O’CONNELL,GILLESPIE,KROSTEK, AND PRAUSNITZ
2000
Diffusivities of Water in Nonpolar Gases
by J. P. O’Connell, M. D. Gillespie, W. D. Krostek, and J. M. Prausnitz Department of Chemical EnQineeTWQ, Universiiy of California, Berkeley, Californla
94720
(Recefved November 1 , 1 9 6 8 )
Vapor-phase diffusivities, obtained by a modified Stefan method, are reported for water diffusing in argon, nitrogen, and inethane at low pressures in the temperature range 10-lOOo. The experimental results are reproduced within experimental error using collision integrals based on the Kihara spherical-core potential function.
-4s a part of a program to determine intermolecular forces in gases containing water, experimental dilute-gas diffusivity measurements were made for water with several nonpolar substances. We report here our experimental method and the data obtained for aqueous systems containing argon, nitrogen, or methane in the temperature range 10-looo. Our data for these systems cover a wider range of temperature than those of earlier workers and it appears that our results are more consistent with theory than those previously reported.
Experimental Method For mixtures where one substance is a liquid a t the temperature of interest, while the other is a gas, the most accurate and rapid method for measuring vaporphase, binary diffusivities is that due to Stefan,l where the liquid is placed in a tube and gas flows over the end of the tube. A number of workers2-6have derived the equations relating the diffusion coefficient to the measurable properties of the system: the gas concentrations a t both ends of the tube, the tube length, and the flux of the liquid component
where N 1 is the flux of liquid component 1 in moles per unit area per unit time, L is the distance from the liquid surface to the end of the tube, CZO is the concentration of gas 2 at the end of the tube, in moles per unit volume, and CZsis the concentration of gas 2 at the liquid surface. In the derivations it is assumed that the concentration of the diffusing component 1 is zero at the mouth of the tube and that the only convection anywhere within the tube is due to the bulk flow compensating for diffusion of gas 2 toward the impenetrable liquid surf ace. In most experimental systems, the ideal gas law has been assumed in order to determine the concentrations of the gas Czoand CZs. At total pressures in excess of 1 atm or in cases where the saturation pressure (and thus the concentration) of component 1 becomes appreciabIe, these approximations can cause appreciable The Journal of Physical Chemistry
errors. An equation of state can be used
CZO = P/xoRT
(2)
Cza = (1 - YI) (P/x,RT)
(3)
where xo and xs are the compressibility factors in the gas stream and at the liquid surface, respectively, and y1 is the mole fraction of the vaporized liquid at the surface, a quantity which may be obtained from the equations of equilibrium. As Lee2p3has shown, curvature of the liquid surface and convection at the mouth of the diffusion tube can significantly affect the value of L to be used in eq 1. I n addition, the diffusivity can be affected by a finite concentration of the liquid component 1 a t the mouth of the tube and by several other considerations pertinent to the experimental system used here. For all effects, the coefficient Sapparent, measured at a given length, L, is related to the theoretical value at by a converging series SRppRreIlt
= %(
f (A x / L )
+ . . .)
( 4)
where AX is a combination of experimental variables having the dimensions of length. Thus, by plotting the measured diffusion coefficient as a function of the reciprocal of the measured length, the zero intercept is the true coefficient. Determination of the exact temperature of the surface of the evaporating liquid is subject to uncertainties.6 For water the change of saturation pressure with temperature is large; therefore considerable error can result if the heat required to evaporate the liquid can be supplied only through an appreciable temperature gradient at the surface. (1) J. Stefan, Wien. Ber. (11), 68, 385 (1873); 98, 1418 (1889). (2) C. Y. Lee, M.S. Thesis, University of California, Berkeley, Calif., 1952. (3) C. R. Wilke and 0.Y. Lee, Ind. Eng. Chem. 46, 2381 (1954). (4) W. Jost, “Diffusion in Solids, Liquids and Gases,’’ Academic Press, New York, N. Y., 1952. (5) R. Schirmer, Verfahrenstechnik. Beth 2. Ver. Deut, Ing., 170 (1938). (6) F. A. Schwera and J. E. Brow, J. Chem. P h y s . 19, 640 (1951).
DIFFUSIT'ITIES OF WATERIN NONPOLAR GASES
2001
Finally, there is the effect of convection. There is a bulk flow in the tube in order to compensate for diffusion of the gas toward the impenetrable liquid surface. If the liquid has a low molecular weight compared to the gas, the density of the vapor a t the liquid surface is less than that a t the mouth of the tube above it and convection could develop in the tube. The measured molecular diffusion coefficient would be too high. At present, the tube size for which convection becomes important is u n k n o ~ n , ~but * ~itt ~is likely that downward diffusion provides the safest solution in systems involving water.
Experimental Apparatus To eliminate as much as possible the potential sources of error encountered in previous experimental syst e m ~ , ~significant ,~,~ improvements were incorporated into the design of the present apparatus. (For details, see ref 7-9.) Schematic diagrams of the main portion of the system are shown in Figures 1 and 2. The major modifications incorporated into the new apparatus are: (1) the evaporating liquid surface is held on a porous metal plate in order to allow maximum heat transfer for evaporation; (2) to minimize convection, diffusion is in a downward direction with the tube length varied by vertically moving the plate; (3) the system can be operated a t pressures different from atmospheric. The last feature allows measurement of the diffusivity of water a t temperatures from about 5 up to 100" by maintaining a partial pressure of water at the surface of evaporation which is relatively low but still sufficient for rapid measurements. The main components of the system are the main tank which contains the constant-temperature bath, the flow system for the gaseous component, the series of tubes for the liquid component, and the peripheral measuring arid controlling equipment. A high-speed fan circulated gas inside the main tank to maintain the system at constant temperature. The temperature was kept constant to within 0.01" while a maximum variation of 0.1" mas found between the two extreme positions of the liquid-containing system. Two windows were installed in the walls of the tank to allow visual access to the interior. In the tank there was no
Vacuum
Figure 1. Schematic flow diagram of t h e vapor diffusion system.
Figure 2. Apparatus for gaseous diffusion of aqueous mixtures.
contact between the flowing gas and the circulating gas bath. The flowing gas reached the tank temperature while inside copper coils and entered a chamber containing a series of plates to force the flow to be straight and laminar past the mouth of the diffusion tube. From the chamber, the gas proceeded to a pressure controller and outlet. The diffusion tube, 0.375 in. in diameter, was located about two-thirds of the may from the entrance to the exit of the chamber. A thermocouple to measure the gas temperature was located opposite the mouth of the tube in the outlet to the pressure gauges. The liquid system was made up of a series of three tubes extending from inside the diffusion tube to outside the tank. The lowest tube was of nickel-plated copper and was fitted with 0 rings to provide a leak-tight seal at the upper end of the diffusion tube. A nickel plate with pores of diameter 1.5-3 p was sealed in the bottom end of the copper tube to support the liquid and to conduct heat to the liquid surface. To prevent formation of a liquid film over the entire surface of the plate which would then have dripped, stearic acid, dissolved in an inert solvent, was applied to the plate and the solvent was evaporated. The result was a nonwettable lower surface of the plate. This treatment should not have significantly affected the vaporization process or the vapor pressure. A thermocouple was in direct contact with the copper tube just above the 0 ring. Since no (7) d. P. O'Connell, Ph.D. Thesis, University of California, Berkeley, Calif., 1967. (8) M. D. Gillespie, M.S. Thesis, University of California, Berkeley, Calif., 1967. (9) W.D. Krostek, M . S . Thesis, University of California, Berkeley, Calif., 1968. Volume 79, Number 6 June 1060
O'CONNELL,GILLESPIE,KROSTEK,AND PRAUSNITZ
2002
temperature difference could be detected between this thermocouple and one temporarily connected directly to the plate, even under high rates of evaporation, a good measure of the actual temperature of the evaporating surface was achieved. A glass tube was inserted into the upper end of the copper tube and sealed with an 0 ring. The lower section was of precision bore while the upper section was connected to a thin side arm to allow gas from the tank to replace the evaporated liquid. Negligible water could diffuse through this side arm. The upper end of the glass tube was inserted into a stainless steel tube which protruded through the top of the tank. Liquid water could be added to the system through a long hypodermic needle inserted into the end of this tube. The whole system of tubes containing the liquid could be moved vertically from outside the tank. Pressures were measured with Bourdon tube gauges accurate to 0.03 psia in the range 0-30 psia and to 0.06 psia in the range 0-60 psig. The system pressure was controlled well within this accuracy. The amount of water evaporated and the diffusion tube length were measured with a cathetometer outside the system. The movement of the water meniscus in the precision-bore tube could be measured to about h0.01 mm. From the distance between a line scribed on the copper tube to one on the stainless steel diffusion tube and from the distances from these lines to the end of their respective tubes, the diffusion path L could be measured from the outside of the tank with an error of less than f 0 . 0 5 mm.
I \
353,2 'K
c
w
2 0.32 o A
z
N2-HzO Ar-H20 (1 a h )
0 0.28 u)
3
LL
LL P
at 2- 0.26 m
04
I
I
I
1
Eqperimental 0 Giilespig
0.3
0 O'Connell t Crider x Schwerz 8; BIOVJ
-
8
-? 5 03
N
c1
Y,
zY P
0.2
0.2
I
275
1
I
325 TEMPERATURE, O K
300
I
350
c
Figure 4. Calculated and observed vapor-phase diffusion coefficients for water in nitrogen.
Procedure To start the measurements, degassed liquid was placed in the liquid system and the tank was filled with the gas of interest. The gas flowed into the system at the appropriate rate and the plate was adjusted to the desired level. After the system reached the desired temperature and steady-state diffusion was established, the liquid level was lowered to within the precision-bore section of the glass tube. Measurements of the meniscus height and the temperatures were made at intervals when the liquid level had changed a t least 1.5 mm. After several measurements of constant rate, the location of the scribed lines was determined and the liquid system moved to a new level. A period was allowed for steady state to be reached and a new set of measurements was begun. The length of the diffusion path was varied more or less randomly from 5 to 15 cm until a sufficient number of values had been taken to establish a straight line for a plot of diffusivity vs. the reciprocal of the diffusion path length. Measurements were repeated a t intervals spaced by several weeks for the nitrogen-water system and good reproducibility was obtained. The total system pressure was always at least 5 times the vapor pressure of water but never more than 30 times over the entire temperature range.
0.24 0
0.05
0.10 I/L
0.15
, cm"
0.20 0.25
Figure 3. Linear extrapola,tionsof diffusion coefficients. The Journal of Physical Chemistry
Data Reduction To obtain the final diffusivity values, the quantities in eq 1-3 were evaluated. The concentrations were
2003
DIFFUSIVITIES OF WATERIN NONPOLAR GASES
Experlmentol 0 Gillespie
I
270
0.22
I 1 i I 310 350 TEMPERATURE, O K
I
I 390
270
310 350 TEMPERATURE,
390 OK
Figure 5. Calculated and observed vapor-phase diffusion coefficients for water in methane.
Figure 6. Calculated and observed vapor-phase diffusion coefficients for water in argon.
calculated from eq 2 and 3 using the virial equation of state (truncated after the second virial coefficient) for calculating vapor-phase nonidealities. The mole fraction of water above the evaporating surface was cal-
culated in the same manner as that described earlier.'" ~ obValues of second virial cross coefficients B I were tained from experiment.1° The measured diffusion coefficients for various path lengths were fitted to straight lines and to parabolas in the reciprocal of the length. The intercept value of a>12 was then converted from the experimental pressure to 1 atm. In every case there was essentially no difference in the deviations from the linear and the parabolic extrapolations. The value of AX in eq 4 was always less than 0.5cm for L between 5 and 15cm and the rms deviation from a straight line was never more than 0.5% with individual deviations always less than 1%. Figure 3 shows representative results. An error analysis of the experimental variables of flux, tube length, and concentrations indicated the known inaccuracies are no more than 0.4%. On the other hand, unknown effects, particularly convection, could markedly increase the inaccuracy. However, we believe the values are accurate to a t least 2oJ,. Based on the results of our analysis, earlier measurements are probably optimistic in their error estimates.
Table I: Binary Gas Diffusivities for Water with Three Nonpolar Gases
Temp, OK
Exptl
cmz/sec--Calcd
Ar
282 298 327.5 353
0.218 0.250 0.295 0.348
0.217 0.245 0.292 0.336
NZ
282 298 327.5 353 373
0.221 0.253 0.305 0.360 0.396
0.223 0.248 0.296 0.340 0.375
CHI
283 303 323
0.234 0.267 0.298
0.228 0.260 0.302
-----312,
Gas
(IO) M . Rigby and J. M.Prausnitz, J. Phys. Chem., 72, 330 (1968). Volume 75,Number 6 June 1969
O'CONNELL,GILLESPIE,KROSTEK,AND PRAUSNITZ
2004
Results and Discussion The diffusivities of water into Ar, N2, and CHI were measured in the temperature range 10-100". The results are given in Table I. There are no previous data available for the Ar-H20 system. Figure 4 compares earlier data for the N2-H20 systemeJ1J2while Figure 5 compares earlier data for the CHI-H~Osystem.6 The deviations between experimental workers are significant but our data agree better with the theoretical temperac ture dependence which, in turn, is nearly independent of the intermolecular potential model. The Kihara potential model with a spherical core was used to calculate the diffusivities from the ChapTable I1 : Pure-Component Parameters for t h e Kihara Potential Function
Spherical-Core
tiubstances
Co:k, "K
u +2a (centercenter), A
2a, A
Ref
HzOa Ar CHI Nt
170 147 231 139.2
2.915 3.314 3 525 3.525
0.265 0.368 0 771 0.705
b
I
I
C
d C
For H20, a dipole-dipole term i s added t o the spherical-core Kihara potential function.'8J8 See ref 13. C A. E. Sherwood and J. M. Prausnitz, J . Chem. Phys., 41, 429 (1964). See ref 15.
The Journal of Phuaical Chemistry
man-Enskog kinetic theory, using the parameters given in Table 11. Customary mixing rules were used. Spherically symmetric induction effects were in~1uded.l~ Comparisons between calculated and observed results are shown in Table I and Figures 4-6. The data are predicted essentially within experimental error and the over-all agreement is much better than when other parameters are used for water.I4 The agreement is as good as or better than that found earlier15for nonpolar, polyatomic gases. The discrepancies at low temperatures are probably due to the neglect of nonspherical effects from induction and rotation which cannot bo included in the kinetic theory formulations. The theoretical implications of these potential parameters for water are explored el~ewhere.'~~'~
Acknowledgment. The authors are grateful to the Office of Saline Water, U. S. Department of the Interior, for financial support, and to the Computer Center a t the University of California, Berkeley, Calif., for use of its facilities. (11) E. T.Nelson, J . A p p l . Chem., 6 , 286 (1956). (12) W.L. Crider, J. Amer. Chem. Soc., 7 8 , 925 (1956). (13) J. P. O'Connell, M. Rigby, and J. M. Prausnitz, I n d . Eng. Chem. Fundamentals, in press. (14) LaMonchick and E. A. Mason, J. Chem. P h y s . , 36,2746 (1962). (15) J. P. O'Connell and J. M. Prausnits in "Thermodynamic and Transport Properties at Extreme Temperatures and Pressures," S . Gratch, Ed., ASME, New York, N. Y., 1965, p 19. (16) J. P. O'Connell and J. M. Prausnitz, I n d . Eng. Chem. f f u n d n mentals, in press.
