GERARDKRAUSAND JOHNW. Ross
3 34
Vol. 57
SURFACE AREA ANALYSIS BY MEANS OF GAS FLOW METHODS. 11. TRANSIENT STATE FLOW IN POROUS MEDIA1 BY GERARD KRAUS AND JOHN W. Ross Applied Science Research Laboratory, University of Cincinnhti, Cincinnati 21, Ohio Received J u l y 69, 1866
Transient state flow rates in the Knudsen flow region have been measured on a series of powders previously investigated by steady state flow methods. Surface areas calculated from the transient flow data using nitrogen were found to agree exceedingly well with nitrogen adsorption areas, even in cases where steady state measurements yield too small a surface area. Helium flow measurements result in somewhat larger areas suggesting that more surface is accessible to the smaller helium molecules than is to nitrogen.
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Introduction It has been shown in the preceding paper2 that
combining equations (1) to (3) and setting k it is found that
=
42,
steady state flow rates of gases in porous media in (4) the region of Knudsen flow may be used to evaluate a surface area which in some instances agrees very closely with the Brunauer-Emmett-Tellera nitro- It is readily shown that this equation is not consistgen adsorption area. In other cases, however, the ent with the specific flow rate deduced from Derjaflow area is found to be definitely less, a behavior guin's theorys as used in the present study2 indicating the presence of blind pores. Ba'rrer and Grove,4 in a recent publication, suggest the possibility of obtaining a total surface, including blind pores, by conducting the flow measurements in the where dn/dt is the flow rate in moles per second, transient state. The present investigation repre- AP is the pressure drop across the porous medium, sents a test of this hypothesis based on comparison and A is the cross-section of the porous medium. with nitrogen adsorption surface areas. By definition
Theoretical In the theory of Barrer and Grove4 the surface area of the porous medium is introduced as follows. The specific flow rate for Knudsen flow (written as a diffusion coefficient) for a straight cylindricad capillary is
It will be noted that the total cross-sectional area of the pores appears as EA,for clearly the diffusion coefficient must be based on the open cross-section of the porous medium. In the flow experiment D is simply D = P/6L
where r is the capillary radius, R is the gas constant, T is the absolute temperature and M is the molecular weight of the gas. The diffusion constant is measured by the time-lag m e t h ~ d ~giving -~ D = k212/6L
(2)
where L is the "time lag" necessary for establishment of steady state extrapolated to zero flow, 1 is the length of the porous medium, and k is a tortuallowing for the osity factor (taken as equal to 4) fact that the capillaries are not straight. Finally the expression (3)
is used to introduce the specific surface area 8, (per unit volume of solid), E being the porosity. By (1) Presented before the Division of Colloid Chemistry of the American Chemical Society, Atlantic City, New Jersey, September, 1952. (2) G. Kraus, J. W. Ross and L. A. Girifalco, THISJOURNAL, 57, 330 (1953). (3) 8. Brunauer, P. H . Emmett and E. Teller, J . Am. Chem. Soc., 60, 309 (1938). (4) R. M. Barrer and D. M. Grove, Trans. Faraday Soc., 47, 826, 837 (1951). (5) R. M. Barrer, "Diffusion in and Through Solids," Cambridge Press, 1951, P. 19.
(7)
without the use of a tortuosity factor, as the internal pore structure is already accounted for in equation ( 5 ) . It follows from equations (6) and (7) that 8v
144 -_
-
e
131--e
which is exactly 18/13 times the surface calculated by the method of Barrer and Grove.4 In view of the experimental results to follow, it is important to note here that both equations (1) and ( 5 ) are based on the assumption of inelastic collisions of the gas molecules with the pore walls.
Experimental The apparatus used (Fig. 1) consisted essentially of two 4-liter reservoirs connected through the cell holding the powder. A mercury manometer D was used to measure the pressure on the high pressure side and a calibrated thermocouple vacuum gage was used to determine the pressure on the discharge side. The actual quantity of gas flowing through the porous medium during an experiment was found to be always quite small, making the use of a manostat on the high pressure side unnecessary. The inlet pressure must, of course, be at all times much larger than the pressure on the outgoing side so that the pressure gradient may be regarded as constant. In operation the entire system was evacuated and flushed twice with the appropriate gas. The system was then pumped down to a proximately 1 to 2 microns of mercury and shut off frome!t pumps by (6) B. Derjaguin, Compt. rend. acad. sci. U.R.S.S., 53, 623 (1946).
TRANSIENT STATEFLOWIN POROUS MEDIA
Mar., 1953
TABLE I TRANSIENT FLOW A N D NITROGEN ADSORPTION AREAS Powder
--
Transient Knudsen flow Helium Nitrogen S, m.z/g. L,min.5 S,m.Z/g.
L,min.o 2.12 0.895
f
closing stopcock G. Stopcocks E and F were closed and the desired inlet pressure established by bleeding gas into reservoir A through H. At zero time stopcock F was opened and the gas allowed to diffuse through the cell C into reservoir B. Figure 2 shows a typical flow rate curve. The time lag L is obtained by extrapolation of the straight line, steady state portion of the curve to the initial pressure in the cell and discharge reservoir. It is not necessary that this pressure be zero.6 Several runs were made in each determination, using various inlet pressures. In order t o ensure essentially ure Knudsen flow, it is important that these pressures be sukciently small to cause the mean free path of the molecules to exceed the pore dimensions. With powders of unknown particle size the pore dimensions are, of course, also unknown. I n such cases it is best to take time lag measurements over a range of inlet pressures and to extrapolate to zero pressure. Both procedures were found to give the same limiting time lag within experimental error.
0.31 0.55 3.89 4.75 7.90
4.8 2.04 1.92 16.0 1.89
0.37 0.64 4.60 5.45 8.85
0.420 CUO 60.32 Glass spheres, I1 42.10 .388 15.31 PbCrOa .652 .86 Bas04 26.94 ,377 7.0 0.80 15.31 ,724 TiOz . a Time lags L based on at least three experiments.
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335
Nitrogen adsorption BET HJ S,m.a/g. S,m.*/g.
0.30 .55 3.75 4.68 7.97
0.31 .59 4.01 4.97 8.26
accuracy of either method, and it is apparent that the method of calculation of Barrer and Grove4 (equation 4) leads to results which are some 28% too low.
200
150
l
h
2 H
5
3 v
4 100
-
TO DIFFUSION
50 TO THERMOCOUPLE
e
GAUGE
I ?.
Fig. 1.-Schematic diagram of transient state flow apparatus. The powders invest,igated were the same as in paper I, and all flow measurements were carried out a t 30 i~ 2". Surface areas from nitrogen adsorption isotherms were calculated by the Harkins-Jura7 relative method as well as by the BET method. The cross-sectional molecular area for nitrogen was taken as 16.2 square lngstromsf
Results and Discussion In Table I are given the flow data for the six powders investigated along with the surface areas calculated by equation (8) and the BET and HJ areas. The agreement between nitrogen flow and nitrogen adsorption areas is easily within the experimental (7) W. D. Harkins and G . J u r a , J . Am. Chem. Sac., 66, 1366 (1944). (8) R. T. Davis, T. W. DeWitt and P. H. Emmett, THISJOURNAL, 51, 1232 (1947).
-
0- -Q
!---
-
,
,
LEO F. EPSTEIN AND MARION D. POWERS
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336
sorption of the gas on the surface, leading to a larger time lag and an apparently higher surface area. It is clear, however, that in the case of helium this is extremely improbable. The other alternative is the possibility of specular reflection of molecules at the walls of the capillaries. Derjaguid has shown that for 100% specular reflection the coefficient 24/ 13 in equation (6) becomes equal to 8/3; in intermediate cases the value of this coefficient may be assumed to lie somewhere between the two extremes. It follows that if specular reflections were involved, D would be larger than indicated by equation (6) and hence the use of equation (8) would lead to surface areas which are too small. Exactly opposite results were observed with helium. Thus, the mechanism advanced in the preceding paragraph appears to be the most plausible explanation for the larger helium flow areas. Despite the success of the transient state flow method in predicting surface areas, it should be mentioned that the use of the time lag method is not always free from objection. The time lag method for measuring diffusion coefficients can only be exact as long as Fick's second law is valid and the diffusion coefficient is a constant. While this is true for transient Knudsen flow through packed particles containing no blind pores, the presence of blind channels must introduce deviations from Fick's second law and produce a time dependent diffusion coefficient. Furthermore, the time lag
VOl. 57
must be a function of blind pore volume as well as area. This objection is probably not a serious one as long as the blind pore volume is small compared to its area, as must be the case with the rather fine powders used in the present study. We may, however, visualize a situation in which there are a few large blind pores such as might occur in some consolidated porous media. Because of the immobilization of gas in the blind pores, the time lag will now be larger than it would be if no blind pores were present. At the same time, however, the internal surface of the medium is hardly affected by the few large blind pores, so that the use of the time lag method will result in too large a value for the surface area. The various physically distinct surface areas which may be derived from gas flow experiments offer an interesting method for the study of surfaces. By combining steady state and transient experiments it is possible to obtain a fairly complete picture of the structure of a porous medium, giving such information as total 'surface, surface area of blind pores (by subtraction of the steady state from the transient state Knudsen area), and at least in some cases geometric surface.2 From these, such quantities as mean particle size, mean pore diameter and surface roughness factor are readily caIculated. Acknowledgment.-This investigation was supported by the Office of Ordnance Research, U. s. Army, under Contract No. DA33-008 ord 123.
LIQUID METALS. I. THE VISCOSITY OF MERCURY VAPOR AND THE POTENTIAL FUNCTION FOR MERCURY BY LEO F. EPSTEIN AND MARION D. POWERS General Electric Company, Knolls Atomic Power Laboratory,' Schenectady, New York Received July 8g, 1868
I t is shown that, contrary to the results of earlier studies, an excellent fit to the data on the viscosity on mercury vapor can be obtained using a Lennard-Jones 6-12 potential, in spite of the fact that X-ray scattering measurements indicate that this substance is more nearly represented by a 6-9 potential. Viscosity measurements, like second virial coefficient studies, do not appear to be sufficient by themselves to determine the repulsive power uniquely. In this study, the 6-12 parameters elk = 851 f 32'K. and ro = 3.253 z!= 0.04r A. are obtained. A detailed survey of the previous studies of the potential function for mercury indicates that the values obtained from the viscosity are quite consistent with the other data. Certain difficulties in the determination of ro and related quantities from X-ray scattering data are pointed out.
Introduction The viscosity of a gas which satisfies a LennardJones 6-12 potential U1&) = E [ ( T o / T ) ~ *- 2(ro/rYl (1) (in which e is t,he depth of the potentia1 energy curve, occurring at T O ) has been shown by Hirschfelder, Bird and Spotz2 in a brilliant study of the transport properties of gases, to be given by 9
= (C/u*)(MT)'/a [V/Wz(2)]
(2)
where the constant C = 266.93 X lo-'
(3)
(1) The Knolls Atomic Power Laboratory is operated f o r the United
States Atomic Energy Commission by the General Electric Company under Contract No. W-31-109 Eng-52. (2) (a) J. 0. Hirschfelder, R. B. Bird and E. L. Spotz, J . Chem. Phya., 18, 968 (1948); (b) J. 0. Hirschfelder, R. B. Bird and E. L. Spotz, Trane. A m . Sac. Mech. Engrs., 71, 921 (1949).
if 71 is in poises, u in A., and Ulz(u)= 0, and is given by u =
elk in
OK.
r0/2'/fi = 0.8909r0
is defined by (4)
M is the molecular weight of the gas. The quantities V and W2(2) are quite complicated functions of elkT which have fortunately been calculated and tabulated over a wide range of the independent variable.2 In attempting to apply eq. 2 to mercury vapor, Hirschfelder and his associates were at once confronted by t,he fact that the experimental evidence seemed to indicate that Hg did not in fact satisfy a 6-12 potential. The liquid mercury X-ray scattering data of Hildebrand, Wakeham and Boyd3 lead to a 6-9 rather than a 6-12 potential function. (3) J. H, Hildebrand, H. R. R. Wakeham and R. N. Boyd, J. Chem. PhV8., 7, 1094 (1939).
