute). This raises both pistons, displacing tracer gas from the upper cylinder a t a rate which can be observed directly from the rate of rise of the piston through use of a millimeter scale engraved on the outside of the upper glass cylinder. The maximum error in measurement of D1* or of Dll, arising from errors in positioning of the sampling capillary, in temperature measurement, tracer metering, and sample analysis, is estimated to be 5%. This causes a maximum error of about 10% in values of D,,computed from Equation 1. As is indicated in the following sections, the experimental diffusivities, together with low temperature values in the prior literature, can be correlated with mean deviations of 2 to 3% for the entire range of temperatures used in this and other studies (200' tq 1800' K.). Binary Diffusivities at High Temperatures
Table I shows all of the data obtained a t high temperatures for the systems COa-Ar and C02-N2. Results for lower tem554
l&EC FUNDAMENTALS
EXHAUST
PROBE
ELECTRIC HEATING
2" TRACER INLET COOLED BURNER
AND
-FUEL
OXYGEN
Figure 1. Schematic diagram of the diffusion apparatus
Lp--$ WATER
DRAIN
INLET
Figure 2.
Tracer gas metering system
peratures, from this work and from prior studies, are listed in Table 11. The prior results have been obtained by a variety of methods. Those of Walker and Westenberg (see 77), which span the gap between low and high temperature data for COL-NZ, were obtained by means of a point-source technique identical in principle to the method used here. Comparison of C02-Nz data in Tables I and I1 sho\vs excellent agreement between the results of Walker and \Vestenberg and those obtained in this work. This is brought out in more detail below. A Spherical Molecular Model
T h e Lennard-Jones (12-6) potential provides a convenient basis for empirical correlation of binary diffusivities. This is a spherically symmetric model, and its application to nonspherical species such as carbon dioxide and nitrogen contrib-
Table 1.
Binary Diffusivities at High Temperatures x z = 0.3
= 0.5 ~x2 _ _
012,
T,
’ K.
1132 1146 1280 1366 1378 1501 1545 1662 1717 1722 1761 1798
rq. c m . / sec.
1.71 1.77 2.1b 2.27 2.31 2.75 2.96 3.32 3.47 3.35 3.50 3.71
T , sq. c m . / a K. sec. System: 1185 1 . 8 3 1388 2 . 1 3 1403 2 . 4 3 1427 2.53 1439 2.47 1495 2.65 1633 3.14
-
x2
012, O
T , sq. c m . / K. sec.
T,
K.
= 0.7
DIZ, sq. em./ sec.
Table II.
012,
Temp.,
1.88
1207 1225 1315 1364 1368 1371 1380 1445 1482 1503 1538 1600 1676 1685
2.01 2.38 2.39 2.59 2.48 2.43 2.66 2.71 2.84 3.08 3.17 3.21
1182
1.96
1.88
1810
1.79 1.98
2.45 2.46 2.67 2.83 2.96 3.33 3.33 3.47 3.73 3.82
1098 1262 1426 1490 1500 1708 1728
1.68 2.08 2.55 2.92 2.77 3.51 3.40
1081 1156 1158 1241 1275 1286 1333 1430 1469 1569 1588
1653 1670
1.64 1.78 1.92 2.07 2.21 2.34 2.26 2.72 2.85 2.99 3.15 3.32 3.43
1164 1191
1.95 2.08
Table 111.
Coejicient a0 a1
utes little to physical understanding. However, the potential does represent the diffusivity-temperature relationship faithfully and allows one to check consistency of various sources of experimental data. Fumerical values of the collision integrals of Hirschfelder, Curtiss, and Bird (72) were used. T o facilitate interpolation a t untabulated values of reduced temperature, the integrals \yere fitted to the fol1ov:ing form.
Here a* is either or Q(2,2)*, the reduced collision integrals for first approximations of diffusivity and of viscosity, respectively (72). The reduced temperature is T* = k T / e , where k is Boltzmann’s constant and E is the depth of the attractive portion of the Lennard-Jones (12-6) potential. The numerical coefficients for diffusivity and viscosity are given in Table 111. Equation 2, with the tabulated coefficients, reproduces Hirschfelder, Curtiss, and Bird’s values to a t least three significant figures for all temperatures in 0.3 < T* _< 400. The aberage deviation on this range is 0.05%; and the maximum deviation, which occurs for the extreme values of Z*, is 0.370. The Lennard-Jones parameters e/k and u, the molecular separation dihtance a t zero potential. n e r e determined so as to minimize the standard deviation between calculated and experimental transport properties. This error criterion is superior to that of minimization of the sum of squares of deviations. Errors associated with transport property measurements increase with increasing temperature. Unless the deviafi(ljl)*
K.
Sq. Cm./ S e t . System: CO2-Ar
Ref.
0.1326 0.14 0.139 0.139 0.1652 System: c 0 2 - N ~
(73)
273 288 288 293 293 293 293 293 205 298 298 298 300 400 500 600 700 800 900 1000 1100
System C O Z - N ~ 1141 1227 1383 1414 1421 1530 1542 1657 1680 1759 1760
a
276.2 293 293 295 317.2
COz-Ar 1181
Binary Diffusivities at low Temperatures
az a3 a4 as a6 a1
a8 a9 a10
(7)
(24) (27)
(73)
0.144 0.158 0.158
(3) (7)
0.163
0.160 0.163 0.16 0.16 0.159 0.167 0.168 0.165 0.17 0.30 0.44 0.61 0.79 0.99
1.21 1.45 1.70
Interpolation Coefficients for Reduced Collision Integrals Q(1,l)
*
0.3646 -0.8434 ( l o w 4 ) 1.8139 -2.1265 1,4964 -0.60835 0.1276 -0.010825 0.6057 -0.8733 1.1612
Q(2,Z)
*
0,4305 -1.00093 (10-4) 1 ,9064 -2.2847 1.7503 -0,75526 0.1610 -0,01355 0.6360 - 1 ,0045 1 ,3064
tion is iteighted, for example, by the inverse measured value, as in the case of the standard deviation, the calculated curve will tend to be relatively far from the experimental points a t low temperatures, where both deviations and measured values are small. Figures 3 and 4 show the smoothed temperature functions for C02-Ar and for c02-S~.Mean deviations are, respectively, 2.8 and 2 1% for the full ranges of temperature. Also shown in Figures 3 and 4 (as solid lines) are smoothed functions for the self-diffusion coefficient of COn (20) and for those of argon and nitrogen. The latter were estimated from Lennard-Jones parameters derived from experimental vircosities (75. 79). For the gas pairs studied here values of 0 1 2 lie between the curves of D11 and 0 2 2 . Any effect of concentration on D I Z should appear as a tendency of experimental values a t low C O S concentrations to cluster near the curve for 0 2 2 and for those a t high CO2 concentrations to be close to the curve for 0 1 1 . There is barely observable confirmation of these tendencies, a t least in the data of rigure 4. Observable concentration effects on Dl2 are not expected for monatomic gas pairs, but an effect could be noticed for polyatomic gases. The dashed lines of Figure3 3 and 4 represent D12 as predicted from Lennard-Jones correlations of experimental viscosities of VOL. 5
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NOVEMBER 1 9 6 6
555
5
I
A
-
4
a
90% 70%
0
50%
I
I
I
I
I
I
I
cot
cop
0 30% REF. I,C13, 0 2 211 24
w 0
-
v)
\ 3
&,.
64'.",
3 0
z
&@,'
-
,e@:/
0
ai*
1
C02
-
DIz(ESTIMATEO F R O M VISCOSITY)
I -
'
0 -
I
I
I
I
I
-
I
I
Figure 3. Binary diffusivities of the system carbon dioxide ( I )-argon (2) tennard-Jones parameters for the smoothed curves o r e 600' K., u = 2.928 A.; 4 2 , e l k = 277 D11, E f k 3.436 A. K., u = 3.208 A.; D22, Efk = 1 1 9' K., u
5
I A
-
4
0 0
0
I
I
I
I
I
I
I
9 0 % COz 7 0 % co2 50% C02 3 0 X C02 REF. 1 , 2 , 3 , 1 1 , 2 1 , 2 2 , 2 3
0 : 3 \
5
-
I -
200
400
I
I
I
I
I
I
I
600
800
1000
1200
1400
1600
l8Oo
TEMPERATURE, O K
Figure 4. Binary diffusivities of the system carbon dioxide ( 1 )-nitrogen (2) Lennard-Jones parameters for the smoothed curves are: D ~ I E, f k = 600' K., u = 2.928 A.; D12, e f k = 172 K., u 3.484 A,; 0 2 2 , e l k 74' K., D 3.754 A.
the pure components. The arithmetic mean of the purecomponent values of u and the geometric mean of the purek used. The displacement of the component values of ~ / were dashed lines from the experimental data indicates that empirical Lennard-Jones potentials for diffusivity and viscosity are quite different. Better agreement can be obtained by use of new mixing rules for the pure-component values of u and e / k . These rules cannot be of the form of the usual rational means, however, such as
wherepl and p2 are either u or Elk for the pure components and r is any real number (7 = 0 corresponds to the geometric mean). For M , must be intermediate in magnitude top1 and pr. LVith both C01-N2 and C02-Ar pure component values of u derived from viscosities are larger than the mean values needed to fit the experimental points of Figures 3 and 4. If the mixing rules cannot be changed in a reasonable way, one can consider changing the potential function so as to take into account the nonspherical nature of the polyatomic species, Nz and COZ. 556
I & E C FUNDAMENTALS
A Rigid Spherocylindrical Model
Curtiss and coworkers (4, 5, 7, 76-78) have studied a kinetic theory for rigid, spherocylindrical molecules. Their final result, a function of molecular size and mass distribution, is the quotient of a transport property of a rigid, spherocylindrical molecule and that of a spherical molecule of equal volume. The spherocylinder model has been previously applied to pure carbon dioxide so as to represent diffusion coefficients, viscosity. and thermal conductivity by the same Lennard-Jones (12-6) potential (70, 74, 79-27). Here the problem of reconciling viscosity and binary diffusivity for mixtures containing carbon dioxide is considered in more detail. To do this we examine data for viscosities and for diffusivities to determine empirically whether quantities can be found which, when divided into the experimental transport properties, give new "diffusivities" and "viscosities" u hich can be correlated satisfactorily by the same spherically symmetric, Lennard-Jones potential. The quantities thus found represent quotients of spherocylindrical and spherical transport properties and can be compared with similar quotients calculated from the theory of Curtiss and co-workers. T h e empirical quotients are temperature-dependent, whereas the theory, which is for rigid molecules, does not suggest temperature dependence. This difficulty may be circumvented in the empirical part of the procedure by choosing a quotient of unity for lower temperatures and stepping to a value other than unity a t some cut-off temperature. I t is reasoned that the outer potential shells of a spherocylindrical molecule will appear essentially spherical to a colliding molecule. Above some cut-off temperature, collisions will be more energetic, penetration of the potential fields will be deeper, and the nonspherical nature of the molecule will begin to play a significant role. The cutoff temperature may be at any point between 400" and 1200' K. for the C02-Ar and C02-N2 systems \vithout affecting seriously the empirical quotients obtained. The empirical procedure depends upon the observation that the value of e/k determines the slope of the diffusivity temperature function, and the value of u determines the magnitude. Consider the data and curves for D12 in Figures 3 and 4 and Z visthe dashed curves representing predictions of D ~ from cosity parameters. For temperatures above 800" K., we divide experimental diffusivities by the number fd so that the smoothed curves for Di2 will be parallel to the dashed curves. Lennard-Jones correlation of these adjusted diffusivities together \$ ith experimental viscosities will yield the same value of ~ / for k both sets of transport properties. Now we find a number f u which when divided into experimental viscosities will give new dashed curves in Figures 3 and 4
Empirical and Theoretical Nonsphericity Factors for Viscosity and Diffusivity
Table IV.
Empirical Results cutoff temp., System
O
K.
coz
fv
fd
Theoretical Predictionsa ~
COP, %
0.955 1.326
-
0.973 1.145
C02-Ar
0.93-0.98 0.90-0 .97 0.93-0.98 0.95-0.99 0.89-0.97
67 50
1.18-1.35 1.13-1.26 1.09-1.18 1.26-1.42
33 0.930 1.061 50 800 0.925 1.059 1200 0.923 1.070 Based on results f o r rigid spherocylinder model (7, 77).
c0z-N~
0
fd
1.27-1.53
1 .o
Ar
N" .*
fu
0.86-0.96
400
1 .o 1 .25-1 .36
\chich coincide with the adjusted curves for 0 1 2 . T h e parameter u has been changed without observable effect on elk. That is, the adjusted viscosities, experimental values divided by fV, and the adjusted diffusivities, experimental values divided by fd> \vi11 have :‘le same :set of Lennard-Jones parameters. Table I V illustrates both empirical results and predictions from the calculations of Curtiss and coworkers. Ranges of values are shoivn for the latter because dimensions and mass distribution of C O ? are available from several sources, not all of which are in agreement. ‘There is good agreeinent between empirical and theoretical values for pure carbon dioxide and for the CO?-Ar system. T h e empirical results i:n the latter case have not been classified as to composition; but the bulk of the experiments \cas carried out with argon concentrations of 507, and higher, for Lvhich case agreement bet\veen experiment and theory is excellent. ‘There is also good a.greement for the viscosity factor of the COZ-S? system. The theoretical diffiisivity factor is high, however, compared ivith the empirical value. It also seems high on the intuitive basis. The accuracy of the formulation or the spherocylinder theory has been questioned (5),and this may be a circumstance in which the numerical results are significantly affected.
Tracer Diffusivities through the Ternary System
In ‘Table V are sho\cn diffusivities of 14C02through various mixtures of carbon dioxide, nitrogen, and argon. These provide a basis for checking consis~encyof results among the three systems, pure ( 2 0 2 , COr-N?, and COa-Ar. If 14C02is present in very small molar concentration, one expects its tracer diffusivity to be related approximately to the self-diffusivity of COS ( D ] , ) ,the binary diffusivity ( 0 1 2 ) of c 0 2 - N ~and that ( 0 1 3 ) of C02-.4r by Equation 4 (6, 25).
Values of D ] * calculated from Equation 4 are compared Lcith observed values a t the six experimental temprratures in Table
Table
V.
1176 1398 1403 1375 1420 1525 a
D1* =
Tracer Diffusivity of l4C02 in Mixtures of Carbon Dioxide, Nitrogen, and Argon Mole Fraction
0.50 0.50 0.50 0.50 0.33 0.50 (xl/D11
0.25 0.25 0.10
0.40 0.33 0.25
0.25 0.25 0.40 0.10 0.34 0.25
1.91 2.40 2.48 2.51 2.35 2.72
+ . Y B / D I+P x a / D l ~ ) - ’ ,sq. cm./sec.
1.84 2.42 2.42 2.40 2.49 2.84
V. The errors shown are generally well within the limits of expected experimental errors for any one of the diffusivities. Good consistency between the two sets of binary diffusion data is thus established. T h e comparison also provides useful evidence in support of the validity of Equation 4 a t high temperatures for the C02-N2-Ar system.
Acknowledgment
This work has had the support of the National Science Foundation, under grant GP-2549, and of E. I. d u Pont de Nemours and Co., Inc., through a fellowship award.
literature Cited
(1) Boardman, L. E., LVild, N. E., Proc. Roy. SOC. (London) A162, 511, 520 (1937). (2) Boyd, C. A,, Stein, N., Steingrimsson, V., Rumpel, W. F., J . Chem. Phys. 19, 548-53 (1951). (3) Chapman, S., Cowling, T . G., “Mathematical Theory of Son-Uniform Gases,” Cambridge University Press, Cambridge, 1960. (4) Curtiss, C. F., J . Chem. Phys. 24, 225-41 (1956). (7) Curtiss, C. F., Muckenfuss, C., Ibid.,2( (8) Ember, George, “Diffusivities of the Carbon Dioxide-Water System at Flame Temperatures,” doctoral dissertation. Universitv of Delaware. Newark. Del.. 1962. ( 9 ) Ember, George, Ferron, J.’ R., Wohl, Kurt, A.Z.Ch.E. J . 10, 68-73 (1964). (10) Ember, George, Ferron, J. R., rb’ohl, Kurt, J . Chem. Phys. 37, 891-7 (1962). (11) Fristrom, R. M., \Vestenberg, A. A , , “Flame Structure,” D. 265. McGraw-Hill. New York. 1965. (12) Hirschfelder, J. 0.: Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” pp. 1126-7, IYiley, New York, 1954. (13) Holsen, J. N., Strunk, M. R., IND.ENG.CHEM.FUNDAMENTALS 3, 143-6 (1964). (14) F a u s , B. J., Ferron, J. R., J . Chem. Phys., in press. (15) . , Lilev. P. E.. Makita. T.. Hsu. H. W.. “Data Book.” Vol. 2. pp. 2043, 2045, 2049,’ Thermophysical Properties ‘Research Center, Purdue University, Lafayette, Ind., 1963. (16) Livingston, P. M., Curtiss, C. F., J . Chem. Phys. 31, 1643-5 (1959). (17) Muckenfuss, Charles, “The Kinetic Theory of Nonspherical Molecules,” doctoral dissertation, University of Wisconsin, Madison, \Vis., 1957. (18) Muckenfuss, Charles, Curtiss, C. F., J . Chem. Phys. 29, 125772 (1958). (19) Pakurar, T . A , , “Diffusivities in the Carbon Dioxide-Nitrogen-Argon System,” doctoral dissertation, University of Delaware, Sewark, Del., 1965. (20) Pakurar, T. A , , Ferron, J. R., J . Chem. Phys. 43, 2917-8 (1965). (21) Pakurar, T. A , , Ferron, J. R., “Proceedings of the Third Conference on Performance of High-Temperature Systems,” Gordon and Breach, New York, in press. (22) IValdmann, L., nhturwissenschaften 32, 223 (194.4). (23) IYaldmann, L., Z. Saturforschung 1, 59 (1946). (24) Waldmann, L., Z. Physik 124, 2-29 (1948). (25) Wilke, C. R., Chem. Eng. Progr. 46, 95 (1950). RECEIVED for review February 10, 1966 ACCEPTED July 20, 1966
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