Acknowledgment We thank W. R. Chappell, T. Wildeman, W. A. Berg, H. P. Harbert, F. Haas, and B. Miller for assistance in collecting the samples, and R. Meglen, C. Crouch, and R. Bernthall for performing much of the analytical work. We also appreciate the helpful comments by the reviewers of this manuscript. Literature Cited (1) Weeks, J. B.; Leavesley, G. H.; Welder, F. A.; Saulnier, B. H., Jr. U.S. Geol. Suru., Prof. Pap. 1974, No. 908,2-5. (2) Pfeffer, F. M. June 1974, EPA Report 660/2-74-067. (3) Rio Blanco Oil Shale Project, Revised Detailed Development Plan, Tract C-a, 1977. (4) Baes, C. F., Jr.; Mesmer, R. E. “The Hydrolysis of Cations”; Wiley-Interscience: New York, 1976. (5) ~, Ward. J. C.: Mareheim. G. A.: Lof. G. 0. G. Water Pollut. Control Res. Sek. 1971, Ript. 14030 EOB. (6) Stollenwerk, K. G.; Runnells, D. D. Proc. 2nd Annu. Pac. Chem. Eng. Congr. 1977,1023. (7) TRW/Denver Research Institute. May 1977, EPA Report 908/ 4-78-003. (8) Fruchter, J. S.; Wilkerson, C. L.; Evans, J. C.; Sanders, R. W. Enuiron. Sci. Technol. 1980,14,1374. (9) Runnells, D. D.; Glaze, M. L.; Saether, 0. M.; Stollenwerk, K. G. “Release, Transport, and Fate of Some Potential Pollutants in Waters Associated with Oil Shale”, in “Trace Elements in Oil Shale”; Progress Report; June 1,1976-May I, 1979; Environmental Trace Substances Research Program, University of Colorado, Colorado State University, and Colorado School of Mines, Feb 1980. (10) Audus, L. J.; Quastel, J. H. Nature (London) 1947,60,263. 111) Bell. A. V. Enuiron. Sci. Technol. 1976,10,130. (12) Wildeman, T. R. 1980, EPA Report 600/7-80-125. (13) Harbert. H. P.: Berg. W. A.: McWhorter, D. B. 1979, EPA Report 600/7-79-188. (14) Wildeman, T. R.; Heistand, R. N. Prepr. Pap-Am. Chem. SOC., Diu. Fuel Chem. 1979,24,271. (15) Bates, E. R.; Thoem, T. L. “Pollution Control Guidance for Oil Shale Development”, Revised Draft Report; U.S. Environmental Protection Agency, 1979; pp 3-70. (16) Runnells, D. D.; Stollenwerk, K. G. “Release, Transport, and Fate of Some Potential Pollutants”, in “Trace Elements in Oil Shale”; Progress Report; June 1, 1975-May 31, 1976; Environmental Trace Substances Research Program, University of Colorado, Colorado State University, and Colorado School of Mines, Feb 1977. (17) Runnells, D. D.; Lindberg, R. D. J. Geochem. Explor., in press. (18) Brunauer, S.;Emmett, P. H.; Teller, E. J Am. Chem. SOC.1938, 60,309. (19) Stollenwerk, K. G. Ph.D. Thesis, University of Colorado, Boulder, CO, 1980. (20) Meglen, R. B.; Krikos, A. June 1980, EPA Report 600/9-80022. \--,
-
-
(21) John, M. K.; Chuah, H. H.; Newfeld, J. H. Anal. Lett. 1975,8, 559. (22) Barasko, J. J. Econ. Geol. 1967,64,732. (23) Alfrey, A. C.; Nunnelley, L. L.; Rudolph, H.; Smythe, W. R. Adu. X-Ray Anal. 1976,19,497. (24) Rasnick, B. A.; Nakayama, F. S. Commun. Soil Sci. Plant Anal. 1973,4,171. (25) Nor, U. M.; Tabatabai, M. A. Soil Sci. 1976,122, 171. (26) Kelly, D. P.; Chambers, L. A.; Trudinger, P. A. Anal. Chem. 1969,41,898. (27) American Public Health Association. “Standard Methods for the Examination of Water and Wastewater”, 13th ed.; New York, 1971. (28) Shih, C. C.; Cotter, J. E.; Prien, C. H.; Nevens, T. D. 1979, EPA Report 66/7-79-075. (29) Glass, J. J. Am. Mineral. 1947,32,201. (30) Ertl, T. Am. Mineral. 1947,32,117. (31) Tom Kuo, M. C.; Park, W. C.; Lindemanis, A.; Lumpkin, R. E.; Compton, L. E. Oil Shale Symp. Proc. 1979,12,81. (32) U. S. Environmental Protection Agency. “National Interim Primary Drinking Water Regulations”; Fed. Regist. 1975, 40, 59566. (33) Saether, 0. M.; Runnells, D. D.; Ristinen, R. A.; Smythe, W. R. Chem. Geol., 1981,31,169. (34) Truesdell, A. H.; Jones, B. F. J . Res. U.S. Geol Suru. 1974,2, 233. (35) Plummer, N. L.; Jones, B. F.;Truesdell, A. H. U.S. Geol. Suru. Water-Resour. Znuest. 1976, No. 76-13, (36) Desboroueh. G. A.: Pitman, J. K.: Huffman C., Jr. Chem. Geol. 1976,17,13.(37) Fleet, M. E. L. Clay Miner. 1965,6,3. (38) Levinson, A. A.; Ludwick, J. C. Geochim.Cosmochim.Acta 1966, 30,855. (39) Lerman, A. Sedimentology 1966,6,267. (40) Couch, E. L. Am. Assoc. Pet. Ceol. Bull. 1971,55,1829. (41) Hendrickson, J. A., Ed. “Synthetic Fuels Data Handbook”; Cameron Engineers, Inc.: Denver, CO, 1975; p 41. (42) Stuber, H. A.; Leenheer, J. A.; Farrier, D. S. J . Enuiron. Sci. Health 1978,13,663. (43) Taylor, 0. J., submitted for publication in U.S. Geol. Suru. Open-File Rep. (44) Colony Development Operation. “Environmental Impact Assessment, Proposed Development of Oil Shale Resources”, 1975. (45) Metcalf and Eddy Engineering, Inc. “Water Pollution Potential from Surface Disposal of Processed Oil Shale from the TOSCO I1 Process”; 1975; Vol. 1. (46) Stephens, D. B.; Siegel, J. Spec. Publ.-N.M. Geol. SOC.1981, No. 10,103. (47) Runnells, D. D.; Esmaili, E. “Release, Transport, and Fate of Some Potential Pollutants”, in “Trace Elements in Oil Shale”; Progress Report; June 1,1980-May 31,1981; Center for Environmental Sciences, University of Colorado, Denver, CO. ~
Receiued for reuiew May 8,1980. Reuised manuscript received April 21,1981. Accepted August 6,1981. This work was supported by the US.Department of Energy, Contract No. EY-76-A-02-4017,through the Center for Environmental Sciences of the Uniuersity of Colorado, directed by Dr. Willard R. Chappell.
Calculation of Adsorption Breakthrough Curves in Air Cleaning and Sampling Devices Otto Grubner* and William A. Burgess Department of Environmental Health Sciences, Harvard School of Public Health, 665 Huntington Avenue, Boston, Massachusetts 021 15
Dynamic adsorption is a special field of physical chemistry that finds application in the evaluation and prediction of gas removal by systems including industrial adsorbers to respiratory cartridges and air sampling devices. Engineering approaches have been published on this subject as long ago as five decades. The fundamentals of dynamic adsorption were proposed during and after World War 11, when the analogy between dynamic adsorption and vapor-solid chromatography was fully recognized and chromatographic theories were developed. Wilson ( 1 ) and DeVault (2) in the United States, 1346
Environmental Science & Technology
Martin and Synge ( 3 ) and Glueckauf (4, 5 ) in England, Damkohler (6) and Wicke (7) in Germany, and Zhukowitskii (8)and Turkeltaub (9) in the USSR must be credited for the present understanding of the principles of dynamic adsorption. In the past few years several papers by Nelson et al. (10-14) have appeared on respirator cartridge efficiency studies. These authors gathered a considerable amount of experimental data studying breakthrough curves of many organic vapors through respirator cartridges filled with activated charcoal. While the
0013-936X/81/0915-1346$01.25/0
@ 1981 American Chemical Society
~~
~
Breakthrough curves of several organic vapors on charcoal beds of protective and sampling devices are calculated and interpreted by the simplified theory of statistical moments. Static adsorption isotherms for diethylamine, methylchloroform, carbon tetrachloride, benzene, hexane, acetone, and
methyl acetate are derived from dynamic experiments. Adsorption data can be correlated by a generalized adsorption isotherm. The breakthrough times of different concentrations of organic vapors through charcoal at various experimental conditions can be predicted without experimentation.
experimental procedures were sound, the author’s interpretation of the experimental data must be regarded with considerable caution. In this paper we intend to show that the experiments can be better interpreted by the basic theories of dynamic adsorption and by the theory of statistical moments (TSM) (15,16)and that the breakthrough curves on charcoal can be predicted on the basis of Lewis’s general isotherm ( I 7).
sents the amount adsorbed and in equilibrium with concentration Ci. If the weight of the adsorbent in the bed is denoted by G and the density of the gas by p, then the amount of sorbate per unit weight of adsorbent, ag,is as follows:
Equilibrium Factors and Adsorption Isotherms The adsorption isotherm represents the dependence of the amount of sorbate adsorbed by a known quantity of solid adsorbent on the equilibrium concentration of sorbate in the gas or liquid phase in contact with the adsorbent. At equilibrium, equal amounts of sorbate are simultaneously being adsorbed and desorbed. Since the rates of the adsorption and desorption processes are the same, no concentration change on the solid or in the gaseous phase can be observed. The adsorbed quantities are usually different with different equilibrium concentrations. In dealing with the adsorption of organic vapors on activated charcoal, it suffices to say that adsorption isotherms are either convex or concave to the axis of equilibrium concentration. The special case between these two extremes is a linear adsorption isotherm. In static adsorption, the pairs of equilibrium values are directly measured point by point. However, in dynamic adsorption experimental results are given by an S-shaped curve of concentrations existing from the bed with time. It must be pointed out that dynamic adsorption experiments may be performed in both the adsorption mode and the desorption mode. In the adsorption mode a mixture of adsorbate and air is introduced into a clean adsorption bed filled with inert gas and the increase of adsorbate concentration at the output is followed with time until input and output concentrations are equal. In the desorption mode clean air is introduced into a bed that is saturated with a mixture of adsorbate with air and already in equilibrium (input and output concentrationslarger than zero and equal), and the decrease of concentration of adsorbate with time at the output is followed until again input and output concentrations are equal to zero. In the first case one follows the front boundary of the zone and the adsorption wave; in the second case one follows the rear boundary of the zone and the desorption wave. There is a point on the S-shaped breakthrough curve characterizingthe static adsorption isotherm with equilibrium concentration equal to that of the input concentration of the breakthrough curve. This point is very close to the center of the S-shaped breakthrough curve where the concentration Cm is equal to 50% of the input concentration, Ci, at a breakthrough time, t50. According to the theory of statistical moments (16), neither concentration Cw nor time t50 is a true equilibrium point. The true equilibrium points are those represented by the coordinates of the mean of the curve. For practical purposes, however, c50 and t50 can be used as equilibrium points. Now, if t m is the time in which concentration Ci would have broken through had there been no kinetic effects, V50 = wt5o represents the volume that would have passed through the bed. The amount of adsorbate in this volume is a, = CiV50 neglecting the void volume of the bed. This entire amount would have been adsorbed by the adsorbent and thus repre-
ag = wpt60CiG
From eq 1it is evident that, once the adsorbed amount ag is known, breakthrough time t50 can be calculated. Thus, if ag is given as weight/weight in the same units as G, w is in L/min, and Ci is the volumetric fraction (dimensionless),then the breakthrough time t50 in minutes of sorbate of molecular weight M (replacing p by M/24.1) can be calculated by t50 = 24.1GaJwMCi
(2)
Nelson et al. (13)measured static adsorption isotherms of several sorbates on charcoal. They also presented breakthrough curves for different concentrations of benzene and methyl acetate on an adsorption bed filled with 28.3 g of charcoal using a flow rate of 26.6 L/min. Experimental breakthrough times t50 calculated by eq 2 are compared in Table I. The agreement is good considering the errors in static and dynamic adsorption measurement. Therefore, eq 2 is supported by experimental evidence. Nelson et al. (12) also measured adsorption isotherms for several other compounds for which breakthrough curves have been determined at a concentration of 1000 ppm. The experimental and calculated values are compared for a concentration of 1000 ppm in Table 11. The knowledge of breakthrough time t 50 can be effectively used to construct static adsorption isotherms which have not been determined experimentally under static conditions. Thus, for example, an adsorption isotherm of vinyl chloride can be recalculated from dynamic data as presented in Table 111. In many cases the dynamic technique may be experimentally simpler than classic volumetric or gravimetric adsorption measurements.
Slope of a n Adsorption Isotherm The slope of an adsorption isotherm is an important parameter in dynamic adsorption since it determines the rate of movement of a zone of a given concentration through the bed according to eq 3. Table I. Comparison of Experimental and Calculated Breakthrough Times for Benzene and Methyl Acetate at Different Input Concentrations breakthrough tlmes, rei min deviatlon, exptl caicd %
concn, ppm
amount adsorbed, g/g
benzene
125 250 500 1000
0.170 0.210 0.263 0.301
430 265 175 92
446 275 172 95
+3.6 4-3.6 -1.7 4-3.0
methylacetate
100 250 500 1000 2000
0.064 0.095 0.127 0.185 0.242
212 105 86 61 41
214 118 88 62 42
+1.0 +12.9 +1.9 +2.1 +2.0
adsorbate
Volume 15,Number 1 1 , November 1981
1347
Table II. Comparison of Experimental and Calculated Breakthrough Times of Several Adsorbates
diethylamine hexane methyl chloroform dichlormethane methyl acetate benzene
0.257 0.291 0.570
122 90 95
103
+18.8
89 109
+1.4 -13.4
0.120 0.185 0.301
30 61 92
36
-20.0
64 95
-4.5 -3.0
%propanol
0.303
0.303
134 109
129 107
+3.8
carbon tetrachloride
+1.9 ~
Table 111. Adsorption Isotherm of Vinyl Chloride on Charcoal Recalculated from Dynamic Experiments ( 12) concn. ppm 50 100 250 500 1000 2000 adsorbed amounts 0.012 0.019 0.039 0.060 0.086 0.082
If the adsorption isotherm is linear, then f'(c) = daldc is a constant, K, over the entire range of concentration and eq 3 becomes t , = to(1
+K )
(4)
a very well-known relationship in gas chromatography. In general, K is much larger than 1and eq 5 can he used. t, = t&
(5)
Equations 3-5 were derived under the assumption that adsorption equilibrium is established instantaneously, i.e., that no kinetic factors play a role. In dynamic adsorption the equilibrium factors are usually dominant and the influence of the slope of an adsorption isotherm, dnldc, is always seen. In the case of the linear adsorption isotherm, eq 4 predicts that all concentration points should break through at the same time as a sharp front. This actually occurs in cases where kinetic effects are negligible, as shown in Figure 1. Let us consider some of the implications of eq 3 for the case of a convex adsorption isothetm and examine its influence on the front boundary in dynamic adsorption. In the region of low concentrations, the slope of a convex isotherm is larger than in the region of high concentrations. Therefore, according to eq 3, zones of low concentrations should move more slowly than those of higher concentrations and a hypothetical front boundary (Figure 1,dashed line) should result. This outcome is, of course, physically i m r n i h l e since higher concentrations cannot exist without their lower proportions. The tendencies cancel each other, resulting in a sharp front boundary. It is important to know that in this case the shape of the front boundary is not determined by the slope of an adsorption isotherm and no judgement on the adsorbed amount and its change with concentration can he made. On the other hand, if a dynamic experiment with a compound having a convex isotherm were performed in the desorption mode, a well-developed rear boundary would travel with a speed directly proportional to the slope of the adsorption isotherm and the adsorbed amount for each concentration point could be calculated by Glueckaufs ( 4 ) equation: n, = [ w ( t , -to)
+ pc1h
1348 Environmental Science (L Technology
(6)
Flgure 1. Influence 01 the shape of the adscfption isotherm on the boundaries of breakthrough curves.
The breakthrough curves for a convex isotherm in both the adsorption and desorption modes are shown in Figure 1. Let us now consider the case of a concave adsorption isotherm. Since the slope in regions of low concentration is small, eq 3 predicts that zones in this region in a dynamic experiment would move faster through the bed than zones of higher concentrations where the slope of the isotherm is large. Therefore, in a dynamic experiment performed in an adsorption mode, the front boundary will be developed as shown in Figure 1,and the adsorption isotherm can be calculated from the breakthrough cuwe by using Glueckaufs equation in which the term is negative. In this case the front boundary can be interpreted, whereas the rear boundary will be sharp for reasons explained. All of the compounds used by Nelson et al. (12) in the breakthrough experiments exhibit convex adsorption isotherms and, therefore, the front boundary should not he interpreted in terms of equilibrium adsorption values.
*,
Kinetic Effects in Dynamic Adsorption Up to this point we have considered the case in which adsorption equilibria are established instantaneously or in which the time needed for the equilibrium establishment is negligible compared to the breakthrough time. This situation, however, does not always hold, and kinetic effectsmust be considered. For the explanation of kinetic effects, the theory of statistical moments (26)will be used. In comparison with other kinetic theories, the TSM is simple, straightforward, and directly related to experimental conditions. It should he stressed that the TSM (as well as other theories) was developed under the assumption that the adsorption isotherm is linear. The isotherms of most compounds on charcoal are not linear. However, in the case of charcoal, we have shown (28)that TSM relationships seem to hold reasonably well even for nonlinear isotherms. Also, in the development of this presentation, many simplifying steps were introduced which may limit application of this theory. Nevertheless, modelsamenahle tomathematical description hy simplification of physical reality must he made to assure progreas of a given disciplineof science. Models must be extensively tested hy practice to find the best possible explanation of a given phenomenon. The establishment of adsorption equilibrium in physical adsorption of any nonporous surface is basically an instantaneous process. However, in the case of charcoal, where every grain contains a maze of an almost infinite number of tiny
pores, the rate of adsorption is controlled by the rate of diffusion of sorbate into the pores. Thus, diffusion is the ratecontrolling step in dynamic adsorption on charcoal. Diffusion, as a random process, is often described in terms of the calculus of probability. As the zone of sorbate moves through the bed, it is affected by chance events, and the resulting breakthrough curve assumes the form of a normal probability distribution of concentration with time. According to the theory of statistical moments, the breakthrough time of a given concentration tc can be described by a simplified relationship: tc = t50(1 + U r x c )
(7)
Table IV. Experimental Values of Standard Deviation (a,)from Breakthrough Curves of Different Adsorbates compd
benzene
u/t50 = k4 = (2/15)l/’4
0.3654
(8)
+ e)D&
av ot d t g o f SD
2000
1 1 1 1 1
0.18 0.20 0.16 0.18 0.15
0.172 f 0.02
1
0.24
1 1
0.20 0.23 0.23 0.25 0.230 f 0.02
125
methyl acetate
acetone
2000 1000 500
250
1
100
1
2000
3 3 3
1000
500 250
)1/2
Fore41
dtso
500 250
where k is a constant pertaining to the shape of the particle of the adsorbent and 4 is given by = R(2(l
carlrldge no.
1000
where ur is the relative standard deviation of the breakthrough curve for a given ratio of the output and input concentrations, C,/Ci, of the breakthrough curve. Simplified treatment of the TSM defines u, as 6, =
concn, ppm
vinyl chloride
3
2000
2 2 2
1000
500
This expression can be transformed to
Thus, it can be seen that the dispersion of the zone by kinetic effects denoted by a, does not depend on the equilibrium constant but on experimental parameters such as particle size and shape, bed geometry, porosity, flow rate, and intrinsic diffusion coefficient D . Therefore, it is possible to minimize the dispersion of the zone by selecting proper experimental conditions and thus to optimize the performance of the bed. We calculated the u values for the breakthrough curve of benzene on charcoal in cartridge I of Nelson and Harder (12, 13) using the following parameters: k = 0.365, w = 440 mL/s, V = 70 mL, ee = 0.42,Di = 0.0932 cm21s,and R = 0.088 cm. As there is a distribution of particle sizes in the bed, the value of the average particle diameter was calculated from the ratio of the geometric surface area to the volume of the particles. The calculated u, is 0.21. If experimental breakthrough curves are available, this experimental a, value can be obtained by measuring the difference between t50 and t16 and expressing it relative to t50. Some experimental values of u, obtained by this procedure from well-developed breakthrough curves of different substrates measured at different concentrations by Nelson are summarized in Table IV. Several important observations can be made from inspecting Table IV. The experimental value of the relative standard deviation for benzene of 17% agrees quite well with the calculated value of u, = 21%. Differences between values of u, for benzene, acetone, and vinyl chloride do not seem to be significant in spite of the difference in cartridges. In a previous paper (18)we suggested the use of two additional concepts to characterize an adsorption bed, namely, the measure of performance, MP, and the lowest retention limit, LRL. When one applies these concepts to experiments tabulated in Table IV, the measure of performance, defined as tso/u, is almost the same for all of the cartridges and has a value of -5.24. The lowest retention limit is less than 3 X 10-7 of the input concentration, indicating that, if an input con-
3
100
250 100
0.22 0.17 0.15 0.16 0.18
0.176 f 0.03
2
0.12 0.19 0.21 0.20
2
0.23 0.188 f 0.04
centration of 1000 ppm were used, 0.0002 ppm would break through with the inert carrier gas. Additional Considerations Additional information can be extracted from the extensive series of experiments conducted by Nelson et al. It is usually a good practice to plot adsorption isotherms in coordinates of adsorbed amount in millimoles of adsorbate per gram of adsorbent, and gas concentration as relative pressure, plp,. This relationship is shown in Figure 2 for the static data on the compounds included in Table IV on Nelson’s paper (13).One notes that the adsorption isotherms of many different compounds do not differ dramatically from each other. All of the isotherms are nonlinear. General Adsorption Isotherm Many attempts have been made to develop a model and a corresponding mathematical description that would encompass the adsorption behavior of almost all sorbates on a given adsorbent at various temperatures. In view of the fact that the adsorbed state is more complex than the liquid state and that, until now, no general theory of liquid phase has been available, one should be skeptical that a general theory of the adsorbed state will soon be developed. Nevertheless, some relatively successful correlations have been found, particularly for charcoal. For example, Lewis et al. showed that, if the adsorbed volume as a liquid is plotted as a function of the logarithin of the relative fugacity of adsorbate multiplied by absolute temperature and divided by the molar volume, a curve can be obtained that accommodates the behavior of several adsorbates on charcoal in a wide range of temperature. This relationship proposed by Lewis, called the general adsorption isotherm, could be very useful in dynamic adsorption on charcoal. It would allow us to predict the breakthrough time t without experimenting. In combination with the TSM, the shape of the entire breakthrough curve could be predicted. Volume 15, Number 11, November 1981 1349
I
,
I
\
I
I
I
I
I
I
I
I
DIETHYLAMINE
6-
\
METHYLCHLOROFORM T E T R A C H L O R I D T-BENZENE
1
5
;1$
I
4
3
2 METHYLACETATE
I
5
Figure 2.
10
15
Adsorption isotherms of various vapors on charcoal at 20
OC.
In reviewing the availability of a general adsorption isotherm, we used Lewis’s approach and plotted the dependence of the adsorbed “volume of liquid per gram of adsorbent” on the term ( p ~ / k f In ) (p,/p) for most of the adsorbates shown in Figure 2. We obtained a curve shown in Figure 3. With the exception of diethylamine, all experimental values seem to be close to this curve in spite of the fact that pressures instead of fugacities were used. Therefore, we tried to find a reasonable analytical expression for this curve using polynomial regression and deduced the following equation:
+
y = 0.749 - 8.307~ 14.826~2
(10)
where y = a,/G = a,/pL and x = ( ~ L / MIn) (p,/p). Since the dimensions of y are mL/g and those of x: are mol/mL, the numerical constants in eq 10 must have dimensions of mL/g, mLVg mol), and mL3/(g mo12). Equation 10 fits the experimental data with a correlation coefficient of 0.98 and a standard error of the estimate of 0.015. When the general adsorption isotherm is combined with the TSM (eq 7-9), the breakthrough time t, of any ratio of C,/Ci can be calculated by eq 11. t, = [(24.1 X 1O9)Gp~y)]/(60MCiw)
(11) where B = 1+ 0.365X,(R/2)(w/V~Si)lf~ and y is defined by eq 10, X, being the argument of the normal probability distribution curve for the ratio C,/Ci, and x = (PL/M)In (p,/p). Time t is in minutes. Constants in eq 11are chosen for C in ppm, G in grams, R in centimeters, V in milliliters, D in cm2/s, w in mL/s, and y is mL/g. M is in g/mol, and other symbols are dimensionless. (See the Appendix for an example of the application of eq 11.) It should be emphasized that our procedure will not cover all possible cases of dynamic adsorption on charcoal. The TSM used in this paper is a considerably simplified version. For more accurate prediction, original papers (15,16)should be consulted. Further, it should be realized that a theory is never better than the model which it describes. Procedures suggested here will be limited by flow rates and the geometry of the bed. Calculations will fail if the linear velocity of the inert gas is outside the range of 10-60 cm/s or if the bed is too short. As a rule, the time needed for diffusioh to travel the distance of a particle radius R should be less than 40% of that given by the ratio w/V. Some authors (19) used the TSM and reported deviation ‘ of calculatedand experimental data. Whether these deviations originated in inadequacy of the theory, improper experimental conditions, or nonlinearity of the adsorption isotherm is difficult to say. At any rate, however, most of the deviations found were of the same magnitude as those reported in this 1350
Environmental Science & Technology
I .03
I .04
I .05
1 .06
I
.07
I
1 08
.09
L t”psP M Figure 3.
General adsorption isotherm on charcoal.
paper, i.e., below 20% of the breakthrough time and mostly in the direction of increasing the adsorption capacity of the bed, meaning that the adsorption bed designed by the TSM would in reality work better than expected by the theory. In our opinion a possible error of 20% is quite acceptable for most practical applications, as is often the case in mechanics. Such an error would be found in simpler systems as, for example, if one attempted to predict the flow of water through an open pipe by using the Poisseuille equation and compared theoretical values with experimental data. Until a better description is found, it has little meaning to argue about differences of this magnitude. Conclusion We believe that the procedure described in this paper presents a solid base for the designer of adsorption samplers or respiratory protective devices. The design techniques may also help the user properly to apply these devices in the field. The extensive experiments performed by Nelson et al. can be reasonably interpreted by the basic concepts of dynamic adsorption and by the theory of statistical moments. Acknowledgment We thank Dr. Avram Gold for valuable comments. Appendix Examples of Breakthrough-Time Calculations. (a) Adsorption Isotherm Is Known. The problem is to calculate the breakthrough time of lo%, i.e., t10, of benzene vapor
through the charcoal bed if the concentration is 250 ppm and other experimental conditions are as follows: weight of charcoal, G = 28.3 g; flow rate, w = 444 mL/s or 26.6 L/min; adsorbed amount of benzene in equilibrium with 1 g of charcoal at 250 ppm, ag = 0.210;molecular weight of benzene, M = 78 g/mol; volume of bed, V = 70 mL; radius of charcoal granule, R = 0.088 cm; external porosity, ce = 0.42; diffusion coefficient of benzene in air Dj = 0.0932 cm2/s;shape constant, k = 0.365. Using eq 2, we first calculate t50: t50 =
24.1Gag -
(24.1)(28.3)(0.210) = 276 min
w M C ~ (26.6)(78)(250X lod6) Next, using eq 9a, we calculate ur:
= 0.204
(0’365)(0*044) Finally, t, will be calculated by eq 7. For this purpose we have to look up X, = Xp(c0/c,)in probability tables. The argument for P(O.l) is not given in the tables directly. The following is valid: Xo.1 = -X(1-0.1) =
mX(0.g)=
-1.28
+
Thus, since t , = t50(1 u,X,), tlo = 275.6[1 - (0.204)(1.28)] = 204 min. The value experimentaly found was 205 min. (b) Adsorption Isotherm Is Not Known. A general isotherm and eq 11will be used. Experimental conditions are as before. For use of eq 11, x must be known. Since x = (pJM) In ( p a l p ) the , pressure p corresponding to 250 ppm must be calculated. Assuming that atmospheric pressure is 760 mmHg, p = (760)(250 X = 0.19 mmHg. The saturated vapor pressure of benzene at room temperature is p s = 72.4 mmHg. Therefore, p$p = 381.0, and x = (0.88/78)(5.94) = 0.067. Then
+
+
y =: 0.794 - 8 . 3 1 ~ 1 4 . 9 ~=~ 0.749 - 8.307(0.067) 14.9(0.00589) = 0.258 mL/g. As B = 0.739, t, can be calculated by eq 11: t, =
(24.1 x 1 0 9 ) ~ p f i y-
6OMCiw
(24.1 X 109)(28.3)(0.88) (0.739)(0.258) = 220 min (60) (78)(250) (444)
which is in good agreement with the experiment and also with the previous calculation. Nomenclature ag = adsorbed amount per gram of adsorbent, wt/wt a,,, = adsorbed amount in general
B = kinetic factor Cj = input concentration usually as volumetric fraction C, = output concentration Di = coefficient of internal diffusion, (length)2/time f’(c) = da/dc = slope of the adsorption isotherm G = weight of adsorbent k = shape factor K = constant L = lengthofbed M = molecular weight
n
= number of moles pressure = saturated pressure = radius of adsorbent particle = breakthrough time of concentration c = time needed to pass void volume of adsorption bed = breakthrough time of 50%of the input concentration
P = ps
R t, to tm
Ci
= linear velocity of carrier gas V = volume of the bed (total) V50 = breakthrough time of 50%of the input concentration Ci w = volumetric flow rate, vol/time x = bed length in eq 6 using Gleuckaufs notation, weight x = variable in eq 10, mol/vol X, = independent variable of the normal probability distribution z = monomolecular layer, vol y = variable in eq 10 and 11, vol/wt u
Greek Symbols E = ratio of internal to external porosity E, = external porosity ci = internal porosity p, = amount of adsorbate p~ = density of liquid u = standard deviation
ur = ~ / t 5 o =
relative standard deviation
4 = kinetic factor w
= adsorption energy
Literature Cited Wilson, J. N. J. Am. Chem. SOC.1940,62,1583. DeVault, D. J. Am. Chem. SOC.1943,65,532. Martin, A. J. P.; Synge, R. L. M. Biochem. J. 1941,35, 1358. Glueckauf,E. Nature (London) 1945,156,748. Glueckauf,E. Trans. Faraday SOC.1955,51,1540. Damkohler,G.;Theile, H. Angew. Chem. 1943,56,353. Wicke, E. Angew. Chem. 1947,19,15. (8) Zhukovickii, A. A. Dokl. Akad. Nauk SSSR 1951,77,3. (9) Turkeltaub, N. M. Zh. Anal. Khim. 1950,5,200. (10) Nelson, G. 0.;Hodgkins, D. J. Am. Ind. Hyg. Assoc. J. 1972,33, 110. (11) Nelson, G. 0.;Harder, C. A. Am. Ind. Hyg. Assoc. J . 1972,33, 797. (12) Nelson, G. 0.;Harder, C. A. Am. Ind. Hyg. Assoc. J. 1974,35, 391. (13) Nelson, G. 0.;Harder, C. A. Am. Ind. Hyg. Assoc. J. 1976,37, 205. (14) Nelson, G. 0.;Johnsen, R. E.; Lindeoken, C. R.; Taylor, R. D. Am. Ind. Hyg. Assoc. J. 1972,33,745. (15) Grubner, 0.;Zikanova, A.;Ralek, M. J. Chromatogr. 1967,28, 209. (16) Grubner, 0.Adv. Chromatogr. ( N . Y . )1968,6,173. (17) Lewis, W. K.; Gilliland, E. R.; Chertow,B.; Cadogan,W. P. Znd. Eng. Chem. 1950,42,1326. (18) Grubner, 0.;Burgess, W. A. Am. Ind. Hyg. Assoc. J. 1979,40, 169. (19) Wood, G.;Kasunic, C., presented at the 15th DOE Nuclear Air Cleaning Conference,paper no. 5, Boston, MA, Aug 7-10,1978, (1) (2) (3) (4) (5) (6) (7)
Received for review September 29, 1978. Revised Manuscript Received February 19,1981. Accepted July 6,1981.
Volume 15, Number 11, November 1981 1351
