Ind. Eng. Chem. Res. 1996, 35, 315-319
315
Moments Analysis of the Zero Length Column Method Stefano Brandani*,† Dipartimento di Chimica, Ingegneria Chimica e Materiali, Universita` de L’Aquila, I-67040 Monteluco di Roio, L’Aquila, Italy
Douglas M. Ruthven‡ Department of Chemical Engineering, University of Maine, Orono, Maine 04469
The first and second moments of the desorption curves of the zero length column (ZLC) response are derived for both intracrystalline diffusion and surface barrier models. Limiting solutions applicable to gaseous systems are obtained, and a simple relationship is developed to discriminate between the two transport models. The integral relationships are discussed and used to extract intracrystalline diffusivity values from experimental curves. Introduction A knowledge of transport properties of the adsorbed species in microporous solids is required for the accurate design of adsorption and catalytic reaction processes. The zero length column (ZLC) method represents a simple and relatively inexpensive technique for obtaining the required physical properties. This method was developed in the late 1980s by Eic and Ruthven (1988). It is essentially a chromatographic technique and has been widely used to measure intracrystalline diffusivities in zeolitic adsorbents and other microporous solids. It has been applied mainly to gaseous systems (for a review see Ka¨rger and Ruthven 1992), but the application to a liquid system has also been demonstrated (Ruthven and Stapleton, 1993). In order to obtain the diffusion coefficients, the solution to the model equations is matched to the experimental desorption curves. This is usually accomplished using the long-time asymptote (LT) of the desorption curve, which is linear on a semilogarithmic plot (Eic and Ruthven, 1988). Recently, Hufton and Ruthven (1993) have discussed the practical advantages and drawbacks of the LT method and have introduced a new approximate method for extracting the relevant parameters from the initial portion (short time, ST) of the desorption curve. These two methods should yield the same results, but the values obtained by Hufton and Ruthven (1993) showed differences of up to an order of magnitude. In this paper we present a new method for extracting intracrystalline diffusivities using moments analysis of the desorption curves. This approach has been widely used to analyze and interpret the response curves from traditional chromatographic measurements (Ruthven, 1984). It has also been used in the analysis of conventional gravimetric and piezometric uptake curves (Ka¨rger and Ruthven, 1992) and in particular to detect the intrusion of extraneous heat effects in the sorption kinetics of benzene on Na-X zeolite crystals (Bu¨low et al., 1984, 1986). Theory Although most applications have been for gaseous systems, we will consider the more general model * Author to whom correspondence should be addressed. † FAX: (+39) 862-432603. ‡ FAX (207) 581-2323.
0888-5885/96/2635-0315$12.00/0
equations which are also applicable to liquid systems. The relevant solutions for a gaseous system can be obtained when the accumulation in the fluid phase is considered equal to zero. The model equations for a diffusion-controlled process are (Ruthven and Stapleton, 1993)
Vs
dq j dt
+ Vf
dc + Fc ) 0 dt
(
)
∂q ∂2q 2 ∂q )D 2+ ∂t r ∂r ∂r q(R,t) ) q* ) Kc;
(∂q∂r)
r)0
) 0;
(1)
(2)
q(r,0) ) q0 ) Kc0 (3)
The solution to this set of equations is given by (Brandani and Ruthven, 1995a):
∞
c ) c0
(
D 2L exp -βn2 t R2
∑
2
n)1β n
)
+ (1 - L + γβn2)2 + L - 1 + γβn2
(4)
where
L)
1 FR2 ; 3 KVsD
γ)
1 Vf 3 KVs
(5)
and βn are the positive roots of
βn cot βn + L - 1 - γβn2 ) 0
(6)
The parameter L can easily be varied by adjusting the flow rate (F), but γ is essentially equal to 1/3K and, as such, depends only on temperature. The limit for γ ) 0 yields the solution applicable to gaseous systems. The aim is to derive the expressions for the first and second moments, of the ZLC response, which are defined by (Ruthven, 1984):
∫0∞ct dt µ) ∞ ∫0 c dt © 1996 American Chemical Society
(7)
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Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996
2
σ
∫0∞c(t - µ)2 dt ∫0∞ct2 dt 2 ) ) -µ ∫0∞c dt ∫0∞c dt
(8)
c0
)
1 1 - γs - xs coth xs s 1 - L - γs - xs coth xs
(9)
where the time has been expressed in reduced form τ ) tD/R2. Taking into account the definition of the Laplace transform (van der Laan, 1957), the following integrals can be evaluated:
I)
∫0
∞
∫0
]
)
where:
A)
1 + 3γ ; 3γ
B)
1 L′ ; 15 γ
5F ; KVsk A+B2 R2 ) - B (19) 2
L′ )
(
)
The solution in the Laplace domain is given by (Ruthven and Stapleton, 1993):
c0
(10)
( ) ( ) [( 2
R2 ) D
2
)
s+A s + (A + B)s + B
(20)
2
]
)
( )
( ) [(
c˜
A 1 ) ∫0∞cc0 dt ) lim sf0 c0 k Bk
1 1 + 3γ 2 + (11) 3L 45L
2 ∞ 2c 1 d c˜ R2 3 t dt ) lim ) 0 sf0 c0 ds2 D c0 2 1 2 1 + 3γ R2 3 1 + 3γ 3 2 + + (12) D 3L 135 L 945 L
]
)
( ) [( 2
1 1 R2 1 + 3γ 1 + 3 D L 5 1 + 3γ
(
)
(13)
]
13 + 60γ 1 + 3γ 2 2 1 + + L 5 L 175(1 + 3γ)2
)
2 6 (1 + 10γ)L σ2 ) 1 + 7 [L + 5(1 + 3γ)2]2 µ2
µ)
dt
) k(Kc - q j)
(
)
A A3 + 2A2B + AB2 - 2AB - B2 (23) Bk3 AB2
1 1 A2 + AB - B 1 1 + 3γ ) 5 + k AB k L′ 1 + 3γ
(
)
(24)
1 A4 + A2B2 + 2A3B - 2A2B - B2 ) k2 A2B2 10 1 + 3γ 2 3γ 2 1 25 +1+ (25) 2 L′ L′ 1 + 3γ k
[ (
(15)
)
(
)]
and
(16)
These two ratios are independent of the time constant and can be used to obtain both the parameters L and γ. The time constant can then be obtained from I, µ, or σ2. For L . 1 σ2/µ2 approaches a constant limit given by 1 + (6/7)(1 + 10γ). For a surface barrier controlled process, the modeling equations are (Ruthven and Stapleton, 1993) eq 1 and:
dq j
(22)
Therefore, the first and second moments are given by:
σ2 ) µ 1 L )1+ I 5 (1 + 3γ)2
)
+ AB - B AB
d2c˜
(14)
and
2
(21)
1 1 ) ∫0∞t2cc0 dt ) lim sf0 c0 ds2 k3 2
µ)
(
dc˜
1 1 A A ) ∫0∞tcc0 dt ) lim sf0 c0 ds k2 Bk2
therefore
1 R2 σ ) 9 D
(
where the time has been expressed in reduced form τ ) kt. The derivation is straightforward.
1 dc˜ R2 lim sf0 c0 ds D
2
][
(A + B) B-A kt cosh(Rkt) sinh(Rkt) 2 2R (18)
c˜
c˜ R2 R2 1 + 3γ ) dt ) lim sf0 c0 D c0 D 3L
∞c
c t dt ) c0
∫
[
exp -
This can be achieved using the solution of the modeling equations in the Laplace domain (van der Laan, 1957), which is given by (Brandani and Ruthven, 1995a):
c˜
c ) c0
(17)
The solution to these modeling equations is given by (Ruthven and Stapleton, 1993):
1 L′ µ )1+ I 5 (1 + 3γ)2
(26)
6γL′2 σ2 ) 1 + µ2 [L′ + 5(1 + 3γ)2]2
(27)
For L′ . 1 this approaches a limiting value of 1 + 6γ. It is interesting to note that, if the Glueckauf approximation, k ) 15D/R2, which is the choice made by Ruthven and Stapleton (1993) to describe liquid systems, is considered, L′ ) L and the first moments derived from the two models are the same, while this is not true for the second moment. The ratio µ/I is linear in L or L′ and therefore in the flow rate. Figure 1 shows the ratio σ2/µ2 as a function of L and L′ for various values of γ. It can be seen that for
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 317
σ02
µ0 L )1+ ; I0 5
a
2
µ0
)1+
6 L 2 7L+5
(
)
(29)
These relationships can be used to obtain the parameter L and the time constant D/R2 from the experimental curves. For surface barrier controlled processes the following relationships are obtained:
I0 )
5 1 ; L′ k
µ0 )
1 5 + L′ ; k L′
µ0 L′ )1+ ; I0 5
σ02 µ02
σ02 ) µ02
(30)
)1
(31)
In this case the ratio σ2/µ2 is independent of the flow rate and always equal to unity. This result offers a simple and useful way to discriminate between the two models at least for gaseous systems. This is particularly true for L values lower than 20, for which the partial loading experiment (Brandani and Ruthven, 1995b) cannot offer conclusive indications.
b
Application to a Sample System
Figure 1. Ratio σ2/µ2 as a function of L (or L′): (a) diffusion control model; (b) surface barrier control model.
diffusion-controlled processes in the range 1 < L < 20 the curves are quite sensitive to L. Therefore, experiments performed in this range should yield accurate values of the model parameters. For high L values, i.e., high flow rates, the ratio tends to be constant; a lower limit estimate for γ can therefore be obtained in this range. For L < 1, the curves for diffusion and surface barrier control are quite similar. This is because, at very low flow rates, the process becomes equilibrium controlled.
1 R2 5 + L ; 15 D L 1 R2 2 25 10 13 + + (28) σ02 ) 225 D L2 L 7
µ0 )
(32)
∫0∞cc0t dt ) ∫0t cc0t dt + ab exp(-bt0)(t0 + b1)
(33)
0
0
0
The relationships for gaseous systems can be obtained in the limit for γ ) 0. In fact, from its definition, γ ≈ 1/3K, and for typical gaseous systems, Henry’s constant K . 1. Almost all ZLC studies have dealt with systems which correspond to this category. Only liquid systems and tracer ZLC experiments (Hufton et al., 1994) may require the use of the general model for a correct interpretation of the experimental desorption curves. For diffusion-controlled processes:
1 R2 ; 3L D
∫0∞cc0 dt ) ∫0t cc0 dt + ab exp(-bt0)
∫0∞cc0t2 dt ) ∫0t cc0t2 dt +
Gaseous Systems
I0 )
The analysis reported in the previous sections is obviously theoretically correct. In practice, however, it is not always possible to obtain an accurate numerical evaluation of the time integrals necessary to calculate the first and second moments. This is because, with a measuring device, we can follow the desorption curve only for a limited period of time until the signal is of the same order of magnitude of the baseline noise. This will inevitably cause tailing errors which will affect the calculated values of the first and second moments. In order to apply the results to real systems, it is, therefore, useful to consider the form of the ZLC desorption curves. In the long-time region the curves follow a simple exponential decay, a exp(-bt). This is actually the basis of the widely used LT method and is true for both diffusion and surface barrier models. The time integrals can therefore be calculated from:
( )(
)
(
)
t0 a 2 exp(-bt0) t02 + 2 + 2 (34) b b b This will be particularly useful for systems where the desorption curve does not change abruptly with time and is therefore useful for systems with relatively low values of the L parameter. The system considered here is benzene on Na-X crystals. In the literature there is a considerable discrepancy between the values of self-diffusivities obtained from NMR (Germanus et al., 1985) and infinite dilution transport diffusivities obtained from ZLC (Eic et al., 1988). This discrepancy amounts to about 2 orders of magnitude. Recently, Shen and Rees (1991) have studied this system through the frequency re-
318
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996
misinterpretation of the experimental desorption curves can be excluded. The validity of the analysis is also confirmed from the good agreement obtained from the LT and moments methods of extracting the relevant parameters. Having eliminated the tailing errors, the moments method appears to be more accurate in the region for L < 10 where the LT method is quite sensitive to the value of the intercept. Using the ratio σ2/µ2 and µ, consistent results are obtained for both flow rates. It appears that the highest uncertainty is in the numerical evaluation of I. For this system, the experimental ZLC data show a diffusion-controlled process with time constant D/R2 ≈ 0.0044 s-1 (D ≈ 1.1 × 10-7 cm2/s) in comparison with the NMR self-diffusivity (≈1 × 10-5 cm2/s) at this temperature. Figure 2. Experimental desorption curves for benzene on Na-X at 468 K: (0) 76 cm3/min; (2) 106 cm3/min; (s) LT asymptotes. Table 1. Parameters Obtained from the Analysis of the Experimental Desorption Curves F, cm3/min a b, s-1 t0, s I, s µ, s σ2, s2 σ2/µ2 La D/R2,a s-1 Lb D/R2,b s-1 Lc D/R2,c s-1
curve 1
curve 2
76 0.6 0.0285 90 22.93 32.3 1214 1.16 2.05 0.0071 3.85 0.0047 2.42 0.0062
104 0.31 0.031 90 11.81 28.0 985 1.23 6.87 0.0041 5.95 0.0044 6.32 0.0043
a Moments analysis with I and µ/I. b Moments analysis with µ and σ2/µ2. c LT analysis.
sponse method and have obtained values which are intermediate between the NMR and ZLC data. A ZLC column was prepared using approximately 1 mg of 100 µm Na-X crystals. Figure 2 shows the experimental desorption curves obtained for this system at 468 K, at two different flow rates, together with the long-time asymptotes. Considering the high flow rates and the small quantity of adsorbent used, the system is clearly isothermal. Benzene is strongly adsorbed on the Na-X zeolite, and therefore the approximation γ ) 0 can be considered valid. For this same reason it is not feasible to reach the conditions required to yield high L values, where the best results are obtained from the LT analysis, and it is also possible to apply the ST analysis proposed by Hufton and Ruthven (1993). Table 1 shows the results obtained from the moments analysis of the experimental curves along with a comparison with the values obtained from the LT method, for which:
a)
β1
2
2L ; + L(L - 1)
D b ) β12 2 R
(35)
The ST method cannot be applied (Brandani and Ruthven, 1995b). The diffusivity values confirm the results of Eic et al. (1988), and the values obtained for the ratio σ2/µ2 tend to exclude the presence of a surface barrier. At this point all of the more obvious reasons for a possible
Conclusion A new method for the analysis of ZLC desorption curves has been derived which is applicable to both gaseous and liquid systems. The utility of the method has been demonstrated by applying it to the analysis of experimental ZLC curves for benzene on Na-X. When the asymptotic form of the desorption curves is taken into account, the method appears to be quite accurate. It is possible, for gaseous systems, to obtain clear evidence which can be used to discriminate between a diffusion-controlled and a surface barrier controlled process. The moments analysis appears to be particularly useful for the analysis of the experimental desorption curves for strongly adsorbed species, such as the system considered here, for which it is difficult to obtain high values of the parameter L. For these systems the accuracy of the LT method is not high and the ST method proposed by Hufton and Ruthven (1993) is also inapplicable. Furthermore, the moments analysis is the only approximate method available which extracts the relevant parameters using the entire desorption curve. Further validation may be needed to assess the applicability of this approach to liquid systems. Acknowledgment Financial support provided by the Petroleum Research Fund is gratefully acknowledged. Nomenclature a ) intercept of simple exponential A ) model parameter defined in eq 19 b ) rate coefficient of simple exponential, s-1 B ) model parameter defined in eq 19 c ) fluid phase concentration of adsorbate, mol/m3 c˜ ) fluid phase concentration of adsorbate in Laplace domain, mol/m3 c0 ) initial fluid phase concentration of adsorbate, mol/m3 D ) diffusion coefficient, m2/s F ) volumetric fluid flow rate, m3/s I ) integral of the desorption curve, s I0 ) integral of the desorption curve for gaseous systems, s k ) rate coefficient of surface barrier model, s-1 K ) equilibrium Henry constant L ) model parameter defined in eq 5 L′ ) model parameter defined in eq 19 q ) adsorbed phase concentration of adsorbate, mol/m3
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 319 q j ) crystal-averaged adsorbed phase concentration of adsorbate, mol/m3 q0 ) initial adsorbed phase concentration of adsorbate, mol/ m3 r ) radial coordinate in crystal, m R ) radius of adsorbent crystal, m s ) independent variable in Laplace domain t ) time, s t0 ) limit of integration, s Vs ) volume of adsorbent in column, m3 Vf ) volume of fluid in column, m3 Greek Symbols R ) model parameter defined in eq 19 βn ) nth root of eq 6 γ ) model parameter defined in eq 5 µ ) first moment, s µ0 ) first moment for gaseous systems, s σ ) second moment, s2 σ0 ) second moment for gaseous systems, s2
Literature Cited Brandani, S.; Ruthven, D. M. Analysis of ZLC Desorption Curves for Liquid Systems. Chem. Eng. Sci. 1995a, 50, 2055-2059. Brandani, S.; Ruthven, D. M. Analysis of ZLC Desorption Curves for Gaseous Systems. Adsorption 1995b, submitted for publication. Bu¨low, M.; Struve, P.; Mietk, W. Experimental Evidence of the Influence of Sorption-Heat Release Processes on the Sorption Kinetics of Benzene in Na-X Zeolite Crystals. J. Chem. Soc., Faraday Trans. 1 1984, 80, 813-822. Bu¨low, M.; Struve, P.; Mietk, W. Intracrystalline Diffusion of Benzene on Sodium X-Type Zeolite Studied by Desorption Kinetics. Z. Phys. Chem. (Liepzig) 1986, 267, 613-616. Eic, M.; Ruthven, D. M. A New Experimental Technique for Measurement of Intracrystalline Diffusivity. Zeolites 1988, 8, 40-45.
Eic, M.; Goddard, M.; Ruthven, D. M. Diffusion of Benzene in Na-X and Natural Faujasite. Zeolites 1988, 8, 327-331. Germanus, A.; Ka¨rger, J.; Pfeifer, H.; Samulevicˇ, N. N.; Zdˇ anov, S. P. Intracrystalline Self-Diffusion of Benzene, Toluene and Xylene Isomers in Zeolites Na-X. Zeolites 1985, 5, 91-95. Hufton, J. R.; Ruthven, D. M. Diffusion of Light Alkanes in Silicalite Studied by the Zero Length Column Method. Ind. Eng. Chem. Res. 1993, 32, 2379-2386. Hufton, J. R.; Brandani, S.; Ruthven, D. M. Measurement of Intracrystalline Diffusion by Zero Length Column Tracer Exchange. In Zeolites and Related Microporous Materials: State of the Art 1994; Weitkamp, J., Karge, H. G., Pfeifer, H., Holderich, W., Eds.; Studies in Surface Science and Catalysis, Vol. 84; Elsevier: Amsterdam, The Netherlands, 1994; pp 1323-1330. Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley: New York, 1992. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984; pp 242-244. Ruthven, D. M.; Stapleton, P. Measurement of Liquid Phase Counter-Diffusion in Zeolite Crystals by the ZLC Method. Chem. Eng. Sci. 1993, 48, 89-98. Shen, D.; Rees, L. V. C. Diffusivities of Benzene in HZSM-5, Silicalite-I, and Na-X Determined by Frequency-Response Techniques. Zeolites 1991, 11, 666-671. van der Laan, E. Th. Notes on the Diffusion-Type Model for the Longitudinal Mixing in Flow. Chem. Eng. Sci. 1957, 7, 187191.
Received for review May 12, 1995 Revised manuscript received September 8, 1995 Accepted September 19, 1995X IE950287M
X Abstract published in Advance ACS Abstracts, November 15, 1995.
