INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1950
represent the over-all process has been written. The hydrocarbon effluent when charging a straight chain C d hydrocarbon always contained butenes and butadiene, suggesting that these materials may be intermediates in the reaction. The somewhat higher yields from butenes than from butane would tend to confirm this. However, these yield data are complicated, especially in fixed bed operation, by the heat effects resulting from the highly exothermic nature of the reactions. ACKNOWLEDGMENT
i
The authors wish to express their appreciation to J. T. Clarke, C. H. Culnane, and B. R. Stanerson for their contributions to the experimental work and to members of the Analytical and Testing Department of the Beacon Laboratories for carrying out much of the analytical work involved.
471
LITERATURE CITED
(1) Auwers, K. V., and KohIhaas, W., J . prakt. Chem., 108,321-31 (1924). (2) Fawcett, F.S., J . Am. Chem. Soc., 68,1420-2 (1946). (3) Fawcett, F.S., and Rasmussen, H. E., Ibid., 67,1705-9 (1945). (4) Lowry, T.,and Nasini, A,, Proc. Roy. SOC.(London), A123, 686-91 (1929). (5) Meyer, V.,and Kreis, H., Bey., 17,1558-63 (1884). (6) Morton, A. A., “The Chemistry of Heterocyclic Compounds,” p. 40,New York, McGraw-Hill Rook Co., Inc., 1946. (7) Rasmussen, H. E., Hansford, R. C., and Sachanen, A. N., IND. ENO.CHEM.,38,376-82 (1946). (8) Shepard, A. F., Henne, A., and Midgley, T., J . Am. Chem. SOC. 56, 1365-6 (1934). (9) Steinkopf, W., “Die Chemie des Thiophenes,” p. 115,Dresden u., Leipzig, Theodor Steinkopff, 1941 (J. W. Edwards, Ann Arbor, Mich., 1944). RECEIVED August 31,1949.
Diffusion Coefficients in Multicomponent Gas Mixtures D. F. FAIRBANKS AND C. R. WILKE University of Callfornia, Berkeley, Calif.
A n experimental study has been made of the diffusion of vapors into multicomponent gases by vaporization of iiquids in a long tube under conditions such that the theory of diffusion in the semi-infinite column is applicable. The results have verified the relation:
D‘A =
1
- yA
YB + y c f Y D
DAB
DAC
+ * * *
DAD
where D‘A is the effective diffusion coefficient of gas A with respect to the total gas mixture; DAB,DAG,DAD, etc., are the respective binary diffusion coefficients: and yA, ye, ye, etc,, are the mole fractions of the components in the mixture.
’
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N A binary system the rate of unidirectional diffusion of a gas, A , through a second stagnant gas, B , may be expressed by the equation
where N A
=
DAB =
P
=
R T PA X
=
pi
=
= =
=
rate of diffusion of A , gram moles per second-sq. cm. diffusion coefficient, sq. cm. per second total pressure, atmospheres gas constant, cc.-atmospheres per gram mole-” K. temperature, OK. partial pressure of component A , atmospheres distance in direction of diffusion, cm. pressure of nondiffusing gas B, atmospheres
For diffusion of A into a multicomponent mixture of stagnant gases it is convenient to express the rat,e of diffusion by an equation analogous to Equation l :
Present addresa, Department of Chemical Engineering, Massachusetts InaCitute of Technology, Cambridge, Mass.
where 0;is some proper effective diffusion coefficient for component A , which will be a function of the gas composition. In this case p i is the sum of the partial pressure of all gases other than A . On the basis of the theories of Maxwell (6) and Stefan (8, 9 ) , Wilke (10)has derived an expression given in Equation 3 for the effective diffusion coefficient and discussed its general use in diffusion calculations. D‘A =
1-
YA
ya+yc + E + . . . .
DAB
DAC
(3)
DAD
where yB, ye, etc., are the mole fractions of components A , B, C, etc., and DAB, DAC,etc., are the respective binary diffusion coefficients of component A with respect to each component of the mixture. The present paper reports the experimental verification of Equation 3 under conditions approximating those assumed in its derivation-namely, the diffusion of one gas into a mixture of stagnant gases. THEORY
On the basis of experimental convenience an apparatus was construeted to operate on the principle of diffusion in the semiinfinite column. I n this method a suitable liquid is allowed to evaporate upward into the gas mixture from the bottom of a long glass tube under conditions such that a negligible quantity of vapor reaches the upper end of the tube during the time of the experiment. This method has been suggested by Arnold ( d ) , who has integrated the differential equations applicable to this case for diffusion in binary systems. These binary equations may be extended to multicomponent systems if the diffusing gas, A , is maintained a t concentrations sufficiently low that the average diffusion coefficient given by Equation 3 remains essentially constant during the diffusion process. This is accomplished in prac-
INDUSTRIAL AND ENGINEERING CHEMISTRY
472
The general procedure in the application of this theory to the determination of the diffusion coefficient is to measure the volume of gas, V , displaced from the top of the tube as a function of time, should be a 6. According to Equation 5 a plot of V zs. straight line of slope (d‘V/d61’2jvc-hich may be determined from the best line through the plot of the experimental data. This formulation for the slope is convenient from the experimental standpoint, because only the time must be referred to a zero value and the volume measurements may be started at a later and more convenient period of the experiment. Equation 5 may be solved for the diffusion coefficient:
TABLE I. FUNCTION F, (2) Vapor Mole Fraction, Y* 0.50 0.55
Vapor
Mole
Fraction, II* 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
R 1 0.9635 0.9268 0.8900 0.8527 0.8152 0.7774 0.7381 0.7004 0.6613
0.60
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1
R 0.6215 0.6810 0.5398 0.4976 0.4540 0.4088 0.3616 0.3112 0.2546 0.1893
0
tice by using liquids of relatively low vapor pressure. Arnold’s development will be reviewed briefly, in so far as it is applicable to the present work. Consider a vapor, A, diffusing at constant temperature and pressure from a liquid surface into a mixture of gases in a long cylindrical tube. The gas space will be assumed initially free of vapor, and the tube will be assumed of infinite length. As vaporization proceeds gas is withdrawn from the upper end of the tube at a measured rate to maintain the condition of constant pressure. For this unsteady state process the following equation is applicable :
(4) where z YA
= distance in direction of diffusion, cm. (measured
from liquid-gas interface)
= mole fraction of A at distance z
mole fraction of A at interface ( 5 = 0), determined by liquid vapor pressure ( ~ Y A / ~ z ) ,= mole fraction gradient a t z = 0 D; . = effective diffusion coefficient for gas A , sq. cm. per second e = time, seconds
y :
Vol. 42, No. 3
=
The boundary conditions for integration are as follows:
APPARATUS AND PROCEDURE
Figure 1 is a schematic diagram of the apparatus. A gas mixture of the desired composition was prepared prior to each experiment in an external mixer (not shown) connected directly to the diffusion tube. DIFFUSIOW TUBE. The diffusion tube was a vertical borosilicate glass tube with an average internal diameter of 1.043 em. and a length of about 200 cm. Liquid entered the tube a t the bottom, the inflow being induced by an initial difference in the liquid level between the two sides of the U-shaped injector section. The high side of the U was connected t o the top of the diffusion column; as the liquid entered the bottom of the column, an equal volume of gas left it at the top, and there was, therefore, no sudden change of pressure in the system at the onset of the experimental run. After the liquid was in, stopcocks were shut, isolating the column from the U. BAROSTAT.The barostat mas a mercurial manometer in a Ushaped tube, and its purpose was to indicate the difference in pressure between the gas in the diffusion system and a nearvacuum maintained in the barostat’s upper chamber. As the liquid in the diffusion tube evaporated, the slow increase of pressure inside the system caused the mercury surface in the upper vacuum chamber of the barostat to rise. A sharp-pointed tungsten wire was sealed into this upper chamber, and electrical connections were made so that the wire and the mercury surface
y = g* at z = 0 for all values of 6 y = 0 at 6 = 0 for all values of z except zero
Integration of Equation 4 with these boundary conditions gives the relation:
where V = volume of vapor formed in time 6, CG. 6 = time of diffusion, seconds a = cross section of tube perpendicular to direction of diffusion, sq. cm. F , = factor correcting for deviation from Fick’s law The correction factor, F,, is defined as follows:
F,
=
2 : Y ~
z/F 9
where 4 is defined by the expressions
Arnold has solved Equations 6 to 8 numerically for F , for various conditions of vaporization and absorption. Values of F, are given in Table I for the present case of vaporization with zero initial vapor concentration in the tube. Because F , is a function of y* only, it is constant for any one experiment.
Figure 1.
Schematic Diagram of Apparatus.
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1950
would form the contact points of a switch which, in turn, controlled a vacuum-tube relay. The relay was constructed after a plan given by Serfass ( 7 ) . The relay operated an indicator light, and, thus, the making and breaking of contact a t the mercury surface were made detectable. No arcing was observable in the upper chamber, and the arrangement was found to be very sensitive in indicating the attainment of the desired pressure. V . 0 ,
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0.7 0.6
473
and minimize temperature differences between jacketed and unjacketed portions of the equipment. PREPARATION OF APPARATUS FOR OPERATION.Before each run the apparatus was carefully brought to temperature, evacuated, and checked for leaks. The gas mixture was then admitted to the apparatus from the mixing section. The procedure of alternate evacuation and filling was repeated several times to sweep the apparatus free of residual gases from the preceding experiment. After final introduction of the gas, stopcocks were closed to isolate the main diffusion section from the U-shaped liquid injection tube, and the injection tube was filled with the liquid to be evaporated. After a few minutes' preheating by means of a portable water bath brought up around the injection tube, the liquid was ready for injection into the diffusion tube. MATERIALS.The source and purity of all gases and liquids used in the experiments are summarized in Table 11.
u 0. 8 U '
10.4
3
>o
0.3
TABLE11. SPECIFICATION OF MATERIALS 0.2
Substance Hydrogen
0.1
0
0
10
PO
30
40
TIME, SEC. 'I2
50
Figure 2. Volume-Time Data for Typical Run
CAPILLARY VOLUMETER.The volumeter was a vertical calibrated capillary tube which had a nominal internal diameter of 1 mm. It was connected a t the top to the top of the diffusion tube and at the bottom through a stopcock to a mercury reservoir in a develing bulb. At the start of an experimental run, the capillary was filled with mercury. As described above, the vaporization of the liquid caused the pressure in the apparatus to increase until, finally, the mercury in the barostat came in contact with the tungsten wire. The time of the contacting was recorded along with the accompanying level of mercury in the volumeter. Then, by releasing some mercury from the volumeter, gas could be withdrawn from the diffusion tube, lowering the pressure there and breaking the barostat contact. The vaporization of the liquid continued, and the pressure of the gas again increased until contact was again made. Once more the time and the volumeter scale reading were noted. The process was repeated until all the mercury had been emptied from the volumeter. With the aid of the volumeter calibration the volume corresponding to each scale reading could be calculated. Thus a series of data giving the increase of vapor volume with time was obtained. Zero volume was arbitrary, only the change in volume being obtained. Zero time was the moment that the liquid entered the diffusion tube. The volume-time data formed the basis for the calculation of the diffusion Coefficient. TEMPERATURE CONTROL.I t was necessary to keep the diffusion tube a t constant temperature to avoid thermally induced convection currents and volume fluctuations. To obtain close control the diffusion tube was encased in a water jacket through which about 2 liters per minute of water were pumped from a 50-liter constant temperature bath. The temperature of the water bath was maintained within 10.01 C. by a sensitive mercury-expansion switch which controlled an electrical heating unit immersed in the bath. The bath was stirred continuously and was well insulated. Corrections to the observed gas volumes were made for slight fluctuations in the water bath temperature. I n order to observe the temperature of the evaporating liquid and detect any evaporative cooling effects, a fine-wire ChromelAlumel thermocouple was sealed through the wall near the bottom of the diffusion tube, and was placed so that the surface of the liquid was about 1 mm. above the thermocouple junction. The apparatus was housed in a windowless, tightly shut, subterranean room to obtain freedom from drafts and room temperature fluctuations. Corrections were made for the effects of small variations in room temperature on the unjacketed tubing in the varostat and volumeter. Experiments were conducted a t approximately room temperature t,o facilitate temperature control
Argon
Source Electrolytic Univ. of Calif. 'Chemistry Dept. Linde Air Products
Air
Atmospheric
Toluene
Eastman Kodak Co.
Ethyl propionate
Eastman Kodak Co.
Boiling Point, O C .
Go.
..
Remarks 99.9% hydrogen
..
99.0% A, 1.0%
..
$"z Dried over PnOs before m e Accepted b.p. 110.62" C. ( 1 ) Accepted b.p. 99.00 c. (5)
110.5
98.799.3
RESULTS
For each run, volumes of gas evolved from the top of the tube were plotted against the square root of time on rectangular coordinates. Figure 2 shows a typical plot, in this case for the diffusion of ethyl propionate into a mixture of 20% air-80% hydrogen. From the slope of the best straight line through the data the diffusion coefficient was calculated by Equation 9. The vapor pressure of ethyl propionate was calculated by interpolation of data from the International Critical Tables (S), and the vapor pressure of toluene was calculated from data published by the National Bureau of Standards (1). For purposes of comparison the diffusion coefficients were converted from their values a t room temperature and prevailing barometric pressure to the standard diffusion coefficient used in the International Critical Tables ( 4 )defined by the relation:
0.35
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0.30 0.25 0.90
d 0.1 5 0.10
0*05
t
01
0
Figure 3.
I
20 40 60 M O L E % OF A I R IN G A S M I X T U R E
80
1I. 100
Diffusion of Ethyl Propionate into HydrogenAir Mixtures
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Vol. 42, No. 3
DISCUSSION
The present experimental techniqoe gives values of D which are self-consistent and reproducible. Furthermore, the straiphtline relation obtained in all the runs in the plot of V us. is in agreement with the theory and supports the validity of the method. It is of interest to compare the iesults obtained for the binary '\ systems with the corresponding results of other investigators. \ For diffusion of ethyl propionate into air Winkleman (4) obtained 0.2 data giving D o = 0.0653 sy ciii. per second compared to an \ average value of 0.069 sq. cin. per second for the present \\orl
