force, AC [expressed a; grams of N H a . A I ( S 0 ~ )12H20 2. per 100 grams of H 2 0 ]; the Sherwood number actually increases with increasing mass flux, the reverse of that predicted by boundary layer theory for constant physical properties ( 7 ) . T h e data in Figure 1, B , have been calculated using the integral diffusivity for the concentration range Co to Cm-i.e.,
p
= density, g./cc.
mass fraction of anhydrous salt, g. anhydrous salt/g. solution R e = P,dv,/w, Sh = kd/p,Bint Sh’ = kd/p,a)., SC PrnjPrnDint SC’ = P,/P,BO
w
=
SUBSCRIPTS m
This modification represents a marked improvement. Further attempts to resolve the data are being made by allowing for the variation of other physical properties with concentration, and also’ for the high mass flux. T h e data for potassium alum follow very closely those given in Figure 1 for ammonium alum, and their correlation is improved to the same extent. It would appear, therefore, that our findings for the dissolution of electrolytes show that the dependence of diffusivity on solution csoncentration is a major factor requiring consideration when correlating mass transfer data of this type. Nomenclature
C d D
k rn v
solution concentration, g. hydrated salt/lOO g. water particle diameter, cm. diffusion coefficient, sq. cm./sec. mass transfer coefficient = m(wt - U I , ) / U J , UJ,, g. hydrated salt/‘(sec.) (sq. cm.) = mass flux, g. hydrated salt/(sec.)(sq. cm.) = mainstream velocity, cm./’sec. = viscosity, g./(cm.)(sec.)
= = = =
o m
t
in mainstream a t surface = arithmetic mean of surface and mainstream concentrations = in transferring solid-e.g., w , = 0.523 for ammonium alum = =
literature Cited
( I ) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” pp. 656-76, Wiley, New York, 1960. (2) Kin C. J., IND. ENG.CHEM.FUNDAMENTALS 5,146 (1966). (3) M ul in, J. W., Cook, T. P., J. Appl. Chem. 15, 145 (1965). ( 4 ) Mullin, J. LV., Garside, J., Unahabhokha, R., Zbid., 15, 502 (1965). ( 5 ) Mullin, J. W., Treleaven, C. R., “Symposium on the Interaction between Fluids and Particles,” p. 203, Institution of Chemical Engineers, London, 1962. ( 6 ) Nienow, A. LV., Brit. Chem. Eng. 10, 817 (1965). (7) Ranz, W. E., Dickson, P. F., IND.ENG.CHGM. FUNDAMENTALS 4, 345 (1965). (8) Rhodes, E.,Zbid., 5, 146 (1966).
7’
-
A . Tt‘. lVienow R a k s rnahabhokha J . TI’. Mullzn
University College London, England
D E T E R M I N A T I O N OF RATES OF SURFACE TRANSPORT SIR: This note is concerned with a binary gas system, A , diffusing through a porous particle counter to B with no gradient in total pressure and under isothermal conditions. A typical example is the Wicke-Kallenbach experiment (4). I t is generally recognized that transport in the porous structure is of two types: T h e first is a diffusion mechanism in which collisions both with the walls of the pores and with other gaseous molecules govern the rate. T h e second is a n adsorption-migration mechanism in which the adsorbed species migrates along the internal surface of the particle. Clearly the diffusion mechanism should predominate when neither gas is adsorbed to an appreciable extent as, for example, in the helium-argon system a t normal conditions of temperature and pressure. T h e flux rat110observed experimentally should be a good test of this; with both bulk and Knudsen diffusion this ratio should be as the inverse square root of the molecular weights over a wide range of conditions and pore structure properties, as we have recently shown (7). I t is probable that both diffusion and surface migration mechanisms are important with most systeims a t many experimental conditions. I t would be most desirable to be able to measure the proportion of total transport due to each mechanism, but this is clearly very difficult. IRivarola and Smith (3) have proposed that the diffusion component could be calculated using a suitable model for dif’fusional transport in porous media. Such computed diffusion components could then be subtracted from measured total transport rates to yield the rate due to the adsorption-migration mechanism. These investigators, working only a t 1 atm., were successful i n relating the surface transport rate thus calculated to properties of the
particle, a surface diffusivity, and an equilibrium constant. O n the basis of this result it is clear there is a prospect of predicting both diffusion and migration transport based on knowledge only of basic properties of the porous solids and of the gases. \Ye have observed, however, some data which cast considerable doubt on the validity of the subtraction step above. Counterdiffusion experiments with the H e - C 0 2 system were carried out in a n experimental arrangement described elsewhere (2) using a porous alumina sample. The sample was a cylindrical plug, 0.787 cm. thick with cross-sectional area of 5.07 sq. cm. and of bulk density 1.01 grams per cc. T h e preparation of the sample, the starting material, and the final specifications were such as to duplicate the pellet of this density reported by Rivarola and Smith. The experimentally measured fluxes and flux ratios are shown in Figures l and 2. A striking feature of these data is the maxima u i t h pressure in the flux of helium a t both temperature conditions; these maxima would not be predicted on the basis of the gas diffusion mechanism alone. Models for the effect of pore structure on transport rates which d o not consider surface migration would predict a gradual rise in helium flux with pressure, eventually flattening ; the almost linear dependence of carbon dioxide flux on pressure would also not be expected. The decrease in flux ratio would certainly not be anticipated from gas phase theory only. Under the conditions of experimentation, it is probable that only C O S ,of the pair, is surface-adsorbed to a significant extent. It is apparent that the surface migration acts to impede the transport of helium, and we suspect that this effect may be VOL. 5
NO. 4
NOVEMBER 1 9 6 6
579
1
THEORETICAL
1
3.32
L
I
0
t 0
4 O
HELIUM
co2
2
4
6
8
IO
12
14
c t
0 0
0
16
2
I
I
I
I
4
6
8
IO
P, A T M .
Figure 1. pressure
T =
- 19.0' C.
12
14
16
P, A T M .
C02-He counterdiffusional flux as a function of Alumina pellet:
1
HELIUM CO2
Figure 2. pressure
C02-He counterdiffusional flux as a function of
p = 1.01
Alumina pellet:
T = 0' C. p = 1.01
most simply pictured as due to an actual reduction in pore area for diffusional transport arising from the thickness of the adsorbed COz layer on the pore walls. I t is difficult to make a quantitative assessment of the differing effects of this in micropore and macropore structures, hov ever. Certainly the dimension of an adsorbed molecule of carbon dioxide is not negligible when compared to the average micropore radius (about 18 A,) reported by Rivarola and Smith, but there is reason to believe that the range of relative pressures of CO2 involved in the experimentation corresponds to multilayer coverage where the thickness of the adsorbed multilayer perceptibly decreases area available for transport in the macropores and essentially blocks completely the micropores. In any event, with adsorbed species on the pore Malls one does not know the correct radius or porosity for predicting gas phase transport. Thus, it is our belief that the subtraction
step used by Rivarola and Smith is open to some question and that it may not be possible to calculate the surface transport rates by simple subtraction of diffusion flow (calculated with a model) from a measured total flow.
SIR: Foster, Bliss, and Butt question the subtraction step for evaluating surface diffusion rates because of the unknown change in porosity and pore radius due to adsorption. However, the experimental data in Table I11 of Rivarola and Smith ( 7 ) show that this effect is not significant a t the operating conditions employed. T h e question is whether the adsorbed species reduces the pore radius sufficiently to affect the volume diffusion rate. This may be ascertained by comparing diffusion rates for hydrogen in the HrN2 and Ha-CO2 systems shown for pellet o-b in Table 111. Hydrogen does not absorb, so that its rate is a measure of the volume diffusion. This quantity is noted to be consistently less in the CO2 system than in the nitrogen case and the reduction may well be due to a decrease in effective radius due to adsorption of CO2. Nevertheless, the decrease is but 270, a figure approaching the reproducibility limit for such measurements.
The reason for the insignificance of the effect proposed by Foster, Bliss, and Butt in the results of Rivarola and Smith is probably the low surface coverage of adsorbed COn (10 to 30% of a monomolecular layer). However, a simple method exists for correcting the subtraction method so that it would be applicable even though the surface coverage were large. This is to measure the diffusion rate of a nonadsorbing gas, first in a binary system with the adsorbable gas and second in a binary system with a nonadsorbing gas. Comparison of the two rates indicates whether the effect is important. If it is, the rate of the nonadsorbing gas in the adsorbing system can be used to calculate the volume contribution of the adsorbing gas by the molecular weight relationship. Thus if the rate of the nonadsorbing gas is N A , the volume diffusion rate of the adsorbing component, B , is
580
l&EC FUNDAMENTALS
literature Cited
(1) Butt, J. B., Foster, R. N., Nature 211, 284 (1966). ( 2 ) Foster, R. N., Ph.D. dissertation, Yale University, 1966. ( 3 ) Rivarola, J. B., Smith, J. M., IND.ENC.CHEM.FUNDAMENTALS 3, 308 (1964). (4)LVicke, E., Kallenbach, R., Kolloid Z. 97, 135 (1941).
Richard N . Foster Harding Bliss J o h n B. Butt Yale C'niversity h'ew Haven, Conn.
