CORRELATION OF SOME GAS ADSORPTION DATA EXTENDING T O LOW PRESSURES AND SUPERCRITICAL TEMPERATURES R
. J.
G RA NT A ND M I LT0 N MA N ES
,
Pittsburgh Chemical Go., Pittsburgh 2.5, Pa.
Adsorption isotherms of nitrogen, methane, argon, hydrogen, and neon in the - 195.8' to -78" C. temperature range have been determined on two activated carbons and a silica gel. The resulting data and published data have been correlated by a modified Polanyi-type correlation. No difficulty was found in correlating data at low pressures (to 1 OP3 torr) or a t supercritical temperatures. Moreover, the method correlated data from each of two published articles in which correlation difficulties were reported. The modified Polanyi-type correlation appears to b e widely applicable and is suggested for the preliminary design of adsorption systems in the absence of direct data.
the low-temperature adsorption of permanent gases on common adsorbents has been studied extensively a t pressures do\vn to 1 nim. of I l g (torr), both for investigating the thermodynamics of adsorption and for determining surface areas and structure of a.dsorbents, the literature contains relatively- few references to adsorption of such gases in the lower pressure region corresponding to concentrations of a few parts per million which is the region of greatest interest for gas purification applications. It also raises the question of which theory to apply in this region. I t is well recognized (2, p. 178) that the Polanyi adsorption potential theory is particularly appropriate for physical adsorption a t relatively low surface coverages on energetically heterogeneous surfaces ---i on common adsorbents--and that the basic assumptions underlying the derivations of the equations of Langmuir and of Brunauer, Emmett, and Teller d o not apply. Moreover. none of the assumptions of the Polanyi theory preclude its extension to low pressures, as long as the adsorption level remains hvithin the limits of some experimentally determined characteristic curve. Furthermore, earlier work in this laboratory ( 3 ) ,although well above cryogenic temperatures, had shown that the Polanyi theory [as modified by Lewis, Gill:iland, Chertoiv, and Cadogan (LGCC) (5)] applies to pressure!; as low as 5 X torr, with no evidence of decreasing accuracy with decreasing pressure. I n addition. the available literature on the adsorption of nitrogen on activated carbons a t IO\Y temperature (4. 6) showed high capacities a t low partial pressures and relative insensitivity of capacity to pressure well into the high-vacuum region, which is what one \vould predict from the Polanyi theory. T h e prospect that the modilied adsorption potential theory would apply to the lo\v-temperature, low-pressure adsorption of common gases was therefore encouraging. I n designing a gas purification system it is important to have adsorption isotherms not only in the region of interest for gas separation. but also a t the higher temperatures required for regeneration of the adsorbent. T h e application of the LGCC method to elevated temperatures often requires the estimation of vapor pressures a t temperatures above critical temperature. I n earlier \cork 011 correlating methane isotherms well above critical temperature 13. 5)> methane vapor pressures were extrapolated above critical temperature and the corresponding vapor fugacities \cere used in the L G C C treatment. in which the adqorbate is assumed to have liquid-like properties. Mas-
A
LTHOLGH
lan, Altman, and Aberth (7) used a similar extrapolation method for oxygen and nitrogen, but substituted the results into a n alternative procedure of their own in which the adsorbate is assumed to be a compressed gas, implying that the L G C C method does not successfully correlate oxygen and nitrogen isotherms above critical temperature. I t was therefore of interest to determine how well the L G C C correlation would apply to the adsorption of permanent gases, not only a t low pressures and temperatures. but also a t supercritical temperatures. Experimental
Adsorption isotherms were determined on the McBain balance apparatus previously described ( 3 ) . Liquid nitrogen and dry ice-acetone baths were used to refrigerate the adsorbents. With the exception of methane, all the gases used were obtained from the Matheson Co. "Extra dry" grade nitrogen was purified by passage through copper at 400' C . and a liquid nitrogen trap. Assayed reagent grade argon and neon were used. Electrolytic grade hydrogen was passed through a Deoxo purifier and a liquid nitrogen trap. Phillips Petroleum Co. research grade methane, 99.65% minimum purity, was used without further purification. T h e adsorbents (4 X 10 mesh) were Pittsburgh Chemical Co. grade PCB and grade BPL activated carbons, and Davison Chemical Co. grade 03 silica gel. T h e surface areas, as determined by applications of the BET equation to the - 195.8' C . nitrogen isotherm, were 11 50, 1040, and 71 5 sq. meters per gram for PCB and BPL activated carbons and silica gel, respectively. Samples were dried a t 110' C. prior to loading in the adsorption chambers. Isotherm determination3 were based. in the case of the carbons, on the dry weight at atmospheric temperature and pressure. and. in the case of silica gel, on the dry weight a t room temperature after outgassing to l V 3 torr. T h e adsorbent samples were degassed a t 400' C . for a t least 2 hours dowm to torr prior to adsorption. Lopez-Gonzalez, Carpenter, and Deitz (6) found they were unable to obtain reproducible isotherms of nitrogen on bone char without several prior cycles of adsorption and desorption. Although this phenomenon was not observed in this ivork on activated carbon and silica gel. the procedure of subjecting the samples to repeated adsorption arid desorption \vas followed routinely prior to the determination of all isotherms. Nitrogen, methane, and argon isotherms a t -195.8' and -78' C . , and hydrogen and neon isotherms a t -195.8' C . were determined on both activated carbons (Figures 1 and 2 ) . Kitrogen and methane isotherms a t -195.8' and -78' C . . and argon. hydrogen, and neon isotherms a t -195.8' C . were determined on silica gel (Figure 3 ) . VOL. 3
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Discussion
The isotherms in Figures 1, 2, and 3 were converted to the LGCC correlation. Saturation pressure, ps, was converted by means of standard activity coto saturation fugacity: is. efficient charts. \+'here the adsorption temperature exceeded the critical temperature of the adsorbate, log vapor pressure was extrapolated beyond critical as a linear function of reciprocal absolute temperature (3. 7). Log vapor fugacity could not be extrapolated in this fashion because of nonlinearity, in most cases. below critical temperature. \\.ith the exception of hydrogen and neon on activated carbon. and nitrogen on silica gel. a fair degree of correlation \vas apparent. However: correlation curves drawn from individual isotherms shoived systematic temperature-dependent shifts, the nature of which suggested the use of a constant (pressure-independent) molar volume as the correlating divisor. It was thought best to compare molar volumes at corresponding states: the normal boiling point was selected for convenience. Substitution of the (constant) molar volume a t the boiling point (designated V ) in the abscissa gave the curves sho\vn in Figures 4: 5. and 6. [.4s in the preceding article ( 3 ) .calculation of these curves \vas facilitated by taking capacity values from the individual isotherms a t conveniently chosen intervals of pressure. Therefore, although the correlation is fully representative of the data, the plotted points do not represent individual determinations. ] The use of a pressure-invariant molar volume improved the correlation. The curves of nitrogen, methane, and argon on activated carbon are more or less superimposable ; hoivever. on silica gel. only the methane and argon isotherms coincide, the nitrogen isotherm being high. T h e latter phenomenon has been ascribed ( 4 ) to interaction of the quadrupole moment of nitrogen with polar hydroxyl groups on the surface of silica gel. T h e adqoiption potentials of hydrogen and neon are anomalously lo\\.. Since they represent the liquids of loivest boiling point in the series, the question arises as to whether the correlation on the basis of molar volume should apply to these t k t . 0 gases and, by extension, to helium. O n e could look for another correlating variable, such as ul'* (lvhere a is the van der \\'sals constant). Holyever, this variable has also been found to result in anomalies ( 2 , p. 110). For the present it appears best not to apply the molar volume correlation to gases boiling much below nitrogen. For these gases the scale factor of the abscissa may be determined experimentally lvithout expressing it in terms of a single property. This suggests. of course. that if there is a single property that correlates the isotherms of all gases on activated carbon, it has not yet been found. It \vas next of interest to re-examine the previous correlation of the normal paraffins on BPL activated carbon in order to ascertain the effect of substituting a constant molar volume. The recalculated points. shoivn in Figure 7 : include new ethane isotherm data at the same adsorption temperatures. There is. of course. a displacement of the curve: but no pronounced improvemrnt in the degree of correlation, and, in fact. the methane points appear to be slightly high. However. this ne\v correlation curve is in excellent agreement \vith the corresponding curve in Figure 5. The utility of molar volume as a correlating divisor in the sa i u probably a consequence of a fortuitous proportionality lvith some temperature-independent properties of specific adsorbates; therefore the molar volume used as a correlating factor should likewise be expected to be invariant. hforeover. rhe cunstant molar volume. in addition to being theoreticall>- preferable, facilitates calculation of isotherms 222
l&EC FUNDAMENTALS
from the generalized correlation, since in effect it reduces the potential term for a single adsorbate to RT In f s , f . O n the other hand, the use of a pressure-dependent molar volume in the ordinate (volume adsorbed) would be in keeping with the expected tendency of an adsorbed liquid to shrink in the presence of a n adsorption force field, more or less in analogy rvith shrinking on cooling. I n any event this is usually a small effect; the major advantage of the LGCC approximation of liquid density is that it always keeps the estimated density within a relatively narrow range. 10 I
100;
1000
0.1
I
I
I
e
HYDROGEN
0.0 I
0.001
100
10
100
PRESSURE,Torr
Figure 1 .
Adsorption isotherms on PCB activated carbon
-Calculated from correlation curve - - - Fitted directly to data I
IO
100 1
100
1000
I
I
-+
j I
c. -78OC.
5 OK-
0.01
Figure 2.
1
1
I
I
I
I
IO
0.I
I PRESSURE, Torr
100
Adsorption isotherms on BPL activated carbon
-- Calculated from correlation curve __-
Fitted directly to data
IOC -I
W
0
a
-u =! v)
IC
0 d
f n m = I
-Y
,I
/
1
HYDROGEN NEON
8 (D
0, a
I k
P 0. 0.
Figure 3.
Adsorption isotherms on silica gel
-- Calculated from correlation curve
- - - Fitted directly to data
100
I 0 NITROGEN A METHANE 0 ARGON e HYDROGEN 0 NEON
0
0
P \
7
-
._
IO
m.
E
0
d W a 0 w n
a W
I
3
> 0
I
1 0
F
>
0 0.1 0
Figure 4. Generalized adsorption correlation on PCB activated carbon
Figure 5. Generalized adsorption correlation on BPL activated carbon
-
10 (T/V)
20
log,, ( f s / f )
30
Figure 6. Generalized adsorption correlation on silica gel IOC
0
172gK
0 2036'K 0 5502'K 0 68 39'K
d
0 0
0
-
P
-._d 10
2 IC
10.
yo.
\
\
--
._ 1
E
0
n
n
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m
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m
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v) 0
v) 0
n
n
a W
W
3 z
I
J
-1 3
0
>
> 0
01'
0
10 lT/V)
20 log,, ( f s / f )
Figure 7. Generalized adsorption correlation of normal paraffins on BPL carbon
A 7969'K. D 8998'K.
D
0.1 0
10 (T/V)
20
30
Figure 8. Generalized adsorption correlation of oxygen and nitrogen on activated carbon (Altman)
We now consider the significance of critical temperature. In correlating the isotherms of nitrogen, methane, and argon, there is no evidence of any difficulty in making the transition through critical temperature. Thus, the isotherms correlate equally well a t -78' C. where the respective reduced temperatures are 1.54, 1.02, and 1.29, and a t -195.8' C. which is well below critical temperature for all three gases. However. some investigators have been reluctant to apply a correlation based on the assumption of a liquid-like adsorbate to data above critical temperature. For example, Maslan et al. (7) chose to regard the adsorbed phase as a highly compressed gas, and with this assumption were able to correlate isotherms of oxygen and nitrogen on activated carbon in the temperature
0.1
log,,(fs/f)
10 lT/V)
20 log,, l f s / f )
30
Figure 9. Generalized adsorption correlation of hydrogen on Carbotox activated carbon (8)
range from 0" to -150" C. However, the method failed to correlate these isotherms between - 150" and - 180' C. and also propane and propylene isotherms between 0" and -30" C.. whereas at least the latter group were successfully correlated by the LGCC method. Moreover? calculations based on the assumption of a gaslike adsorbed phase overestimated, by an order of magnitude, the occupied pore volume of the activated carbon. Furthermore, as shown by Figure 8, all of the oxygen and nitrogen isotherms of Altman ( 7 ) can be correlated by the modified LGCC correlation. The resulting correlation. like that of Maslan et al., makes no distinction between data above and below critical temperature. However, it also goes VOL. 3
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smoothly to the boiling point, and, in addition, leads to a realistic estimate of the occupied pore volume. It does, however, lead to separate curves for oxygen and nitrogen. Of further interest in this regard are the data of van Dingenen and van Itterbeek (8) on the adsorption of hydrogen on Carbotox activated carbon over the temperature range 17' to 90' K. They correlated their low-temperature data by the Polanyi theory available a t that time but reported they could not correlate their high-temperature data, whereas Maslan et ai.,as we have seen. correlated their high-temperature data but could not correlate their low-temperature data. Figure 9 shows the successful correlation of all the isotherms of van Dingenen and van Itterbeek, using the modified LGCC method. Although there is no overlap of data above and below critical temperature, the correlation curve shows a smooth transition between the two regions and again a realistic estimate of the occupied pore volume. This hydrogen correlation resembles the corresponding hydrogen curve on BPL carbon (Figure 5)? suggesting a similarity between the two carbons. The extrapolation above critical temperature, therefore, appears to work for hydrogen as well as for oxygen, nitrogen, methane, and argon, and apparently has general applicability. There appear to be good reasons for supposing that a liquid-like state may persist on a n adsorbent a t temperatures far above the critical temperature of the free liquid-i.e., for supposing that the critical temperature of an adsorbed liquid is to be sharply distinguished from that of a free liquid. T h e attractive force of the adsorbent for the molecules may be considered as reinforcing the attraction of these molecules for each other, so that the over-all effect is in the direction of increasing the magnitude of the van der Waals constant. a, a t constant b . Lvhich immediately raises the critical temperature.
An increased value of a also can account for liquid-like properties for the adsorbate. If we assume a liquid that retains its liquid-like properties-i.e., enthalpy, as well as density-as we raise the temperature beyond the (normal) critical temperature. we may (by the Clausius-Clapeyron equation) similarly justify the extrapolation of its vapor pressure. Under these considerations. any expectation of a discontinuity in adsorption behavior as one passes through the (normal) critical temperature is based on neglect of the influence of the force field on the properties of the liquid phase. 'These effects were implicitly recognized in the LGCC assumption that one of the properties of the liquid phase-Le., its densityshould be evaluated a t some temperature below the adsorption temperature. Acknowledgment
The experimental assistance of J. R. Krause is gratefully acknowledged. literature Cited (1) Altman, M., D.Sc. thesis, New York University, New York;
1952. ( 2 ) Brunauer, S., "Adsorption of Gases and Vapors," Princeton University Press, Princeton, N. J., 1943.
(3) Grant, R. J.. Manes, M., Smith, S. B., A.I.Ch.E.J. 8, 403 (1962). (4) Kaganer, M. G., Dokl. Akad. .VQU~ SSSR 138, 405 (1961). (5) Lewis. \V. K., Gilliland. E. R., Chertow, B., Cadogan, LV. P., Ind. Eng. Chem. 42, 1326 (1950). (6) Lopez-Gonzalez, .J. de D., Carpenter, F. G., Deitz, V. R., J . Res. .Vat/. Bur. Std. 5 5 , 11 (1955). (7) Maslan, F. D.; rlltman, M., Aberth, E. R., J . Phys. Chem. 57, 106 (1953). (8) van Dingenen, \V., van Itterbeek, A , , Physic0 6, 49 (1939). RECEIVED for review October 7, 1963 , ~ C C E P TMarch ~ 27, 1964
MATRIX CALCULATION OF MULTICOMPONENT MASS TRANSFER IN ISOTHERMAL SYSTEMS WARREN E. STEWART A N D R I C H A R D P R O B E R 1 Department of Chemical Engineering, 1 %.iversity of Wisconsin, Madison, Wis.
Matrix methods are developed for solving multicomponent diffusion problems in terms of related binary problems. The diffusion coefficients of the binary problems are determined b y the eigenvalues of a multicomponent diffusivity matrix, and are shown to be real and positive. When the diffusivity matrix i s known, multicomponent concentration profiles and mass transfer rates can be calculated for systems of any geometry for which the corresponding binary functions are available. The treatment holds for nonideal mixtures in laminar or turbulent isothermal flow with moderate concentration differences and no homogeneous chemical reactions. The theory i s tested b y comparison with exact boundary-layer calculations for mass transfer between a flat plate and a three-component gas stream.
M
diffusion problems are important in many chemical and physical processes. Exact treatment of such problems is seldom feasible? since the governing differential equations are complicated and nonlinear. For small concentration differences. however. a linearized treatment of the differential equations leads to simple and accurate ULTICOMPONENT
1
Present address. Shell Dewlopment Co., Emeryville. Calif.
224
I&EC FUNDAMENTALS
results. The present paper develops and tests this treatment for isothermal systems. Tlvo concepts of linearization are applied jointly in this work. First. we use Onsager's postulate (73! 7 4 that in ordinary diffusion the chemical potential gradients and diffu. sion fluxes are linearly related. Second, we write the differential equations in their limiting forms for nearly constant physical properties. The use of these concepts to analyze multi-
