95
Ind. Eng. Chem. Fundam. 1985, 2 4 , 95-101
support of this work and to Professor J. R. Grace for drawing to their attention the pertinent work of Poggi et al. (1969) on entrainment of mercury droplets by argon bubbles from a pool of mercury into an overlying aqueous phase. Literature Cited Beeckmans, J. M. In "Fluldization", Kunll, D.; Toei, R., Ed.; Engineering Foundation Conference: New York, 1984; p 177. Beeckmans, J. M.; Bergstrom, L.; Large, J. F. Chem. Eng. J . 1084, 28, 1. Burgess, J. M.; Fane, A. G.; Feii, C. J. D. Proc. Pac. Chem. Eng. Congr. 1077, 2, 1405. Cheung, L.; Nienow, A. W.; Rowe, P. N. Chem. Eng. Sei. 1074, 2 9 , 1301. Chiba, S.; Kobayashi, H. J . Chem. Eng. Jpn. 1077, 70, 206. Chiba. S.; Tanimoto, H.; Kobayashi. H.; Chiba, T. J . Chem. Eng. Jpn. 1070, 72, 43. Chiba, S.; Nienow, A. W.; Chiba, T.; Kobayashi, T. Powder Techno/. 1080, 26, 1. Davenport, W. G.; Bradshaw, A. V.; Richardson, F. D. J . Iron Steel Inst. 1067, 205, 1034. Fane, A. G.; Nghiem, N. P. J . Chinese Inst. Chem. Eng. 1083, 1 4 , 215.
Gibihro, L. G.; Rowe, P. N. Chem. Eng. Sci. 1074, 29, 1403. Hussein, F. D.; Maitra, P. P.; Jackson, R. Ind. Eng. Chem. Process Des. Dev. 1081, 20, 511. Kunii, D.; Levenspiei, 0. Ind. Eng. Chem. Fundam. 1068, 7 , 446. Fiaimer, N. S.; Chiba, T.; Nienow, A. W. Chem. Eng. Sci. 1082, 3 7 , 1047. Nienow, A. W.; Rowe, P. N.; Agbim, A. J. Trans. Inst. Chem. Eng. 1073, 57, 260. Nienow, A. W.; Rowe, P. N.; Cheung, L. Y.-L. Powder Techno/. 1978, 20, 89. Rowe, P. N.; Nienow, A. W.; Agbim, A. J. Trans. Inst. Chem. Eng. 1072, 50, 310. Rowe. P. N. Chem. Eng. Sci. 1076, 3 7 , 285. Poggi, D.; Minto, R.; Davenport, W. G. J . Met. 1960, 2 7 , 40. Tanimoto, H.; Chiba, S.; Chiba, T.; Kobayashi, H. In "Fluidization",Grace, J. R.; Matsen, J. M., Ed.; Plenum: New York, 1980; p 381. Tanimoto, H.; Chiba, S.;Chiba, T.; Kobayashi. H. Chem. Eng. Jpn. 1981, 7 4 , 273. Yang, W.-C.; Keairns, D. Ind. Eng. Chem. Fundam. 1982, 2 7 , 228. Yoshida, K.; Kameyama, H.; Shimizu, F. In "Fluidization",Grace, J. R.; Matsen, J. M., Ed.; Plenum: New York, 1980; p 389.
Received f o r review October 25, 1983 Accepted July 30, 1984
Gas Adsorption Isotherms by Use of Perturbation Chromatography Sang H. Hyunt and Ronald P. Danner' Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
Perturbation chromatography techniques have been used to determine gas adsorption equilibria on 13X molecular sieves. A critical analysis of these techniques is presented. The theoretical difference between the concentration boundary and the species velocities is analyzed, and the difference is illustrated experimentally. Pure-component isotherms of ethylene and ethane have been determined at subatmospheric pressures by the tracer-pulse technique with helium as a diluent. These pure-component data are in excellent agreement with the proven static data. Nonlinear isotherms can be treated successfully with the tracer-pulse method. A method consisting of a combination of the concentration-pulseand tracer-pulse techniques is shown to produce reliable gas-mixture adsorption isotherms in systems where the concentration-pulse technique cannot. This combined method has both economic and practical advantages.
A wide variety of methods and apparatus has been used to determine adsorption equilibria of pure gases or gas mixtures on solid surfaces. In general, the chromatographic methods have the advantages of simplicity, rapidity in producing data, and suitability for higher temperatures and pressures than the more conventional static methods. During the past few decades, considerable attention has been given to the development of chromatographic techniques for physicochemical measurements. Adsorption equilibria data have been obtained principally by frontal chromatography or by elution chromatography, but these methods are limited to the infinitely dilute solute concentration range and cannot be extended easily to multicomponent systems at finite concentrations. Newer chromatographic approaches have been developed which can be applied over the entire concentration range. Perturbation chromatography, which includes both the tracer-pulse and concentration-pulse techniques, is one such approach. Since the concentration-pulse method is more convenient in experiments than the tracer-pulse method, the theory of concentration-pulse chromatography has been dealt with in the literature by a number of authors. Generalized theories of concentration perturbation chroCollege of Engineering, Yonsei University, Seoul, Korea. 0196-4313/85/1024-0095$01.50/0
matography for systems with interfering species have been well established by Helfferich and Klein (1970), Deans et al. (1970), Glover and Lau (1983), and Hyun and Danner (1982b). Only a few researchers, however, have used the concentration-pulse method and compared their results with data collected by proven static methods. In a number of these cases, comparisons were made only for the infinitely dilute region. Hyun and Danner (1982b) analyzed in detail the theory of the concentration-pulse technique and then demonstrated the limitations of this method by comparing gas-mixture adsorption data obtained by a static (volumetric) method with those predicted by the concentration-pulse method. This method has formidable analytical difficulties in obtaining the required number of equilibrium partial derivatives from the limited data provided by the concentration-pulse retention volumes as shown by Hyun and Danner (198213). On the other hand, the tracer-pulse technique is theoretically a very simple and versatile method for gas adsorption measurements. It has been used for the determination of gas adsorption equilibria by many authors including Gilmer and Kobayashi (1965),Peterson et al. (1966), Haydel and Kobayashi (1967), Everett and Kobayashi (1978), Rolniak and Kobayashi (1980), and Danner et al. (1980). None of these authors, however, has shown that this method is reliable for determining multicomponent gas adsorption equilibria over the entire concentration range. Furthermore, it has not been successfully applied below atmospheric pressure. 0 1985 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
If the accuracy of the technique for multicomponent systems as well as for low-pressure data can be verified, this method should receive wide-spread acceptance as a valuable alternative to conventional static methods for adsorption measurements. It is capable of handling nonlinear isotherms just as easily as linear isotherms. In this paper, a definitive analysis of both the species velocity used in the tracer-pulse technique and the concentration boundary velocity in the concentration-pulse method is presented. The application of the tracer-pulse technique is shown to give accurate low-pressure adsorption data. In addition, a new approach combining the tracer-pulse and concentration-pulse methods is described.
Theory In perturbation chromatography, an adsorption bed is brought to equilibrium with a flowing gas mixture at the vapor composition of interest. The system is then perturbed by either a small pulse of one of the pure components (concentration-pulse method) or by a small pulse of a detectable isotope of one or more of the components (tracer-pulse method). In the former method one is measuring the velocity of a concentration boundary. The experimental method and theoretical analysis for this method were described in a previous paper by Hyun and Danner (1982b). The tracer-pulse technique, on the other hand, involves the observation of the movement of individual, distinguishable molecules (Le., measuring the species velocity). A pulse of a detectable isotope, in general, does not introduce any change in the adsorption properties of the adsorbing species. The retention volume of the isotope is measured and used to calculate the partition coefficient (i.e., adsorption equilibrium constant) through the tracer retention volume equations. To analyze the column behavior the following assumptions are made: (a) Axial dispersed plug flow exists. (b) There are no end effects at the column inlet or outlet. (c) The ideal gas law is assumed to be valid. (d) The pressure drop through the column is negligible. (e) The temperature is constant. (f) Flow is one-dimensional. The differential mass balance for a solute in a small volume element of the packed column can then be written as -DL,,(
s)t
+ U( $ ) t
*):(
+ Ci*(
+
This equation can be further simplified if one assumes that the injection of a tracer pulse does not alter the overall adsorption equilibrium, and thus there is no sorption effect. This assumption is always true for pure-component systems and for multicomponent systems if the injected pulse has a composition identical with that of the gas mixture stream. If the composition is not identical but the injected pulse is very small, this sorption effect will still be negligible. That is, (Ci*/C)(&/at), in eq 3 is equal to zero. The resultant differential mass balance for the tracer component i is
If there is no isotopic fractionation effect, the equilibrium distribution coefficient of the labeled species is equal to that of the unlabeled species C,*/C,* = C,/C, = constant (5)
For all practical purposes, this is always true unless extreme conditions are sjlected in an attempt to separate the isotopes. While C, and C, are constant at a given composition of the mobile phase, C,* and C,* are functions of z and t. Equation 5 becomes
aC,*/ac,* = C L / C 1
(6)
or
(G*/ a t ) ,
= cC,/c,) (dC,*/ a t ) ,
(7)
Using the relationship (dC,*/dt), = -(ac,*/dz),(dz/at)c~*
(8)
and substituting eq 7 and 8 into eq 4 leads to
2)+
(%),a
With these changes eq 1 becomes
(az/at),* = 0 (1)
The first term in eq 1 represents migration of tracer component i along the column axis in a diffusion-like manner and the second term represents transport by bulk flow. The third term describes the variation of the mobile phase velocity. The last two terms represent the holdup caused by changes in vapor phase and adsorbed phase concentrations. In adsorption equilibrium studies only the retention time or the f i s t moment is required, and thus the axial diffusion effects can be neglected on the grounds that they result only in peak broadening but do not alter the retention time as long as instantaneous adsorption equilibrium is established (Buffham, 1973; Deans et al., 1970). The velocity variation of the mobile phase (aU/az),, caused by the sorption effect a t a constant vapor phase total concentration, C, can be obtained by writing eq 1 for the total concentration again ignoring axial diffusion.
= U / [ 1 + (C,/C,) (V8/VJ1
(9)
Since C,* varies with distance z and time t , ( ~ ? z / d t )is~ ~ , the instantaneous species velocity. That is
U, = (az/at)ct* = U / [ l +
(e,/c,)(v,/v,)l
(10)
Equation 10 illustrates that the species velocity is always less than the mobile-phase velocity because the chords of adsorption isotherms, C,/C,, are always positive. The specific equations useful for determination of adsorption equilibria by the tracer-pulse method can be directly obtained through the same manipulations as used by Danner et al. (1980) and Hyun and Danner (1982b).
c,/C, = (V*R, - Vg)/Vs
(11)
Y , m v * R I - V,)O x, =
RPW
(12)
and n
W = (P/RP)Cy,(v*R, - Vg)' r=l
(13)
where V*& is the average retention volume of tracer component 1 and W is the total amount adsorbed of the given gas mixture.
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985 97 RADIOACTIVE GAS AND VACUUM
RADIOISOTOPE
DUCTlVlTY CELL ADSORBENT CONSTANT TEMPERATURE BATH
Figure 2. Schematic of the apparatus. C;'
C;
M O B I L E PHASE CONCENTRATION, Ci
Figure 1. Relationship between chord and slope values for different types of physical adsorption isotherms.
The concentration velocity in a multicomponent system is given by Hyun and Danner (198213) as their eq 9. For a binary system this concentration velocity is givei
u,= uc, = u,, = and the net retention volume is (VR1 - VJO =
Po
[
Y2(
2) +
Yl(
a)]
For a single-adsorbing component system, eq 14 at constant pressure becomes U u, = (16) 1 + (1 - YJ(dCi/dCi)(V,/Vg)
This equation says that the con_centrationvelocity depends on the slope of the isotherm, dC1/dC,, and the vapor-phase mole fraction, yl, at the given total pressure. Equation 10 for the species velocity is quite similar to eq 16 for the concentration velocity. The factor (1- yl) in eq 16 arises from the sorption effect. Only when y1 approaches zero could the concentration velocity be written as U u, = (17) 1 + (dCi/dCJ(V,/Vg) In this concentration region the slope of the puce gas isotherm, dCl/dC1, is equal to the isotherm's chord, C1/C1, and thus the concentration velocity is identical with the species velocity. In general, physical adsorption isotherms might be classified into the five types of isotherms shown on Figure 1. For type I isotherms, the slopes of the chords are always larger than or eqyal to the isotherm slopes, and thus (1 - yl) dCl/dC1 5 C1/C1 at all vapor phase concentrations. This means that the tracer-pulse peak comes later than or at the same time as the concentration-pulse peak for single-component adsorption cases. For type I11 isotherms, although the slopes are larger thgn the chords (1 - yl) de, /dC1 is not always larger than C1/C1,so which velocity is greater depends on the vapor-phase mole fraction, yl. Types 11, IV, and V isotherms can be divided across the inflection points into the type I and type 111regions and then the same argument as above can be applied.
For the binary mixture system, the concentration velocity depends on the vapor-phase mole fractions and slopes of the two mixture isotherms, but the species velocity depends only on the mixture isotherm chord of the tracer component. It is instructive to compare the species velocity and the concentration velocity under the same experimental conditions. In the case of a helium(89%)ethane(ll% )-13X molecular sieve system, a small pulse of radioactive ethane diluted with normal ethane was pulsed, thus simultaneously obtaining a concentrationpulse and a tracer-pulse peak. Hyun and Danner (1982a) have shown that ethane on 13X molecular sieve exhibits a type I isotherm and thus the concentration velocity should always be larger than the species velocity regardless of the composition. In this experiment the concentration-pulse retention time was only 2.96 min compared to the tracer-pulse retention time of 3.47 min. The same relationship between the retention times was observed regardless of the composition of the gas mixture. In the case of an isobutane-ethylene-13X molecular sieve system which contains two adsorbable components, the species velocity of ethylene depends only on the chord of the mixture isotherm of ethylene, but the concentration velocity depends on the slopes of the mixture isotherms of both isobutane and ethylene and on the mixture compostion as shown by eq 14. In the Experimental Results section, it is shown that the mixture isotherm for ethylene is not type I, and therefore, one may anticipate that there may be a shift in the relative values of the retention times of the concentration-pulse and tracer-pulse peaks depending on the concentration. When the mixture contained 53% isobutane, the retention time of the concentration-pulse peak was shorter (2.28 vs. 3.14 min). When the mixture contained only 17% isobutane, however, the retention time of the tracer-pulse peak was shorter (2.56 versus 3.81 min). These experimental observations are consistent with the theory.
Experimental Section A schematicdiagram of the apparatus is shown in Figure 2. The overall system was constructed with 1/8-in.0.d. copper tubing with the exception of the column which was 1/4-in.0.d. copper tubing. Pure gases were introduced through gas driers and electronic nonlinear mass flow meters, which were calibrated with a soap bubble meter for the gas of interest. The flow rates were accurate to *2%. The desired compositions were established by setting the constant flow rates of each of the gases with fine metering valves. After blending, the gas was split into two parts-one which passed through the adsorption column and one which served as the reference for the thermal conductivity detector. The system pressure was controlled with two expansion valves installed in the
98
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
column and reference gas streams after the adsorption column. The pressure of the system was directly measured with a 0-500-psi Heise-Bourdon gauge with an accuracy of f0.2 psi. The column was immersed in a mineral oil bath whose temperature was measured with a mercury thermometer within 0.1 K. The gas pulses were injected through a rotary gas-injection valve which had a sample loop volume of 0.3 cm3. The thermal conductivity detector was a nanokatharometer flow cell and the radioactivity detector was a vibrating reed electrometer equipped with an ionization cell of an internal volume less than 1 cm3. The absorbent samples were intially regenerated by heating to 530 K under better than 3 pmHg for a t least 48 h in order to determine the true sample weight. Before each run the adsorbent was regenerated in the column by bringing the column to the above regeneration conditions for 10-12 h. For a run the column was brought to equilibrium with the desired vapor phase. A very small (low-pressure) gas sample was then introduced through the gas-sample valve and the response of the column to the pulse was recorded on a dual pen recorder which simultaneously monitored both detectors. The actual response peaks observed in this work always showed slight asymmetries. Assuming that there are no nonlinear isotherm effects over the very small concentration perturbation, the only cause of asymmetry is the normal nonideal band-spreading in a column of relatively few plates. In this case, the retention volume is still accurately given by the time corresponding to the mass center of the peak. Two methods of locating the mass center were applied. The steepest-slope tangent method (Conder and Young, 1979) was employed in the concentration-pulse data analysis, and the well-known numerical-integration method in the tracer-pulse data. The steepest-slope method could not be applied to the tracer-pulse data analysis since the response peaks of the ionization cell had a lot of background noise. For the concentration-pulse peaks, the integration method could not be used because in some cases the fronts of the peaks overlapped with the output response of the pressure and flow rate fluctuations caused by the injection of the pulse. The accuracy of the steepest-slope method in this work turned out to be comparable to the numerical-integration method. The retention time caused by adsorption corresponds to the net retention volume, ( V*Ri- V,). Since helium is used in the tracer-pulse method to determine the free gas volume, V,, two detectors are required for determination of the net retention volume-the ionization detector for the isotope and the thermal conductivity detector for the helium. To obtain the net retention time caused by adsorption it is necessary to eliminate all of the effects of the system dead volume and the time lags of the detectors. T o do this a bypass shunt was installed between the inlet and outlet of the column and the following equation was used to determine the correct retention volume for use in eq 11. (v*RL
- V J = (v*R,
-
v*R/BY)
-
(vP- *?By)
(18)
The first term on the right-hand side eliminates the effect of the system dead volume between the thermal conductivity detector and the ionization detector. The second term accounts for the difference in the internal volume of the packed column and that of the bypass column. Further details of the apparatus and the calculation procedures for both the concentration-pulse and tracer-pulse methods are described by Danner et al. (1980), Hyun and Danner
0
OPEN POINTS
- S T A T I C DATA ( HYUN AND - TRACER P U L S E DATA
DANNER, 1982 a )
4
SOLID P O I N T S
20
40
I 60 PRESSURE
1
I
I
1
SO
100
120
140
1
Ik P 0 )
Figure 3. Pure-gas isotherms for ethylene and ethane on 13X molecular sieves at 323 K.
(1982b), and Al-Ameeri and Danner (1984). The adsorbent used was 13X molecular sieves with 20% binder (Lot No. 13945390174)manufactured by the Linde Division of the Union Carbide Corp. As received, the sieves were in 1/16-in.pellets having a BET surface area of 420 m2/g and an apparent density of 2.31 g/cm3. The pellets were crushed in a porcelain mortar and sized using U.S. standard screens. The particles used had an average particle diameter of 0.046 cm. A 1/4-in.0.d. copper tube in the shape of a U was packed with 1.5824 g of crushed adsorbent yielding a column length of 14.37 cm. Since the inside diameter of the copper tube was 0.4826 cm, the particle diameter to column diameter ratio was l / l o and the column length to column diameter ratio was 30/1. The adsorbate gases used were research grade ethane (99.99%), ethylene (99.95%), and isobutane (99.99%). Radioisotopes of ethane I4CzH6manufactured by New England Nuclear Co. and of ethylene 14C2H4manufactured by Amersham Corp. were diluted with pure ethane and ethylene, respectively, and then were used as tracer samples. After dilution, the concentration of the radioisotopes were 20.73 mCi/g-mol of ethane and 25.18 mCi/g-mol of ethylene. Pure-Gas Results Pure-gas isotherms of ethane and ethylene at several temperatures have been determined by the tracer-pulse method in this laboratory by Hetrick (1982). These data were in perfect agreement with the static data of Hyun and Danner (1982a); thus the tracer-pulse technique was experimentally proved to be an extremely promising method for the determination of adsorption equilibrium data. All of these data, however, were collected at pressures higher than atmospheric. Considering the importance of the low pressure data, particularly for the development of adsorption models, the use of the tracer-pulse method to obtain low-pressure data was investigated. Since the dynamic adsorption apparatus used in this laboratory was restricted to operation above atmospheric pressure, a different operation method was needed. The simplest alternative is to use the helium-dilution method, as was used in the concentration-pulse method to obtain pure-gas isotherms (Hyun and Danner, 1982b). The determination of the amount adsorbed did not require the complex integration procedures used in the concentration-pulse method. The amount adsorbed could be directly calculated from eq 12 which was developed for the tracer-pulse retention volume of the gas mixture. Pure-gas isotherms for ethylene and ethane on 13X molecular sieves a t 323 K were determined in the ranges of partial pressure of 35.5 to 92.5 kPa and 1.8 to 104.6 kPa, respectively. As shown in Figure 3, the adsorption isotherms of ethylene and ethane at low pressures, deter-
Ind. Eng. Chem. Fundam., Vol. 24,
2Bc
No. 1, 1985 99
I / -
'
0
E X P E R I M E N T A L DATA
-.-
i
___
ORDER OF REGRESSED POLYNOMIAL
- 3R0
POINTS-STATIC DATA ( H Y U N AND DANNER, 1982 0 ) RDER OF REGRESSED POLYNOMIAL 2ND
1
4TH
1.I
0
02
06
04 Yl
- C,
OS
IO
0
02
HIO
Figure 4. Net concentration-pulse retention volumes in the isobutane-ethylene-13X molecular sieve system at 298 K and 138 P a .
mined by the tracer-pulse method, are in excellent agreement with the static data. These results confirm that the tracer-pulse method is accurate for the determination of the pure-gas isotherms in any pressure range.
Gas-Mixture Results In the previous paper of Hyun and Danner (1982b) the isobutane-ethylenel3X molecular sieve system was used to evaluate the concentration-pulse technique in conjunction with the data reduction method of Van der Vlist and Van der Meijden (1973). The concentration-pulse retention volumes, however, were not measured experimentally but regenerated from the static mixture isotherms by using eq 15 in reverse. In order to predict the mixture isotherms, the data reduction procedure requires that these retention volumes be fitted with a low-order polynomial (Hyun and Danner, 1982b). A third-order polynomial fitted the data quite well and the resulting predictions were quite good. In the current study concentration-pulse retention volumes were determined experimentally by injecting the radioisotope of ethylene (see Figure 4). These volumes were regressed with second-, third-, and fourth-order polynomials in order to try to predict the mixture isotherms by the Van der Vlist-Van der Meijden method. Figure 4 shows that while the second- and third-order polynomials fit the data unsatisfactorily, the fourth-order polynomial gives a good curve fit. The mixture isotherms predicted in all cases including the fourth-order polynomial, however, are unacceptable as shown in Figure 5 where they are compared to the volumetric data of Hyun and Danner (1982a). The above demonstrates that although the data reduction method could fit the data when hypothetical values of the concentration-pulse retention volumes were used, the method was not successful when used with actual experimental data. This is then another example of the inadequacy of the concentration-pulse method for systems having more than one adsorbable species. Previous proponents of the concentration-pulse method (Van der Vlist and Van der Meijden, 1973; Ruthven and Kumar, 1980) have considered only binary gas mixtures in which both components are weakly adsorbed and have not compared
06
04 YI-
c,
08
I O
HI,
Figure 5. Prediction of the gas-mixture isotherms for the isobutane-ethylene-13X molecular sieve system at 298 K and 138 kPa by the concentration-pulse method.
their results to independently determined data. The tracer-pulse method has the disadvantage of requiring radioisotopes for all components present in the mixture. If one of these isotopes is unavailable or very expensive, an approach which combines the concentration-pulse method with the tracer-pulse method may be useful. Consider the case of a binary mixture. Equation 15 for the retention volume of the concentration-pulse contains two isotherm partial derivatives which are unknown and the retention volume which is determined experimentally. The failure of the concentration-pulse method is a result of there being inadequate information from the experimental procedure to define the two derivatives. If one of these derivatives can be determined independently (by the tracer-pulse method, for example),the other derivative can easily be calculated from the concentration-pulse data. The isobutane-ethylene-13X molecular sieve system was a good candidate for using this approach-the concentration-pulse method had failed and we did not have a radioisotope of isobutane available. First the mixture isotherm of ethylene was calculated with tracer-pulse retention volume data according to eq 12. The results shown in Figure 6 are in good agreement with the static data indicated by the solid points. In fact, the tracer-pulse data appear to have better consistency than the static data. The isotherm slopes, d WC2H4/dyCZH4, were then measured graphically from the smooth curve interconnecting the ethylene mixture isotherm data in Figure 6. Next the isobutane isotherm slopes were calculated from the concentration retention volume data for the corresponding mixtures of isobutane and ethylene. Equation 19, which is a revised form of eq 15, was used to calculate the isobutane isotherm slopes. -I
I =
100
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
''k
total number of unknown partial derivatives is n(n - 1) n - 1 sets of concentration-pulse retention volume data are available. As long as n is three or greater an analytical solution of the retention volume equation cannot be easily obtained. If all the species are injected as isotopes, this is simply the tracer-pulse technique itself.
1
-
SOLID POINTS STATIC DATA (HYUN AND DANNER, 1 9 8 2 0 1 W T R A C E R - P U L S E DATA UlCONCENTRATION-TRACER7
I
- m. But to solve these derivatives only
0 W
m 0
i
U
02
3
06
04
08
IO
- c,H,,,
'i
Figure 6. Prediction of the gas-mixture isotherms for the isobutane-ethylene-13X molecular sieve system at 298 K and 138 kPa by the tracer-pulse and concentration-pulse methods. I0
SOLID P O I N T S
~-__ STATIC
AND D A N N E R ,
DAA-Ff
19820)
M CONCENTRATION - TRACER-
S 8 -
PULSE D A T A
-- C O N C E N T R A T I O N - PULSE PREDICTION ( 2ND ORDER POLYNOMIAL 1 -1
0 06 I 0t
-,
04
Conclusions This study has shown that the tracer-pulse technique can be used successfully for the determination of pure-gas isotherms. These isotherms can be nonlinear and the technique can be applied in all pressure ranges including the subatmospheric range by using helium dilution. This has been demonstrated by measuring isotherms for ethylene and ethane on 13X molecular sieves at 323 K. A comparison of species velocities (observed with tracer-pulse experiments) and concentration boundary velocities (observed with concentration-pulse experiments) showed theoretically and experimentally that for type I pure-gas isotherms, the species velocity cannot be greater than the concentration boundary velocity. For other types of pure-gas isotherms and for binary mixtures, the species velocity can be greater or smaller than the concentration velocity depending on the vapor phase composition and the slopes and chords of the gas-mixture isotherms. Excellent gas-mixture data for the highly nonideal isobutane-ethylene-13X molecular sieve system have been obtained by a combined concentration-pulse-tracer-pulse technique. Acceptable data for this system could not be obtained by the concentration-pulse method nor by the tracer-pulse method since no radioisotope was available for the isobutane. The use of this combined procedure should be quite valuable for other binary systems whether one of the tracer gases is unavailable or too expensive. For systems containing three or more adsorbable species, however, only the tracer-pulse method can be used to obtain gas-mixture equilibria data over the entire concentration range. Acknowledgment
02 /
Support of the National Science Foundation under Grant No. CPE-8012423 is gratefully acknowledged. I - CqH
Figure 7. Adsorption phase diagram for the isobutane-ethylene13X molecular sieve sytem at 298 K and 138 kPa.
Finally, in order to generate the isobutane isotherm, the slope data were integrated. Areas under the isobutane isotherm slope curve were determined graphically. The total area up to a given composition corresponds to the amount of isobutane. The total adsorption amount obtained in this way is given in Figure 6 and the adsorption phase diagram in Figure 7. Figure 7 shows clearly that this is a very nonideal system-an adsorption azeotrope is formed. It also demonstrates that the combined concentration-tracer-pulse method can produce excellent agreement with the static data. Unfortunately, the combined concentration-tracer-pulse method does not offer any advantage for systems containing three or more adsorbing components. For a system containing m adsorbing species, the total number of partial derivatives in the concentration-pulse retention volume equation is n(n - 1). An injection of a mixture of m tracer components will product n - 1 concentration-pulse peaks and m tracer-pulse peaks. For each tracer-pulse set of peaks one partial derivative can be determined. Thus, the
Nomenclature
C = total concentration in the mobile phase, mol/cm3
C = total concentration in the stationary (adsorbed) phase, mol/cm3 C, = concentration of component i in the mobile phase, mol/cm3 = concentration of component i in the stationary phase, mol/cm3 C,* = concentration of radioactive component i in the mobile phase, mol/cm3 c,*= concentration of radioactive component i in the stationary phase, mol/cm3 DL,, = effective logitudinal dispersion coefficient, cm2/s F = volumetric flow rate of the mobile phase at column conditions, cm3/s m = number of the radioactive isotopes injected n = total number of components present in the mobile phase Po = pressure at standard conditions, kPa R = universal gas constant t = time or elution time, s tR= retention time, s 7"' = temperature at standard conditions, K U = interstitial mobile phase velocity at the column conditions, cm/s Uc, = concentration velocity of a species i, cm/s U, = species velocity of a species i, cm/s
e,
101
Ind. Eng. Chem. Fundam. 1985, 2 4 , 101-105
Literature Cited
V = free gas volume, cm3 v", = retention volume of the species i through the adsorption column, cm3 V*& = retention volume of a tracer species i through the adsorption column, cm3 (VRi - Vg)O = net retention volume of a component i at standard conditions, cm3 STP V*R/By= retention volume of tracer species i through the bypass shunt, cm3 vHe - retention volume of helium through the column, cm3 = retention volume of helium through the bypass shunt, cm3 V , = volume of the stationary phase, cm3/g W = total amount adsorbed, kmol/kg W i= amount adsorbed of component i, kmol/kg x i = mole fraction of component i in the adsorbed phase y i = mole fraction of component i in the mobile phase z = axial distance from the column inlet
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Received for reuiew October 28, 1983 Registry No. Ethylene, 74-85-1; ethane, 74-84-0.
Accepted July 24, 1984
EXPERIMENTAL TECHNIQUES Image Processing Techniques for the Estimation of Drop Size Distributions Randy D. Hazlett,+Robert S. Schechter,*t and J. K. Aggarwalt Deparfment of Chemical Engineering and Laboratory for Image and Signal Analysis, The University of Texas at Austin, Austin, Texas 78712
Digital image processing techniques have been developed for automated estimation of drop size distributions from photomicrographs of emulsions. The system and algorithms used are outlined and the computer results are compared with direct measurement. Results with test samples showed good agreement in the distributions.
Introduction Particle size analysis is a rapidly growing field of interest in environmental science, pharmacology, microbiology, and engineering. The particle size and size distribution influence dispersion properties such as viscosity, heat transfer, and stability (Becher, 1965). Furthermore, changes in particle size distribution with time are a direct indication of dispersion instability. The wide variety of methods currently in use for particle size analysis fall into two categories: direct and indirect. Light scattering and electrical conductivity measurements are examples of indirect techniques, while direct methods include optical and electron microscopy (Groves and Freshwater, 1968). We are currently using differential interference microscopy to size relatively unstable emulsions of systems containing surfactant stabilizers which may be of importance in the field of oil recovery (Bourrel et al., 1979; Wasan et al., 1979). The nature of our systems necessitates
photomicrographic analysis which can be a very tedious process if done by hand. Some commercially available automatic sizing apparatuses are described elsewhere (Allen, 1981). Most processes involve a variable aperture light pen or a digitizing tablet in which a drop is either illuminated or traced. We have significantly decreased the time necessary for analysis with the aid of digital image processing techniques (Castelman, 1979; Duda and Hart, 1973; Pratt, 1978). Although our application involves emulsions, the algorithms described are fully applicable to all dispersion of roughly spherical matter: solid particles, drops, or bubbles. In this work, differential interference photomicrographs of dispersions were digitized and processed by several computer routines: a direct measurement, a semiautomated procedure, and a fully automated process. The methods are outlined and compared for rapid estimation of drop size distributions. Background In another chemical engineering application (Schrodt and Saunders, 1981), image processing was used to compute bubble sizes, which are important to mass transfer
Department of Chemical Engineering. Image and Signal Analysis.
t Laboratory for
0196-4313/85/1024-0101$01.50/0
0
1985
American Chemical Society
