ENGINEERING, DESIGN, AND EQUIPMENT
Diffusion in Carbon Dioxide at EIevated Pressures HAROLD A. O'HERN, Jr.l,
AND
JOSEPH J. MARTIN
University o f Michigan, Ann Arbor, Mich.
AS
result of a successful kinetic theory, and nearly 100 years ofA experimental investigation, the laws governing the diffusion of gases a t low pressure are well known. The mutual diffusion coefficient, DI2,of gases 1 and 2 is given to the close approximation by an equation of the form
'3 d&+ z2 ps:, d&l+ RTf12(T)
1
1
C1202, by a different method, are reported here. These were made a t pressures to 25 atmospheres a t 0" C., 98 atmospheres a t 35' C., and 205 atmospheres a t 100' C. Two-chambered diffusion cells were used in this study
(l)
The diffusion cells employed in this investigation were related to several earlier units, especially to that of Hutchinson (7). These cells consisted of two ionization chambers connected by where fiz( 7') is a function of the temperature, T, and of the gases some form of diffusion opening. With a radioactive component involved, which can be evaluated by the methods of Hirschfelder diffusing, the ionization currents from the two chambers provided and coworkers (5),Gillilaiid ( 4 ) , and others. a continuous measure of the diffusion occurring through the open7~ = molal density of the gas, gram mole/l. ing. Unlike most earlier cells of this general type, these units MI and Mz = molecular weights of gases 1 and 2 incorporated no means of closing the diffusion opening in order P = total pressure of gas mixture, atm. to fill the two chambers with different gases. Instead, the initial R = ideal gas law constant, (1.) (atm.)/(g. mole) ( O K.) concentration difference was obtained by adding a small amount s12 = effective collision diameter, cm. of radioactive gas to one chamber, after filling the entire cell with nonradioactive carbon dioxide. Concentration ratios as In the region where the ideal gas law applies, the indicated dehigh as 2 0 : l were obtained in this way. I n practice, a short pendence of the diffusion coefficient on molal density or presperiod was allowed a t the beginning of each run for the dissipasure has been verified by many investigators. tion of temperature inequalities and convection currents resulting On the other hand, comparatively little is known about the from charging, and for the attainment of a linear concentration diffusion of gases a t elevated pressures. The dense gas theory gradient in the diffusion opening. of Enskog (S), based on the hard sphere molecular model, is a t Two different diffusion cells were used. The second of these, present the most useful theoretical treatment. Recently, piodesignated cell B, is shown in Figure 1. The diffusion path in neering experimental work in the field of diffusion a t elevated this cell was provided by a plug of Porex Grade 4, a porous bronze pressure has been reported by Drickamer and his coworkers. of porosity 0.40, manufactured by the Moraine Products Division, Robb and Drickamer ( 1 6 ) and Timmerhaus and Drickamer General Motors Corp. The first cell used, designated cell A, (18, 19) measured diffusion rates in the system C1402-C1202 was similar to cell B, except that four drilled holes, 0.037 inch a t temperatures from 0' to 50" C. and a t pressures to 1000 in diameter, provided the connection between the ionization atmospheres. Similar measurements for the system CH4-TCHs, chambers. Because of its strength and electrical resistivity a t a t pressures to 300 atmosphere, have been reported by Jeffries high temperature, Teflon was used for all insulators in both difand Drickamer (8). The results for methane are consistent with fusion cells. The pressure seal was obtained by installing the the predictions of the Enskog theory, but agreement is less satisTeflon under an initial compressive stress, and by the further factory in the case of carbon dioxide, presumably because of the compression caused by internal pressure. This seal was espelack of spherical symmetry in that molecule. Additional measurements of diffusion in the system CI4O2cially good a t the higher pressures, permitting 3-day diffusion runs with pressure loss well below 1%. The diffusion cell, together with all the RETAINING RING auxiliary equipment containing radioactive carPOSITIVE ELECTRODE POROUS M E T A L bon dioxide, was housed in a high velocity hood, to guard against exposure to radioactive GUARD RING gas. A Keleket radiation monitor, with a NSULATORS 1.4 mg. per sq. cm. window Geiger tube, was located in the hood to give warning of escaped radioactive gas. The system of pressure vessels, valves, and tubing, used in charging the diffusion cell, is shown in Figure 2. I n this diagram, the diffuSOLDERED sion cell is 1, with equipment handling radioJ O I N T ONNECTION FOR active gas on the right and that handling E L D E D CABLE inactive carbon dioxide on the left. Cylinders Diz =
=
CTION FOR TUBING
Figure 1. October 1955
Diffusion cell B
1 Present addreas, Engineering Research Laboratory, E. I. du Pout de Nemours & Co., Wilmington, Del.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING, DESIGN, AND EQUIPMENT
121
Figure 2.
The radioactive carbon dioxide was generated from a supply of barium carbonate containing 20 microcuries of carbon-14 activity per gram, prepared by the Atomic Energy Commission Laboratories. Carbon dioxide of known radioactive strength was generated under vacuum by the addition of dilute hydrochloric acid to weighed quantities of reagent grade barium carbonate and the radioactive barium carbonate. The generated gas was dried with magnesium perchlorate and phosphorus pentoxide and was condensed in small steel cylinders with liquid nitrogen. The fraction of C1402 in the radioactive gas was actually very low, usually of the order of 10-5. Both the radioactive and nonradioactive gases were filtered and dried in entering the diffusion apparatus. Several checks of the vapor pressure of carbon dioxide samples indicated high purity.
Diagram of charging system
U
2 and 4 were used for storage of inactive and radioactive carbon dioxide. These gases were compressed for the diffusion runs a t high density by successive cooling and heating of cylinders 3 and 5. Tubes 6 and 7 contained glass wool and phosphorus pentoxide to filter and dry the gases entering the system. Four calibrated pressure gages, of ranges 200, 600, 2000, and 5000 lb. per sq. inch were used alternately in position 9 for pressure measurement, Steel tubing, 1/8 inch in outside diameter, was used for all pressure connections, and the sections adjacent to the diffusion cell were filled with wire to minimize the area for diffusion. The dotted lines of Figure 2 indicate the limits of the air chamber used for temperature control. The temperature in the chamber was controlled by a small heater, actuated by a mercury expansion switch. Although the air temperature varied noticeably with the control heater cycle, the temperature changes in the diffusion cell were less than the 0.02" C. detectable with the potentiometer and the thermocouple used. During most of the high pressure runs, the diffusion cell was insulated by a Transite box and glass wool wadding to reduce further the temperature fluctuations. The diffusion cell temperature was normally constant within 0.1 O C. throughout a diffusion run. The four runs a t 0 O C. were made with the aid of dry ice cooling. The measurement of the small ionization currents, usually in the range 10-14 to 10-11 amp., was extremely critical to the accuracy of the diffusion measurement. The primary measuring instrument was a Beckman Ultrohmeter. A shielded switch permitted the selection of the current from either ionization chamber for measurement. Shielded cables with polyethylene, polystyrene, and Teflon insulation were used for connections between the cell and the measuring instrument. The polystyrene insulation was most satisfactory a t room temperature, but Teflon was of course essential for operation a t 100 C. Dry batteries giving 135 or 270 v. provided the ionization chamber potential. The output from the Beckman Ultrohmeter was recorded on the 10inch strip chart of a Brown recording potentiometer. This was extremely convenient, both in providing a time-current record and in permitting a good average reading of the currents, which were subject to fluctuation. Although the absolute accuracy of the instrumentation is unknown, the current measurements are reamp. and 0.5% a t low1* producible within about 0.3% a t amp. A cylinder of 99.96% pure carbon dioxide, obtained from the Matheson Clo., Inc., supplied the nonradioactive gas used in this investigation.
Diffusion cell i s brought to temperature, charged with differing C L 4 0 2content in the two cells, and ionization current i s recorded
The procedure for a diffusion run consisted of bringing the diffusion cell to the desired temperature, charging it with a substantial difference in the Ci402content of the two chambers, and recording the ionization currents for a suitable length of time. The record of the run consisted of the recorder chart with the time-ionization current curves, together with periodic readings of the cell temperature and pressure.
O
2082
0
Figure 3.
IO
20
30
TIME,
e.
40
50
60
70
HOURS
Plot for diffusion run B 1 7 at and 88.3 atmospheres
35' C.
The quantitative treatment of the diffusion run data required the concentrations of C Y 4 0 2 , or of radioactivity, in the two chambers, rather than the measured ionization currents. In view of the recombination of ions known to occur in high pressure ioniza-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 10
ENGINEERING. DESIGN. AND EQUIPMENT
-
tion chambers, an assumption of proportionality between the currents and radioactive contents could not be made. Therefore, the current-concentration relationship for each ionization chamber was determined by direct calibration with known mixtures of CIZOz and C140z, a t various temperatures and pressures. Although exact proportionality between current and radioactive carbon concentration was not found, the rela.tionship was such that the mathematical treatment of the diffusion data was not difficult. Hutchinson ( 7 ) and Jost (IO)have given the following derivation of the relation for the two-chamber diffusion cell, based on the assumption of an essentially linear concentration gradient in the diffusion opening: Let C1 and Cz be the concentrations of a diffusing component in chambers No, 1 and 2, and L be the length of the opening connecting them. Treating the concentration in each chamber as uniform, the linear concentration gradient in the opening is
d C / d x = (C,
-
C2)/L
(2)
The expression for the rates of change of concentration in the two chambers can be written, using Fick's law
,-
0
200
400
600
800
1000
1200
1400
1600
T I M E , MINUTES
(3)
where V Iand V Zare the chamber volumes, D is the diffusion coefficient, and a is the cross-sectional area of the opening. Subtracting Equation 4 from Equation 3:
This is readily integrated to the form In (Cl
- C Z )=
- KDO + constant
(6)
where (7) Since only the resistance of the connecting opening was considered in this derivation, Equation 7 may be slightly in error when the diffusion opening is large in comparison with the chambers. From the experimental relationships between the ionization currents, II and Iz,and the concentrations, C1 and CZ, a plot of In (C, - Cz)versus the time, e, can be prepared for each diffusion run. Hutchinson has shown that if such a plot is linear, the slope must represent the product of the diffusion coefficient and a cell constant. If the cell constant, K , is known, the diffusion coefficient can be obtained from the slope, -KD. The ionization current calibrations showed that, for a fixed amount of C14O2 in the chamber, the ionization current decreased with increased pressure and increased with temperature. At fixed temperature and pressure, however, the current was nearly proportional to the concentration of radioactive material. Since diffusion runs were carried out a t fixed temperature and pressure, it was possibIe to represent the current-concentration relationships by very simple equations. The runs with cell B gave plots which were straight lines, in accordance with Equation 6. Figure 3 is a plot of this type for run B17 a t 35' C. and 88 atmospheres pressure. Here the concentration difference in microcuries per liter is plotted on a logarithmic scale against the time in hours. The semilog plots for the cell A runs, however, showed a tendency toward curvature, with the steepest slope a t the beginning October 1955
Figure 4.
Cell A diffusion run plots showing curvature
of a run. This effect was not serious a t pressures below 25 atmospheres, where straight lines could be drawn for all but the first 10 to 20% of the diffusion period. Figure 4 gives the plots for two runs at pressures near the limit of measurement with cell A; these show moderate curvature, but reasonably good straight lines can be drawn through the end portions. At higher pressures, the curvature was so serious that no reliable estimate of the end slope could be made. The reason for the curvature was not established. Since it myas pressure-dependent and did not occur in the cell B measurements, it could not have resulted from the approximation used in deriving the cell equation. One possibility is that small density differences between the two chambers, resulting from the presence of impurities, caused circulation through the drilled holes of cell A. In any case, the use of the porous metal diffusion medium in cell B eliminated this difficulty and permitted measurements a t pressures approaching 200 atmospheres without curvature. Robb and Drickamer ( 1 6 ) also observed accelerated diffusion during the first part of diffusion runs, even though porous mediums were used. For a diffusion cell with a small cylindrical connecting opening, the cell constant, K , can be calculated by means of Equation 7 from the physical dimensions of the cell. This was not possible for cell B, with its porous metal diffusion opening. Even for cell A , the errors in the measurement of the small drilled holes could lead t o an error of several per cent in the cell constant. Therefore, the low pressure measurements of three other investigators were used to determine the constants, K A and KB, for cells A and B. These were the measurements of Amdur and coworkers ( I ) , on C140z-C1202;of Winn (do), on C1302-C120z; and of Winter ( $ I ) , on C1201601*-C120216. Converted to C120z self-diffusion at 1 atmosphere, the various data points were plotted versus temperature. They were in good agreement, giving a value of D = 0.1202 sq. cm. per see. a t 35" C., or Dn = 4.753 X 10-3 (cm.2 per sec.) (gram mole) per (liter). The values of KADn and KeDn from the 35' C. runs were plotted separately versus the molal density, n, and the resulting curves were extrapolated to zero density. For low pressure (0 to 3 atmospheres) the values obtained were KADn = 2.675 X 10-6 and KBDn = 1.007 X Using Dn = 4.753 X 10-8,the constants KA = 0.005628 ern.-, and K B = 0.02119 cm.-Z resulted. The above
INDUSTRIAL AND ENGINEERING CHEMISTRY
2083
ENGINEERING, DESIGN, AND EQUIPMENT Not including the run B11 measurement, which appears to be about 25% high, the average deviation of the cell B results from the curves in Figure 5 is just under 1%. The high deviation for run B11 is believed to be the result of some unusual factor, such as the presence of air in the diffusion cell. Absorption by the Teflon insulators appeared to be a negligible factor a t the high pressure used with cell B. Also, diffusion through the connecting tubing was probably not a source of error in the cell B runs, since the higher cell constant meant relatively rapid diffusion within the cell. Runs B9, B16, and B18, a t approximately equal densities, but with a sixfold range of Cl4O2 concentration, indicated that there was no effect of radioactivity on the diffusion rate in cell B. Several tests were made for thermal convection or thermal pumping effects in the high density runs with cell B by changing the insulation on the diffusion cell and connecting tubing. These tests gave no indication that such an error was present. The theoretical relations of Pollard and Present (16), for diffusion in small capillaries, indicate that wall effects are probably negligible in the porous metal of cell B, which had pores about 0.001 inch in diameter.
O
TEMPERATURE, C ' 0 ' 350 IO00
MOLAL DENSITY, n, G . M O L E / L
Figure 5.
Experimental results
Values do not agree with Enskog dense gas theory predictions
constant for cell A, corrected back to C1402-C1202diffusion, checked that calculated from its dimensions within about 3%. The diffusion results are presented in Figure 5 and in Tables I and 11. I n Figure 5, values of the diffusivity-density product, Dn, are plotted against the molal density, n. Curves have been drawn through the points representing Oo, 35", and 100" C. Here the darkened circles refer to cell A measurements, and the open circles to cell B runs. The numerical values are for C1202 self-diffusion, since the cell constants were calculated on that basis. Cell A values reflect lower accuracy than cell B values
The values obtained with cell A are considerably more scattered, reflecting a lower accuracy. Part of this inaccuracy may be attributed to the curvature effect mentioned previously, but the absorption of carbon dioxide by the Teflon insulators and diffusion through the connecting tubing of the cell were also probable sources of error. The high values of the diffusion coefficient for runs A21 and A22 a t 100° C. may have resulted from thermal convection.
Table I. Run NO.
0
Temp.,
c.
A1 34.9 A2 34.9 A3 34.8 A4 35.0 34.9 A5 A6 35.0 35.0 A7 A8 34.9 A9 34.9 A10 34.9 All 35.0 34.9 A12 A13 35.0 35.0 A14 35.0 A15 35.0 A16 35.0 A17 A18 35.0 35.0 A19 AZO 20.0 100.1 A2 1 A22 100.1 A23 100.0 100.1 A24 Severe ourvature of
2084
The Enskog dense gas theory, based on the hard sphere molecular model, has been mentioned previously ( 3 ) . This treatment makes use of a correction factor, Y , which is given for a one-component gas by the following series:
Y = 1
+ 0.625 (I 7rh7.9) + 0.2869
(5
rNs3)'
0.115
(5
+ +
T N s ~ ) ~(S)
in which N is the number of molecules per unit volume, and s is the effective collision diameter (3, 6 ) . For the system in which self-diffusion is approximated, the diffusion coefficient in a dense gas is given by the equation
D = Do/Y
(9)
where Do is the diffusion coefficient calculated by a relation which applies a t low pressure. Diffusivities were calculated by direct application of Equations 8 and 9, using collision diameters which were obtained from the low pressure diffusion coefficients a t 35" and 100" C. The calculated curves for the diffusivity-density products a t 35" and 100" C. are shown, with the experimen-
Diffusion Results with Cell A Press., Atm.
n,
G ram LMoles/ Liter 0.477 0.339 0.581 2.090 1.921 1.029 1.027 0.0782 0.0796 0.0768 1.216 1.493 1.024 1.904 1.890
11.42 8.25 13.74 41.30 38.76 23.07 23.05 1.960 1.984 1.925 26.66 31.67 22.98 38.45 38.23 14 0 .'BO9 14.36 6 6.165 0:2515 24.75 32.75 1 .'i$3 25.08 0.872 6.175 0.2047 14.21 0.481 semilog plot.
D,
EmA2
sec. 0.00061 0.01338 0.00769 Excessive Excessive 0.00482 0.00475 0.0641 0.0601 0.0585 0.00302 0.00316 0.00461 Excessive Excessive Excessive 0.00778 Excessive 0.01903 Excessive 0,00626
0,00746 0.02804 0.01201
Dn X 105 4.58 4.54 4.47 Curvature Curvature 4.96 4.88 5.01 4.78 4.40 4.77 4.72' 4.72 Curvature Curvature Leakage 4.74 Leakage 4.79 Curvature 7.27a 6. 50a 5.74 5.78
0
2
4
6
8
MOLAL DENSITY,
Figure 6.
IO
12
14
16
18
n , G.MOLE / L
Comparison with Enskog dense gas theory
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 10
ENGINEERING, DESIGN, AND EQUIPMENT ~
Table II. Run No.
Temp., O
c.
0.0 0.0 0.0 0.2 34.9 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Diffusion Results with Cell B
Press., Atm. 14.30 6.06 25.50 7.80 11.20 40.98 58.48 24.79 69.95 89.28 78.59 77.57 79.34 98.05 86.01 69.68 88.26 69.81 21.61 41.29 30.98 71.11 102.2 135.7 204.8 171.7 14.54
n,
Gram Moles/ Liter 0.713 0.282 1.430 0.369 0.4675 2.069 3.534 1.117 5.180 15.85 9.14 8.04 10.20 16.62 15.22 5.128 15.74 5.152 0.745 1.500 1.095 2.822 4.495 6.69 11.37 9.26 0.492
D,
cm? sec.
0.00598 0.01501 0.002902 0.01126 0.01016 0.002347 0.001378 0.004285 0.000942 0.0002869 0.000649 0.000594 0.000457 0.0002732 0.000309 0.000932 0.0002973 0.000926 0.00768 0.00390 0.00529 0.002088 0.001316 0.000888 0.000497 0.000638 0.01121
Table 111. Dn X 108 4.26 4.23 4.15 4.16 4.75 4.86 4.87 4.79 4.88 4.55 5.93 4.77 4.66 4.54 4.70 4.78 4.68 4.78 5.72 5.85 5.79 5.89 5.92 5.94 5.65 5.91 5.52
tal curves, in Figure 6. The deviation from the experimental curves is substantial; for example, the theoretical prediction is about 45% below the experimental curves a t 35" C. and 9 gram moles per liter. Enskog and others ( 3 , 12) have successfully calculated dense gas viscosities by the use of Y factors obtained from the volumetric behavior of dense gases. This is, in a sense, a semiempirical procedure, since it may be equivalent to assigning a variable collision diameter. It seems preferable to calculate the collision diameters first, and then to see how these might be applied to the calculation of diffusivities. The Enskog equation of state is
where b is a constant. By partial differentiation, with the assumption that b and s do not vary with temperature
This equation, together with Equation 8 for Y, was used to evaluate collision diameters from volumetric data for several gases ( 2 , 11-14, 17). The calculated collision diameters are plotted against the molal densities of the gases in Figure 7 . While the calculated diameters of the nitrogen and methane molecules are nearly constant, those of carbon dioxide, n-butane, and n-heptane decrease significantly with gas density. I n spite of inconsistencies in the method of calculation, there is little doubt that large variations in the calculated diameters indicate unsatisfactory representation of volumetric data by the Enskog treatment. The gases which show the largest variations in the collision diameters are those with the higher critical temperatures, and also those which deviate most from spherical shape. The calculated collision diameters of carbon dioxide were used to calculate Y factors, as listed in Table 111. The diffusivities calculated by direct application of these Y factors in Equation 9 are not in agreement with the experimental values. The calculated diffusivity is about 33% below the experimental a t 9 gram moles per 1. This is only a minor improvement over the previous calculation, which gave 4570 error a t this point. There is another way in which the collision diameters obtained from volumetric data might be used to compute the diffusivities of dense gases. If the effect of collision diameter in Equation 1, for low pressures, is combined with the Enskog correction, the diffusivity density product would be expected to vary as l/s*Y, Table I11 gives values of l/sZY obtained from the collision diam-
October 1955
~~~
Collision Diameters and Other Quantities for Carbon Dioxide at 35" C.
Molal Density, n Gram Moles/l. 0.842 1.472 2.322 5.045 6.91 10.19 15.83 18.97
Atm./OK. (IS,14)
Collision Dism., s Cm. X 108
Factor,
Quantity, l/slY
0.0775 0.1467 0.2555 0.691 1.042 1.627 3.375 5,456
4.73 4.68 4.56 4.24 4.03 3.64 3.47 3.61
1.074 1.130 1.197 1.382 1.470 1.527 1.841 2.23
4.16 4.03 4.02 4.02 4.18 4.95 4.50 3.44
(bP/bT)n
Y
x
10-14
eters a t various fluid densities. This function is nearly constant to about 7 gram moles per l., in reasonable agreement with the experimental behavior. Beyond this, the function l/s*Y increases to a maximum in the vicinity of the critical density, 10.6 gram mole per 1. and then falls off. The limited number of terms in the series for Y makes it inaccurate in the high density region, but it appears t o converge reasonably well to the critical density. All these methods of correlating diffusion coefficients are of doubtful value. It seems probable that the Enskog dense gas theory is not sufficiently general t o represent diffusion in carbon dioxide a t elevated pressure. Values at lower densities agree with those of Robb and Drickamer
The experimental points of Robb and Drickamer (16) are shown in Figure 8, superimposed on the experimental curves from Figure 5. Allowing for the difference in calibration, the results are in essential agreement to about 9 gram moles per 1.; beyond this density the values of Robb and Drickamer are below those obtained here. Their experimental diffusion coefficients are about equal to those calculated from the Enskog theory a t 17 or 18 gram moles per l., and are lower than theoretical a t higher densities. The measurements of Timmerhaus and Drickamer (18) a t still higher densities, well beyond the range covered in this investigation, showed diffusivities far below the theoretical. The possibility that the results of this investigation are high, as a result of some effect promoting mixing, must always be considered. An error of another sort appears possible in the results of Robb and Drickamer. These investigators used a modified Loschmidt method, with packed columns of different lengths, and a scintillation crystal for concentration measurement. I n changing to a different porous medium and shorter column IO 9
0
I
2
3
4
5
6
7
MOLAL D E N S I T Y . n. G.MOLE
Figure
7.
8
9
IO
II
I2
L
Collision diameters calculated from volumetric data
INDUSTRIAL AND ENG INEERING CHEMISTRY
2085
ENGINEERING, DESIGN, AND EQUIPMENT
0.007
I
I
I
I
l
0006
l
1
!
t; 3
g
0 005
Conclusions
m
2 -
0.004
Y)
w z
n l 0 003
>
b
2
0 002
:-: a
oo,
0: I DC 20'26*C
0
I
l MOLAL
Figure 8.
+*
0
0
3O'-3Z0C
9
4S46'C
i
I
DENSITY,
n.
l
I
G MOLE / L
Comparison of results with those of Robb and Drickamer
length for runs at higher density, they obtained the effective path length of each new arrangement by overlapping the lower density determinations made with the previous column arrangement. The results of Robb and Drickamer for densities above 9 gram moles per 1. were obtained with their fifth column arrangement, and are denoted by the filled-in circles in Figure 8. Two of the three values obtained with this column arrangement a t lower density are much lower than those obtained with previous column arrangements, denoted by the open circles. The accuracy of the calibration for the fifth arrangement, therefore, appears to be questionable. Further investigation of the behavior in this range seems desirable, although Robb and Drickamer ( 1 6 ) and Timmerhaus and Drickamer (18) are no doubt correct in finding a rapid decrease in the product Dn a t still higher densities. Jeffries and Drickamer (9) also found this decrease in Dn a t high pressure for the diffusion of Cl4Oz in mixtures of carbon dioxide and methane. Thus, in general, the results of Drickamer and his coworkers tend to be consistent on this point, which if correct would indicate that our diffusivities are too high in the neighborhood of 16 gram moles per 1. On the other hand, Robb and Drickamer have some low pressure data points which differ from each other by 50% or more, so that questions of accuracy are certainly involved. Therefore, it is difficult from the data presented here and in the literature to ascertain the precise shape of the Dn versus n curves. Additional experimental work is indicated if more accurate values of diffusivity a t high pressure are to be obtained. In Table IV, the observed temperature variation a t low pressure, from Figure 5, is compared with that calculated from theory and with the variation shown by other experimental investigations. The theoretical values were calculated by the method of Hirschfelder and coworkers ( 5 ) . The experimental values were obtained from a curve drawn through the data of Amdur and coworkers ( I ) , Winn (do), and Winter ( $ I ) , with a small
Comparison of Temperature Variation at LOW Pressure Product, Dn Temp.,
Table IV.
O
C.
0 35
100 a
extrapolation for the 100' C. point. The 35" C. value from this curve was used for the cell calibrations, and is, therefore, in exact agreement. Both the theoretical and experimental variations from 0' to 100' C. are in reasonably good agreement with the variation observed here, although the theoretical values are somewhat lower at all temperatures.
Figure 5 (n
= 0) Other Investigators 4.310 X 4.205 X 10-3 4.753 x 10-3 4.753 x 10-95 5.715 X 10-3 5.700 x 10-8
Calibration point.
2086
Theoretical
4.100 X 10-8 4.653 x 1 0 - 8 5.405 X 10-8
The temperature and density dependence of the diffusion coefficient in the Ci20z-Ci402system have been determined between 0" and 100" C., and to 400 times the density of 1 atmosphere. The results indicate that the diffusion coefficient is, to the first approximation, inversely proportional to the density within the range studied. Agreement with the predictions of the Enskog dense gas theory is poor, but the results are consistent with those of Robb and Drickamer to about 200 times atmospheric density. Above 200 times atmospheric density the diffusivities obtained here are considerably higher than those of Robb and Drickamer. Further experimental work is needed to determine the true situation in this range. The two-chamber diffusion cell is believed to have advantages in simplicity which are especially valuable in measurements a t elevated pressure. Ac knowledg ment
The assistance of the Michigan Memorial Phoenix Project, which provided funds for the experimental equipment, is gratefully acknowledged. Fellowships from the Allied Chemical and Dye Corp. also assisted in supporting this work. Nomenclature
cross-sectional area of diffusion opening, cm.Z constant in Equation 10 concentrations of radioactive gas in chambers No. 1 and 2 of diffusion cell diffusion coefficient, cm.z/sec. diffusion coefficient calculated from low pressure relation, cm.2/sec. mutual diffusion coefficient of gases 1 and 2, cm.2/sec. function of temnerature and of gases 1and 2 ionization currents from chambers No. 1 and 2 of diffusion cell cell constant for diffusion cell, cm.-2 constants for diffusion cells A and B, cm.-2 length of diffusion opening, cm. molecular weights of gases 1 and 2 molal density, gram moles/l. molecular density, molecules/cc. total pressure, atm. molecular collision diameters, cm. ideal gas constant, 1.-atm./gram mole-OK. absolute temperature, OK. volumes of chambers No. 1 and 2 of diffusion cell, cm.a correction factor in Enskog relations length, cm. time, sec. literature cited (1) Amdur, I., Irvine, J. W., Mason, E. A . , and Ross, J., J . Chem. Phuls., 20,436 (1952). (2) Beattie, J. A., Simard, G. L., and Su, G. J., J . Am. Chem. SOC., 61,26 (1939). Enskog, D., Svenska Akad. Handl., 63, No. 4 (1921). Gilliland. E. R.. IND. ENG.CHEM..26. 681 (1934). Hirschfelder, J. O., Bird, R. B., and Spotz, E. L., Trans. Am. Soc. Mech. Engrs., 71,921 (1949). Hirschfelder, J. O., Curtiss, C . F., and Bird, R. B., "Moleculrtr Theory of Gases and Liquids," p. 635, Wiley, New York, 1954. Hutchinson, F., J . Chem. Phus., 17, 1081 (1949). Jeffries, Q. R., and Drickamer, H. G., Ibid., 21, 1358 (1953).
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 10
ENGINEERING, DESIGN, AND EQUIPMENT (9) Ibid., 22, 436-7 (1954).
(IO) Jost, W., “Diffusion in Liquids, Solids, Gases,” Academic Press, New York, 1952. (11) Keyes, F. G., and Burks, H. G., J . Am. Chem. SOC., 49, 1403 (1927). (12) Michels, A.,and Gibson, R. O., Proc. Roy. SOC.(London),A134, 288 (1931). (13) WIichels, A., and Michels, C., Ibid., A153, 201 (1935). (14) Michels, A , , Michels, C., and Wouters, H., Ibid., A153, 214 (1935). (15) Pollard, W.G., and Present, R. D., Phys. Rev., 73,762 (1948). (16) Robb, W.L.,and Drickamer, H. G., J. Chem. Phys., 19, 1504 (1951).
(17) Smith, L.B.,Beattie, J. A., and Kay, W. C., J . Am. Chem. SOC., 59, 1587 (1937). (18) Timmerhaus, K. D., and Drickamer, H. G., J. Chem. Phys., 19,1242(1951). (19) Ibid.,20,981 (1952). (20) Winn, E.B.,Phys. Rev., 80, 1024 (1950). (21) Winter, E.R. S., Trans. Faraday Soc., 47, 342 (1951). RECEIVED for review May 21, 1954. ACCEPTEDJune 20, 1955. Division of Industrial and Engineering Chemistry, 125th Meeting, ACS, Kansas City, Ma., March 27, 1954.
l o w Frequency Bubble Formation at Horizontal Circular Orifices ROBERT J. BENZlNGl
AND
JOHN E. MYERS
School of Chemical and Metallurgical Engineering, Purdue University, W. Iafayetfe, Ind.
G
AS bubble formation a t horizontal submerged orifices is characterized by two different mechanisms which are related to the frequency of formation. At low frequencies, generally below 100 bubbles per minute, the frequency of bubble formation is almost directly proportional to the rate of flow of gas to the orifice while the bubble size is almost constant. This is referred to as the region of static bubble formation. At higher rates, generally above 500 bubbles per minute, the volume of the individual bubbles formed increases with increasing rate of flow of the gas supplied to the orifice while the frequency of formation remains almost constant. This is known as the dynamic region. The work described in this paper is limited to studies made in the static region. While the bubble is growing but still attached to the orifice, the adhesive force due to surface tension is
ad
COS
e
(1)
where d is the orifice diameter, u is the liquid-vapor surface tension, and 8 is the contact angle between liquid, solid, and vapor. For all the systems used, the liquid wetted the orifice readily so that the contact angle was assumed to be zero and cos 8, therefore, equal to unity. The buoyant force may be written as V(P1
-
P,)9
(2)
where V is the bubble volume, p1 is the liquid density, p g is the gas density, and g the acceleration due to gravity. At instant of rupture these forces may be considered equal:
V
(PI
-
p,)g
=
(3)
If the volume of the bubble is considered to be equal to the volume of a sphere of diameter, D, and p g is small compared with p l . then n- 0
3
--g-
pig =
nd u
(4)
a constant for any particular gas-liquid system a t a fixed temperature. For air bubbles formed in water a t 20” C. V / d = 0.231 sq. cm., ideally. However, Datta, Napier, and Newitt ( 1 ) report results for this system ranging between 0.17 and 0.44. In the same paper a comparison is made of the results of Maier (6),Owen (Y), and Swinden (8). Considerable variation is also evident in their work which gave values of V/d extending from 0.04 to 1.0. Eversole, Wagner, and Stackhouse (3) measured bubble volumes a t rates of formation between 40 and 90 bubbles per second. Under these conditions the mechanism of static bubble formation was apparently not applicable. This is indicated by their results which gave values of V/d between 0.114 and 0.628 sq. cm. Guyer and Peterhans ( 4 ) investigated the effects of fluid properties on bubble size using 20 liquids. Most of their data are taken a t a frequency of one bubble per second. For the airwat,er system values ranged from V/d = 0.136 sq. cm. a t an orifice diameter of 0.264 cm. to V / d = 0.227 a t an orifice diameter of 0.0045 cm. The over-all trend in their data indicates that V / d increases with decreasing orifice diameter. However, for orifice diameters between 0.1 and 0.014 cm. the values of V / d decrease from 0.180 t o 0.168sq. cm. for no apparent reason. Recent work by van Krevelen and Hoftijzer ( 5 ) measuring bubbles formed from capillaries a t rates less than one bubble per second check the theoretical results much better. Twentytwo runs with air and water gave values of B/d between 0.207 and 0.270 and an average value of 0.246 sq. cm. Davidson ( d ) , who reports data over a wide frequency range, is one of the few investigators who has studied the effect of orifice geometry in bubble size. His results indicate a relationship between the size of bubble formed and the upstream volume of the gas chamber leading to the orifice. Experimental equipment and technique are outlined
This equation may be rearranged and expressed in terms of dimensionless groups
The experimental equipment is shown schematically in Figure
A number of observers have studied gas bubble formation a t circular orifices. -4convenient method of comparing their results is to express them in terms of the ratio of the bubble volume, V , to the orifice diameter, d . This ratio V / d (Equation 3 ) is
The backflow check is used to prevent the liquid in the humidifier from flowing back to the surge tank. The general control valve, 4, gives an approximate setting while the fine control valve, 5, gives close adjustments and uniform flow control. The length of pipe between control valve, 5, and the orifice chamber is approximately 3 feet: f/8-inch galvanized pipe is used throughout the system. The liquid bath consists of a large glass container.
1. The gas enters the surge tank a t approximately atmospheric pressure after being reduced from pressures ranging between 35 and 1500 pounds per square inch. It then flows to the humidifier - - 1.82 (1)1’3 (5) where it is saturated with the liquid being used in the system. d Sd2 P1
1
Preaent address, Wright-Patterson Air Force Base, Dayton, Ohio.
October 1955
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