SELF=DIFFUSION IN LIQUID CARBON DIOXIDE AND PROPANE RICHARD C. ROBINSON’ AND WARREN E. STEWART Department of Chemical Engineering, University of Wisconsin, Madison, W i s .
The self-diffusivities of liquid carbon dioxide and propane are determined as functions of temperature and density, by radioactive tracer measurements. Results are given for reduced temperatures of 0.80, 0.90, and 0.97,and pressures up to 182 atm. The data are correlated in corresponding-states form, and are used to test several theories of diffusion in liquids.
IFFUSIVITYmeasurements in
dense fluids can provide useful
D information for process engineering and for improvement
Table I. Selected Physical Properties (Bird et al., 1960; Pitzer et al., 1955) Carbon Dioxide Propane T,, K. 304.2 370.0 P o , atm. 72.9 42.0 Vc,cc./g.-mole 94.0 200 z, = p C ~ J R T c 0.275 0.277 w 0.225 0.152
of molecular transport theories. I n this paper, self-diffusivity measurements and correlations are given for the compressed liquid region near the critical temperature; this region is of special interest for testing current theories of liquids, and for determining the transition of the diffusional behavior between the liquid and gaseous states. Carbon dioxide and propane were chosen for this study in order to evaluate and extend existing data for these substances. T h e self-diffusivity of carbon dioxide has been measured by Drickamer and associates (Robb and Drickamer, 1951 ; Timmerhaus and Drickamer, 1951) in the dense-gas and liquid regions, and by O’Hern and Martin (1955) in the dense-gas region; self-diffusivities at lower densities are available for both fluids from the work of Kuether (1965). The data of Kuether for carbon dioxide agree reasonably well with those of O’Hern and Martin, but not with those of Drickamer and associates; hence it seemed advisable to make new measurements for carbon dioxide in the liquid region. A second important consideration in the choice of substances is their suitability for predicting the behavior of other fluidsfor example, by use of a principle of corresponding states. T h e critical compressibility factors and acentric factors of carbon dioxide and propane (see Table I) are close to the values for many other substances; this should be advantageous in making corresponding-states estimates of diffusion coefficients, and in future testing of such estimation methods by measurements on other fluids.
F
11 nH
Diffusion Cell
The diffusion cell is shown in a cross-sectional view in Figure 1. I t consists of four rotatable disks with precisionbored holes 0.125 inch in diameter, mounted inside a pressure vessel. Each of the four disks is geared to a control rod which extends out of the cell. By means of these rods, and a set of positioning stops machined on the disks and container, the holes in the disks can be aligned for diffusion or for filling and sampling. The diffusion holes when aligned form a column 2.011 inches in length, which is appropriate for measuring diffusivities of 10-5 to sq. cm. per second. One side of each disk is superpolished; the other side is covered with a 0.010-inch layer of Teflon sheet bonded with epoxy cement. T o prevent leakage between the flat surfaces during operation, the stack of disks was clamped from below with hydraulic oil pressure about 700 p.s.i. greater than the fluid pressure in the diffusion holes. Further insurance 1
90
Present address, Chevron Research Co., Richmond, Calif. l&EC FUNDAMENTALS
Figure 1. A. B. C.
D. E. F. G.
H.
Cross section of diffusion cell
Axle Upper container Diff usion column Diffusion disks End disks Filling and sampling lines Gear Control rod
J. K.
I. M. N.
0. P.
Clamping piston lower container Outer nut Hydraulic oil pressure source Hydraulic system nut O-rings Teflon packings
against leakage was provided by filling the cavity around the diffusion disks with nontracer fluid at the same pressure as within the holes. The container of the diffusion cell is a pressure vessel made of No. 303 stainless steel. A Teflon O-ring was used for the stationary seal between the upper and lower containers. Teflon V-ring packings were used to seal around the control rods and the hydraulic clamping piston. T h e cell was designed for pressures of 1100 to 10,000 p s i . , and temperatures from -5OOto +130° C. Experimental Procedure
The diffusion cell was mounted in a constant-temperature bath of circulating air. The air temperature was controlled to *O.0lo C. with a proportional controller and resistance heaters, and measured with a platinum resistance thermometer. For operation at room temperature, the bath was cooled by water flowing through a coil and the temperature was maintained with electrical control heat. For temperatures below room temperature, a constant flow of liquid nitrogen was used as coolant. T o prepare for a run, the diffusion cell and the tubing system shown in Figure 2 were: pressurized with liquefied, nontracer fluid using a mercury p:iston compressor. T h e pressure gages were calibrated for ea.ch run with a Barnet dead-weight balance. Radioactive tracer gas was then drawn into the evacuated tracer injection bomb, diluted with nontracer fluid, liquefied, and pressurized. All four of the diffusion holes were then turned to align with the filling lines in the cell for tracer injection as shown in Figure 1. By simultaneously screwing in on one hand-operated piston pump and out on the other hand pump, tracer fluid was swept from the injection bomb into the diffusion holes under a constant pressure. T h e disks were then realigned so that the top diffusion hole (No. 4) could be swept clean and filled with pure fluid. Standard samples were taken from diffusion holes 3 and 2 as indications of initial tracer concentration, and these disks were filled with pure fluid. T o start the diffusion, disks 4, 3, and 2, which then contained pure fluid, were rotated into precise alignment with disk 1, which contained tracer fluid. T h e diffusion was allowed to proceed for an allotted time; then the four diffusion holes were separated by rotating disks 2 and 4 out of alignment. This terminated the diffusion process and divided th(e column of fluid into four segments, ready for sampling. Sampling was accom:plished a t system pressure by pushing fluid into the cell with one hand screw pump while drawing the sample out into the other hand pump. T h e pressure was thus held constant within 1 to 2 p.s.i. during sampling. Sample volumes of 5 cc. were found sufficient to recover 99.9570
of the tracer from each disk. T h e samples were removed from the pump by expansion into a n evacuated bomb which was cooled in liquid nitrogen. T h e tracer concentrations of the samples were determined by ionization counting in the vapor state. Each sample was expanded into a 4300-cc. Cary-Tolbert ionization chamber operated a t about atmospheric pressure. Ionization currents were measured with a Cary Model 31 vibrating-reed electrometer and a voltage recorder. T h e current was determined from the rate of charging of a IO-"-farad capacitor. Diffusion experiments were performed with C1402in CO2 and with H3C-C14H2-CH3in C3H8. The nontracer fluids were obtained from the Matheson Co. with purities of 99.99% for carbon dioxide and 99.5% for propane. T h e tracer substances were purchased from the Volk Radiochemical Co. Evaluation of Diffusivities
The data taken for a diffusion run consisted of the temperature, the pressure, the diffusion time, and the ionization currents for the samples. I n all there were six samples: standard samples from disks 2 and 3, and samples taken after the run from all four diffusion disks. T h e standard samples indicated the uniformity of the initial tracer concentration, and gave a material balance against the sum of the four final samples. Material balances on the total tracer count agreed within 1 or 2%. T h e data were analyzed in terms of Fick's second law of diffusion,
in which c * is the tracer concentration and a)* is the tracer diffusion coefficient. The solution of Equation 1 for a bounded column of length L , initially filled with tracer from x = 0 to x = L / 4 , is
in which LO* is the initial tracer concentration in the first section (Crank, 1956). Equation 2 was integrated over each of the four disks to obtain the dimensionless average concentrations C I * / C Q * , L Z * / C O * , C3*/cO*, and c4*/co* as functions of a*t/L2, and a numerical table of these four functions was computed (Robinson, 1965). Preliminary values of D* were estimated by fitting the four final tracer counts with values from this table. Equation 2 holds for the following ideal diffusion conditions: The tracer is initially in disk 1 in uniform concentration; no tracer is initially present in disks 2, 3, and 4 ; no mixing occurs upon aligning or separating the four diffusion holes; and no convection occurs during the diffusion run. T o test for deviations from these ideal conditions, two or three runs of different durations were made at each temperature and pressure. T h e preliminary estimates of D*, described above, decreased slightly with increasing run time; this trend indicated that some mixing had occurred upon aligning the disks. Plots of these preliminary results as 3*t/LZ against t were found to be linear a t each temperature and pressure, so that the effect of the initial mixing could be eliminated by obtaining D* from the slope of the line. A more precise method of calculating a*,particularly for sets of more than two runs, is to introduce a time correction, t,, into Equation 2 and fit the observations directly. T h e equations which predict the sample counts are then as follows (Robinson, 1965) :
MI! Q
-----
I
I I I
I
I
II
0
I
--A
i. Figure 2. A. B. C.
D. E. F.
I
\
- !
h
U'
K@
I
1
Schematic flow diagram
Diffusion cell Temperature bath Tracer injection bomb Differential pressure instrument Dead weight balance Radioactive tracer storage
G.
H. J.
K. I.
Pressure g a g e Mercury piston compressor Liquefied gas source Vacuum pump Sample bomb
L
+ ... VOL. 7
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1968
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I n this relation, cj* is the tracer count for the j t h disk and A j , B j , and E3 are known constants. Terms beyond E, in the series are negligible for the runs reported here. The final diffusion coefficients and confidence limits were determined by fitting Equation 3 to all the data at each temperature and pressure. T o allow for some uncertainty in the total tracer count, c ~ *for each run was made an adjustable parameter along with D* and t,. Thus for each set of three runs there were five parameters (a>*, t,, COI*, COZ*, and c03*) to be found from 12 observations of final tracer counts. For each set of two runs there were four parameters to be fitted from eight observations. T h e diffusion coefficients and other parameters were evaluated by least squares, using the nonlinear estimation program developed by Guttman and Meeter (1964). This computer analysis gave the best values and statistical confidence limits for all the parameters. For carbon dioxide, the 9570 confidence limits of the diffusivity averaged f 9 . 8 % for sets of three runs and f15.6% for sets of two. For propane, the confidence limits averaged =k 5.8% for sets of three runs and =kl3.5% for sets of two. Experimental Results
T h e diffusion coefficients are presented in Table 11. These values are reported as measured-i.e., as binary diffusivities, D*, of the tracer in the corresponding nontracer fluid. Within the experimental accuracy a)* may be equated to the selfdiffusivity, D, of the nontracer fluid. T h e product of molar density and diffusivity, cD*, is also listed for subsequent comparisons. T h e molar densities were obtained from the data of Michels et a l . (1935) and Amagat (1926) for carbon dioxide, and the data of Reamer et a l . (1949) for propane. Several previous investigations on carbon dioxide selfdiffusion at high densities may be compared with these results. In the gas phase a t 35' C., the data of O'Hern and Martin (1955), supported by Kuether (1965), indicate that CD is nearly constant at about 4.7 x 10-6 gram-mole cm.-l set.-' up to gas densities of 16 gram-moles per liter. O n the other hand, Robb and Drickamer (1951) and Timmerhaus and Drickamer (1951) reported self-diffusion coefficients which decreased more rapidly with density. Their data indicated a twofold decrease in c D over the above range of density, and a continued downward trend in the liquid region. T h e liquid self-diffusivities reported for carbon dioxide in Table I1 are about twice as large as those found by Drickamer and associates. This is comparable to the discrepancy noted by O'Hern and Martin a t high gas densities. Thus the carbon dioxide self-diffusivities reported by Robb and Drickamer and
by Timmerhaus and Drickamer appear to be significantly low in the dense-gas and liquid regions. T h e values reported by O'Hern and Martin for carbon dioxide in the compressed gas phase are compatible with our values for the liquid, as shown below. Corresponding-States Correlations. Dimensional analysis and kinetic theory both indicate that the function
in given units (see Nomenclature) should be a function only of pr and T , for heavy isotopic molecules that obey a twoparameter principle of corresponding states. T h e present data on liquid carbon dioxide and propane are reasonably well correlated in this manner, as shown in Figure 3. We see also that A is expressible as a function of reduced density alone, within the precision of the data. Thus for these liquids a t this is constant density, c D * varies approximately as somewhat weaker than the temperature dependence typically found for self-diffusion in the gaseous state (Bird et a l . , 1960;
13
16
20
22
24
Reduced Den: t p , pr Figure 3. data
Corresponding-states plot of liquid self-diffusion
Table II. Experimental Diffusivity Values" Ca)
*,
G.-Moles/
a
92
c, G.-Moles/ N o . of a)*, Sq. Cm./Sec. C m . Sec: Substance T, O K. T, p, A t m . c c . x 703 Runs x 104 x 706 295.67 0.972 69.1 17.60 3 1.97 f 0.23 3.47 coz 109.2 19.32 2 1.86 k 0.46 3.60 145.8 20.24 2 1.59 f 0.40 3.21 182.3 20.92 2 1.56b 3.26 273.77 0.900 72.8 21.78 3 1.18 f 0.09 2.58 coz 72.8 21.78 2 1.17 f 0.05 2.56 145.6 22.73 2 1 .OO f 0.08 2.28 C3H8 295.95 0.800 14.5 11.34 2 1.22 f 0.21 1.38 83.9 11.78 3 1.20 f 0.05 1.41 168.2 12.11 2 0.99b 1.20 CaH8 332.96 0.900 29.5 9.87 3 1.97 =k 0.13 1.94 83.9 10.47 2 1.80 f 0.19 1.88 CaHs 358.85 0.970 42.1 8 -44 3 3.08 rf: 0.22 2.60 83.9 9.41 1 2.34b 2.20 1.87 168.0 10.31 2 1.82 f 0.18 a)* column gives di@usivity values obtained by least squares, and 95% confidence intervals. b Confidence interval omitted, since adjusted t , used.
l&EC FUNDAMENTALS
Hirschfelder et ai., 1954; O’Hern and Martin, 1955; Tee et ai., 1966). Another method of correlating the data is shown in Figure 4, where the quantity
is plotted as a function of p I and T,. This plot illustrates the strong temperature dependence of c D * in liquids at constant pressure; comparison with Figure 3 shows that this dependence is due largely to the thermal expansion of the fluid. T h e lines on these two figures are regarded as tentative, but should be useful for estimation of self-diffusivities of other normal fluids with w values near 0.2. Figure 4 is a refinement of a plot given by Tee et 121. (1966). Results of three related investigations are included in Figure 4. T h e line for T , = 1.01 is based on the tracer measurements of O’Hern and Martin (1955) for carbon dioxide, and of Kuether (1965) for carbon dioxide and propane; the line for T , = 0.7 is based on the spin-echo measurements of Douglass, McCall, and Anderson (1961) for neopentane. T h e
3.0
-0 2 .o
(D
X
co
I
T.=OBO
0
0
I.o
2.0
Reduced Pressure,
3.0
I
4.0
p,
Figure 4. Corresponding-states plot with results of other investigators T = 1.01 (Kuether, 1 9 6 5 ; O‘Hern and Martin, 1 9 5 5 ) . Tr = 0.70 (Douglass et al., 1961 1
Table 111. Substance
T, K .
coz
273.77
coz
295.67
CsHe
295.95
CIHS
332.96
O
358.85
tracer data of Naghizadeh and Rice (1962) for argon, krypton, and xenon at T , = 0.7 would lie from 10% below to 3001, above the bottom line of Figure 4; this difference is within the scatter of their data but will bear further study, since it may indicate a need for additional variables in the correlation. Comparison with Hydrodynamic Theory. T h e hydrodynamic theory of diffusion gives the following result (Li and Chang, 1955; Sutherland, 1905) for self-diffusion of spherical molecules of diameter d:
a = -kT
2 npd
If one replaces d by the intermolecular spacing calculated for a cubic lattice, Equation 6 becomes:
(7) Equation 7, with the approximation D = a*, predicts the tracer diffusivity with an average deviation of *12% for many liquids (Li and Chang, 1955). T h e present data permit a test of this equation at temperatures nearer the critical. Viscosity values a t the conditions of Table I1 were obtained from Michels et ~ l (1957) . and Stakelbeck (1933) for carbon dioxide, and from Smith and Brown (1943) for propane. Molecular diameters, d, for Equation 6 were estimated from Lennard-Jones collision diameters (Hirschfelder et al., 1954), u = 3.996 A. for carbon dioxide and u = 5.061 A. for propane. T h e molar volumes were obtained from Table 11. T h e calculated results are shown in Table 111. Equation 6, with d = u, deviates significantly from the trends of the measured values. Equation 7 gives much better predictions; its average deviation of 6.5y0 is close to the experimental accuracy. Activated-Complex a n d Significant-Structure Theories. Eyring and his coworkers have presented equations for D based on a n activated-complex theory (Glasstone et ~ l . 1941) , and, more recently, a significant-structures theory (Eyring and Ree, 1961; Eyring et ai., 1962; Ree et d., 1964). T h e activated-complex theory leads to Equation 7 with the constant 27r replaced by unity; the significant-structures theory gives Equation 7 with 27r replaced by a coordination parameter, E. Commonly E is set equal to 6 ; this gives slightly smaller diffusivities than Equation 7. T h e present data are best fitted with { = 6.30; this agrees closely with the constant 27r in Equation 7 .
Comparison of Hydrodynamic Theory with Diffusion Data Dzffusivities, Sq. Cm./Sec. X lo4 p , Atm. El. 6 El. 7 Obsd.
72.8 145.6 69.1 109.2 145.8 182.3 14.5 83.9 168.2 29.5 83.9 42.1 83.9 168 .O
1.28 1.12 2.53 2.06 1.84 1.54 1.53 1.18 1.02 1.92 1.69 3.38 2.46 1.94
1.21 1.07 2.22 1.88 1.70 1.44 1.44 1.14 1.01 1.77 1.58 2.95 2.23 1.81
VOL. 7
1.18, 1.17 1 .oo
1.97 1.86 1.59 1.56 1.22 1.20 0.99 1.97 1.BO
3.08 2.34 1.82
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Comparison with Free-Volume Theory. Cohen and Turnbull (1959) have given an expression for 3 of the form
in which g is a geometric factor, a* is a length approximately equal to the molecular diameter, yp* is a constant approximately equal to the molecular volume, and v,is the free volume per mole. Use of the hydrodynamic theory, with Doolittle’s empirical formula (Doolittle, 1951) for the viscosity, gives a similar result, except that T then appears to the first power in the pre-exponential term. T h e development given by Cohen and Turnbull permits some latitude in the evaluation of a* and 8*/8,. We have taken 8* as Riedel’s “zero-point volume,” which is the molar volume obtained at T = 0’ K . from his equation of state for saturated liquids (Riedel, 1954, 1957). Correspondingly, the “free volume” 8, is taken as P - PO. T h e quantity a* may be considered proportional, for molecules of a given the latter choice shape, either to (V0/&‘)1/3 or to seems more in accord with Cohen and Turnbull’s derivation and also gives the better fit of the present data. With these substitutions, Equation 8 becomes :
(v/,v)1/3;
or
where A = loglo g W 2 N-1/3 and B = y/ln 10. Figure 5 shows the present data plotted according to Equation 10, with P O = l .624 grams per cc. for carbon dioxide and p o = 0.812 gram per cc. for propane as given by Riedel (1957). A single straight line fits the data for both substances well; the maximum deviation is 15% and the average deviation is 4.9%.
The constants of the straight line in Figure 5 are A = -4.335, = 0.384; the latter value corresponds to y = 0.884. Cohen and Turnbull’s values of y for the substances they studied are appreciably lower: yv*/vo = 0.86 for carbon tetrachloride, 0.70 for 2,3-dimethylbutane, 0.665 for 2,2-dimethylbutane, and 0.69 for isobutyl bromide. Cohen and Turnbull’s calculations are not strictly comparable to ours, since they obtained both yP* and po from diffusion data; however, their results do indicate a significant variation of B with molecular structure. In view of this, we d o not recommend the line of Figure 5 for other fluids. I t appears likely, however, that this form of correlation can be extended to other fluids by corresponding-states techniques.
B
Conclusions
T h e diffusivity data given here are fitted almost equally well by Figure 3 or 4 (corresponding states), Equation 7 (hydrodynamic theory), or Equation 10 (free-volume theory), The significant-structures theory, when fitted to these data, gives essentially the same results as Equation 7. The activated-complex theory overestimates D by about a factor of 6, as noted earlier by Li and Chang (1955). Equation 7 is recommended for prediction of D for other liquids. This equation has been applied successfully to a wide range of materials (Li and Chang, 1955), including water, alcohols, and liquid metals as well as nonpolar substances. For normal fluids, Figure 3 or 4 may be used for making estimates when viscosity data are lacking. Equation 10 is useful mainly for curve-fitting at the present time. Acknowledgment
Appreciation is expressed to the National Science Foundation for financial support of this investigation under Grant No. G-14812, and to the University Research Committee for providing computing time at the University of Wisconsin Computing Center. We also thank Gary F. Kuether for helpful advice on many aspects of this work.
-4.6 Nomenclature
a* c c*
-4.7
GO*
co1* D
a” g
k L M M*
fl
p pE p,
-5.0
R t
t,
T T,
-5.I 3
1.0
1.2
1.4
1.6
1.8
2.0
P
(p,-p) Figure 5. Correlation of liquid self-diffusivities according to Equation 10 94
I&EC FUNDAMENTALS
Tr V*
Vf 8,
characteristic jump distance for self-diffusion, cm. molar density, g.-mole/cc. = tracer concentration, g.-mole/cc. = initial tracer concentration, g.-mole/cc. = initial tracer concentration in first run of a set, g.-mole/ cc. = self-diffusivity, sq. cm./sec. = diffusivity of tracer in corresponding nontracer species, sq. cm./sec. = parameter in Equation 8, dimensionless = Boltzmann constant, ergs/’ K. = length of diffusion column, cm. = molecular weight of nontracer fluid = molecular weight of tracer = Avogadro’s number, g.-mole-’ = pressure, atm. = critical pressure, atm. = p/pc, reduced pressure = gas constant, consistent units = time from start of diffusion, sec. = time correction in Equation 3, sec. = temperature, O K. = critical temperature, O K . = T/T,, reduced temperature = molar volume, cc.g.-mole-’ = characteristic volume for self-diffusion (Cohen and Turnbull, 1959) cc. g.-mole-‘ = critical volume, cc. g.-mole-l = free volume, cc. g.-mole-l = zero-point volume (Riedel, 1954, 1957), cc. g.-mole-l = =
= distance from bottom of diffusion column, cm. = critical compressibility factor, dimensionless
x t,
GREEKLETTERS Y
parameter in Equation 8, dimensionless = function defined in Equation 4 = T,ll2A,function defined in Equation 5 viscosity, g. cm.-’ set.-' = = 3.1415. . . = density, g./cc. = p / p , , reduced density = zero-point density (Riedel, 1954, 1957), g./cc. = Lennard-Jones collision diameter, A. = Pitzer acentric factor (1955), dimensionless = coordination parameter, dimensionless =
A 6 I . (
a P PT
PO u
w
€
Literature Ciled
Amagat, E. H., in “International Critical Tables,” Vol. 111, p. 38, McGraw-Hill, New York, 1926. Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” !%ley, New ’York, 1960. Cohen, M. H., Turnbull, ID., J . Chem. Phys. 31,1164 (1959). Crank, J., “The Mathenlatics of Diffusion,” Oxford University Press, London, 1956. Doolittle, A. K., J . A p p l . P h p . 22, 1471 (1951). Douglas, D. C., McCall, D. W., Anderson, E. LV., J . Chem. Phys. 34, 152 (1961). Eyring, H., Henderson, ID., Ree, T., “Progress in International Research on Equilibrium and Transport Properties,” Masi and Tsai, eds., Academic Press, New York, 1962. Eyring, H., Ree, T., Proc. .Vat/. Acad. Sei.47, 526 (1961).
Glasstone, S., Laidler, K. J., Eyring, H., “The Theory of Rate Processes,” McGraw-Hill, New York, 1941. Guttman. I.. Meeter. D. A.. DeDt. of Statistics. Universitv, of Wisconsin,’Madison, Rept. 37 (1564). Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liouids.” Wilev. New York, 1954. Kuether, G. F., Ph.D. thesis,’ University of Wisconsin, Madison, 1965. Li, J. C . M., Chang, P., J . Chem. Phys. 23, 518 (1955). Michels, A., Botzen, A., Schuurman, W., Physzca 23, 95 (1957). Michels, A., Michels, C., Wouters, H., Proc. Roy. Sac. A153, 214 (1935). Naghizadeh, J., Rice, S. A , , J . Chem. Phys. 36, 2710 (1962). O’Hern, H. A,, Jr., Martin, J. J., Ind. Eng. Chem. 47, 2081 (1955). Pitzer, K. S., Lippman, D. Z . , Curl, R. F., Huggins, C. M., Petersen, D. E., J . A m . Chem. Soc. 77, 3433 (1955). Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem. 41, 482 (1949). Ree, T. S., Ree, T., Eyring, H., J . Phys. Chem. 68,3262 (1964). Riedel, L., Chem.-Ing.-Tech. 26, 259 (1954). Riedel, L., Kaltetechnzk, Band 9, Heft 5, S.127 (1957). Robb, W. L., Drickamer, H. G., J . Chem. Phys. 19, 1504 (1951). Robinson, R. C., Ph.D. thesis, University of !$’isconsin, Madison, Wis., 1965. Smith, A. S., Brown, G. G., Ind. Eng. Chem. 35,705 (1943). Stakelbeck, V. H., Z . Ges. Kalte-Znd. 40, 33 (1933). Sutherland, !$‘illiam, Phtl. M a g . 9, 781 [ 1905). Tee, L. S., Robinson, R. C., Kuether, G. F., Stewart, LV. E., American Petroleum Institute, Refining Division, Houston, May 1966. Timmerhaus, K. D., Drickamer, H. G., J . Chem. Phys. 19, 1242 (1951). RECEIVED for review February 10, 1967 ACCEPTED July 24, 1967
VARIATION OF NONISOTHERMAL D I FFUS I C) N COEFFICI ENTS S E R G I O D I C A V E ’ A N D A L D E N H . E M E R Y , JR.2 School of Chemical Engineering, Purdue UniuersiQ, Lafaq’ette, Ind.
The diffusion coefficient measured in thermal diffusion is typically 20 to 40% larger than that measured in isothermal diffusion, and increases with temperature gradient. Mass-transfer theory does not account for the magnitude of the effect.
N THE
development of the phenomenological equations for
I thermal diffusion (Enskog, 191l ) , people naturally assumed
the minimum complexity needed to handle the data then available. One of the assumptions always made is that the diffusion coefficient measured in isothermal experiments, Di, is meaningful also in the nonisothermal conditions involved in thermal diffusion. Here attention is’ called to the data of Agar and Turner (1960a, 1960b) and data are presented that indicate that the diffusion coefficient measured in thermal diffusion, D,,is usually higher than Di. Previous Work
T h e effect was first called to our attention by I. Prigogine, who noted that Jeener and Thomaes (1954) had observed a D , 1
Present address, University of Rome, Rome, Italy.
* Present address, Cniversity of Leeds, Lceds, England.
in their open cell which was significantly larger than D,in the same cell. Several pieces of research in this laboratory (Childress, 1957; Maniotes, 1959; Tope, 1960; Wilsbacher, 1964) followed. Results indicated that there may be a significant effect, but the precision of the work was not high enough for us to be sure. .411 of these researches were done in a two-compartment cell with a sintered-glass membrane of the type used by Drickamer (Saxton et al., 1954), with the hot side on top. This type of cell yields only one point per run, and the cell must be cleaned and reloaded for another point, which gives poor precision in the resulting diffusion coefficient. Cells that record continuously during a run, such as the one of Jeener and Thomaes which used an optical analysis, give much better precision. Agar and Turner (1960a, 1960b) used such a technique in work on dilute aqueous solutions in a n open cell, in which they fo1lov;ed the course of the diffusion by a conductometric
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