Langmuir 1999, 15, 3965-3971
3965
Determination of Diffusion Coefficients from Sorption Kinetic Measurements Considering the Influence of Nonideal Gas Expansion Richard Schumacher,*,† Klaus Ehrhardt,‡ and Hellmut G. Karge Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin-Dahlem, Germany Received November 24, 1998. In Final Form: February 8, 1999
The influence of gas-phase kinetics on sorption kinetic measurements under constant volume/variable concentration conditions was investigated by theoretical means. Therefore, an analytical solution of Fick’s law under appropriate boundary conditions was developed taking into account the finite rate of gas expansion in real experiments. This equation allows us to predict the influence of the valve effect on sorption kinetic curves. Reasonable criteria for the applicability of sorption kinetic methods are derived. The equation is suitable for the evaluation of experimental curves with respect to Fickian diffusivity.
1. Introduction Sorption processes in porous materials of various types have been studied extensively. Especially the investigation of sorption kinetics in zeolites has developed into a field of its own due to its importance in catalysis and gas-phase separation processes (see, e.g., Haag et al.1). Nevertheless, different classes of porous materials such as charcoal,2 carbon molecular sieves,3 or aerogels4 have been thoroughly examined, too. Standard methods for the experimental observation of sorption kinetic processes have been the subject of review articles and monographs several times.5,6 Among the most widely used techniques are those methods which operate in a static mode with a constant amount of sorbate within a finite volume. These techniques monitor the response of a sorption system to a stepwise change of the sorbate concentration in the surrounding gas phase by gravimetric or barometric devices. The step-change is usually produced by expanding a well-known quantity of sorbate from the dosing chamber to the sorption chamber. Generally, sorption kinetics is studied in order to determine transport coefficients. An important aspect of the quantitative determination of transport coefficients is the method of evaluation of the experimental data. Most of the earlier studies made use of the xt-approximation7 * To whom correspondence should be addressed. † Present address: Schuit Institute of Catalysis, Fac. of Chem. Engineering, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. ‡ Institut fu ¨ r Angewandte Chemie, Rudower Chaussee 5, 12489 Berlin-Adlershof, Germany. (1) Haag, W. O.; Lago, R. M.; Weisz, P. B. Discuss. Faraday Soc. 1981, 72, 317. (2) Czepirski, L.; Laciak, B.; Holda, S. In Proceedings of the Vth International Conference on Fundamentals of Adsorption; Asilomar Pacific Grove, California, USA, May 13-18, 1995; Knaeble, K. S., LeVan, M. D., Eds.; Kluwer Academic Publishers: Boston, 1996; pp 219-226. (3) Chihara, K.; Suzuki, M.; Kawazoe, K. J. Colloid Interface Sci. 1978, 64, 584. (4) Stumpf, C.; v. Ga¨ssler, K.; Reichenauer, G.; Fricke, J. J. NonCryst. Solids 1992, 145, 180. (5) Post, M. F. M. In Introduction to Zeolite Science and Practice; van Bekkum, H., Flanigen, E. M., Jansen, J. C., Eds.; Elsevier: Amsterdam, 1991. Post, M. F. M. Stud. Surf. Sci. Catal. 1991, 58, 391. (6) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites; John Wiley: New York, 1992.
or the method of moments.8 In more recent work, transport coefficients have generally been determined by fitting a solution of the differential equations under appropriate boundary and initial conditions to the experimental data. This became a convenient method due to the advent of modern desktop computers. However, experimental uptake curves may be influenced by different effects in addition to intracrystalline diffusion. These processes may give rise to experimental errors in the determination of transport coefficients if they are not accounted for during the evaluation procedure. In the case of heat dissipation, several authors were able to present a theoretical analysis of the problem based on the analytical solution of the combined model of mass and heat transfer.9,10 Another effect which is unavoidable in real experiments is connected with the process of gas expansion. It is obvious that an ideal step of the sorbate concentration will not be possible in a real experiment. This may be neglected if the uptake rate is slow enough. However, a considerable error will be caused if the uptake process is fast. This effect has been referred to as valve effect in previous publications.11,12 In the more recent literature, two approaches for the theoretical treatment of this effect have been published. It is possible to solve the complete set of differential equations for the gas phase dynamics and the sorption process simultaneously on the basis of the Volterra formalism.12,13 However, this approach makes the use of complex mathematical procedures necessary, and direct analytical criteria for the limits of the applicability of sorption kinetics techniques for the investigation of transport processes have not been developed. Ro¨denbeck et al.14 have investigated the influence (7) Barrer, R.; Brook, D. Trans. Faraday Soc. 1953, 49, 1049. (8) Radeke, K. H.; Struve, P.; Ehrhardt, K. Z. Phys. Chem. (Leipzig) 1978, 259, 568. (9) Kocirik, M.; Struve, P.; Bu¨low, M. J. Chem. Soc., Faraday Trans. 1 1984, 80, 2167. (10) Lee, L.; Ruthven, D. M. J. Chem. Soc., Faraday Trans. 1 1979, 75, 2406. (11) Bu¨low, M.; Struve, P.; Finger, G.; Redszus, Ch. Ehrhardt, K.; Schirmer, W.; Ka¨rger, J. J. Chem. Soc., Faraday Trans. 1 1980, 76, 597. (12) Micke, A.; Kocirik, M.; Caro, J. J. Chem. Soc., Faraday Trans. 1, 1990, 86, 3087. (13) Micke, A.; Struve, P.; Kocirik, M.; Zikanova, A. Collect. Czech. Chem. Commun. 1994, 59, 989.
10.1021/la9816454 CCC: $18.00 © 1999 American Chemical Society Published on Web 05/07/1999
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Schumacher et al.
φtotal(t) ) [Vgascv + Vsorbentq j (t ) 0)] + [Vgas(c0 - cv)] ) 0; t < 0 φv + φ0 × (5) 1; t g 0
{
Figure 1. Scheme of a sorption kinetic apparatus operating at constant volume and variable pressure.
of the valve effect on tracer exchange experiments14 in some detail. However, their study has been focused on single-file systems only. A more quantitative investigation of the influence of the valve effect on conventional sorption uptake experiments has so far not come to the attention of the authors. In this publication, we will present an analytical solution of the model equations assuming sorption into a spherical sorbent controlled by Fickian diffusion under constant volume/variable pressure conditions and taking into account the valve effect. This enables us to study the influence of the valve effect on the sorption curves in detail and to derive quantitative criteria for the influence of the valve effect on sorption kinetic measurements. Additionally, an improved method for the determination of diffusivities is provided. 2. Model
where Vgas ) Vdos + Vsorpt. In eq 5, φv is the amount of sorbate which was included in the apparatus before the experiment, while φ0 is the amount of sorbate added during the experiment. If we assume the sorbate to expand linearly into the sorption volume, the valve effect can be accounted for in the following way:
{
0; t < 0 φtotal(t) ) φv + φ0 × t/tδ; 0 e t < tδ 1; tδ e t
(
)
(1)
where q(r,t) is the sorbate concentration within the spherical crystallite, and r and t are the space and time coordinates, respectively. Concerning the abbreviations used in this work, the reader may also refer to the glossary provided at the end of this article. The boundary conditions are given by eqs 2 and 3.
(6)
For mathematical convenience, the derivative of eq 6 is used in the following calculations. It is obvious that the set of equations has to be solved for the different time ranges separately. The first part is trivial as the system is generally assumed to be in equilibrium prior to the beginning of the experiment.
q(r,t) ) qv ) Kcv ) const; t e 0
(7)
This is also used as the initial condition for the solution in the second time range. The initial conditions for the third time range have to be derived from the solution for the second time range.
The principle of operation of a sorption kinetic apparatus of the constant volume/variable pressure type is shown in Figure 1. In the following derivation, the uptake process is assumed to be controlled by intracrystalline diffusion according to Fick’s law:
∂q ∂2q 2 ∂q )D 2+ ∂t r ∂r ∂r
}
c(3)(t)tδ) ) c(2)(t)tδ)
(8)
q(3)(r,t)tδ) ) q(2)(r,t)tδ); 0 e r e R
(9)
3. Solution of the Model Equation 3.1. Period of Gas Expansion. To simplify further calculations, the variables are transformed to reduced variables. The space coordinate is written as
y)
r ; 0eye1 R
(10)
the normalized time is
τ ) βt
(11)
D R2
(12)
where
∂q | )0 ∂r r)0
(2)
q(r ) R,t) ) Kc(t)
(3)
β)
where K is the differential adsorption constant and c(t) represents the sorbate concentration in the gas phase. According to the diagram presented in Figure 1, the following equation can be given for the total amount of sorbate, φtotal, within the apparatus at any time, t:
The sorbate concentration within the sorbent is transformed into
j (t) ) φtotal (4) Vdoscdos(t) + Vsorptcsorpt(t) + Vsorbentq
and the reduced sorbate concentration in the gas phase is
where q j (t) is the average concentration of sorbate within the crystallite. In earlier models, an instantaneous concentration jump cv ) c(t ) 0-) f c0 ) c(t ) 0+) was assumed. (14) Ro¨denbeck, C.; Ka¨rger, J.; Hahn, K. Collect. Czech. Chem. Commun. 1997, 62, 995.
Q(y,τ) )
r(q(r,t) - qv) RK(c0 - cv)
P(τ) )
q(r,t) - qv
)y
c(t) - cv c0 - c v
K(c0 - cv)
(13)
(14)
Equations 1-3 and 6 can now be written in terms of the reduced coordinates. Laplace-transformation according to eq 15a/b is performed.
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Langmuir, Vol. 15, No. 11, 1999 3967
H(s) ) L{P(τ)}
(15a)
h(y,s) ) L{Q(y,τ)}
(15b)
Finally, for Fick’s law eq 16 is derived from eq 1. For the mass balance in the second time range, eq 17 can be derived from eq 6.
sh(y,s) )
∂2h(y,s)
(16)
∂y2
∂h -h s2H(s) ) -3Rs ∂y
[
1 + y)1 βtδ
]
(17)
where R ) KVsorbent/Vgas. The boundary condition eq 3 transforms into
h(1,s) ) H(s)
(18)
The solution of these equations is based on the following approach:
h(y,s) ) A(s) sin(x-sy)
where the value of un is determined by
tan(un) )
h(y,s) ) sin(x-sy) 1 (20) βtδ [3Rs]x-s cos(x-s) + [s2 - 3Rs] sin (x-s)
q j (t) ) qv +
∑n ress [h(y,s)es ]
(21)
n
n
K(c0 - cv)
(1 + R)βtδ
c(2)(t) ) cv +
(c0 - cv)
{
r
R r
9R(1 + R)un2 + un4 sin(un)
∑n
2
(
}
2βt
n
(25)
{
R
+
∑n ane-u
}
2βt
n
(26)
In both equations, the parameters an are given by
an )
6(1 + R)
(27)
9R(1 + R)un2 + un4
3.2. Range of Constant Amount of Sorbate. From eqs 23 and 26 the appropriate initial conditions for the third time range can be calculated according to eqs 8 and 9. c and q have to be substituted as
Q ˜ (y,τ) )
r q - q1 R K(c1 - cv)
(28)
and
P ˜ (τ) )
c - cv c 1 - cv
(29)
)
( )
τ ) β(t - tδ) c1 ) c(2)(t ) tδ) q1(r) ) q(2)(r,t ) tδ) For the sake of clearness, the reduced parameters of this time range will be written as P ˜ and Q ˜ . Other substitutions are identical to those defined above. Fick’s law is written as
2
+ 6R (1 + R) 1 1 R βt + 2 1 + R (1 + R) 6 10 r sin un 6R R βtδ
ane-u ∑ n)1
βt - R
(22)
1 - sn/3R
K(c0 - cv)
15(1 + R)
+
where
x-sn
When the variables are resubstituted according to eqs 10-14, the solution for the concentration profile within the sorbent is given as eq 23. For convenience, sn is substituted by sn ) -un2.
q(2)(r,t) ) qv +
∞
1
βtδ(1 + R) 15(1 + R)
The poles sn are determined by the s values at which the denominator of eq 20 vanishes; this means
tan(x-sn) )
{
βt -
This is the quantity which is, for example, monitored in a gravimetric experiment. In case of barometric experiments, the pressure in the surrounding gas phase is monitored. At constant temperature it is proportional to the sorbate concentration as given by
This equation can easily be transformed from the Laplace domain into the time domain by Heaviside’s theorem (see, e.g., Doetsch15).
Q(y,τ) )
(24)
1 + un2/3R
The average concentration of sorbate within the sorbent can then be calculated after integrating eq 23 with respect to r.
(19)
Equation 16 is fulfilled directly. In eq 17, H(s) can be substituted by the relation given in eq 18 and A(s) can then be determined by inserting eq 19 into the combined equation. The mathematical procedure is trivial but tedious and will therefore not be given in detail. The solution of eqs 16-18 is then
un
e
}
-un2βt
{
}
∂2q1 2 ∂q1 ∂2Q ˜ y ∂Q ˜ ) 2 + + ∂τ y ∂y K(c1 - cv) ∂y2 ∂y
(30)
the mass balance is
(23)
(15) Doetsch, G. Handbuch der Laplace-Transformation; Birkha¨user: Basel, 1972.
∂q1 3R ∂Q ˜ ∂P ˜ )| -Q ˜ - 3R ∂τ ∂y K(c1 - cv) ∂y y)1
[
]
y)1
(31)
and the boundary conditions at the surface of the sorbent
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Schumacher et al.
are
Q ˜ (y ) 1,τ) ) P ˜ (τ) - 1
(32)
Transformations and substitutions similar to those described above lead to
(3)
q (r,t) ) qv +
{
βtδ
βtδ
1+R
∑ r n
r sin un 6R R -un2βt e 9R(1 + R)un2 + un4 sin(un)
( )
K(c0 - cv)
R
-
6R
×
9R(1 + R)un2 + un4 r sin un R -un2β(t-tδ) -un2βtδ e [e - 1] sin(un)
( )
}
K(c0 - cv) 1+R
{
1-
1 βtδ
∞
an[e-u ∑ n)1
n
r sin un R -un2βt ) e lim Rf0 3R(9R(1 + R)u 2 + u 4) u cos(u ) n n n n r sin un 2 R -un2βt (39) lim 3 e Rf0 u cos(un) n
(33)
2
}
( )
From eq 24 it is obvious that un ) nπ in the limit of R f 0. Therefore, eq 23 becomes
(34)
while the sorbate concentration in the gas phase is given by
q(r,t) ) qv +
1+R
{
1+
R
∑an[e-u βt n
2βt δ
n
- 1]e-un β(t-tδ) 2
δ
}
βtδ r
(35)
where un and an are determined according to eqs 24 and 27. 4. Discussion 4.1. Consistency of the Derived Solutions. It is instructive to check the formulas for some of the limiting cases. If βtδ is small, the exponential term in eq 34 can be approximated up to the second order in βtδ by
exp(-un2βtδ) - 1 ≈ (1 + (-un2βtδ) + (-un2βtδ)2 - 1) ) (-un2βtδ(1 - un2βtδ)) (36) where the factor -un2βtδ is canceled by the identical factor in the denominator of eq 34. In the limit of βtδ f 0 this leads to
q j (t) ) qv +
K(c0 - cv) 1+R
{
1+
∞
∑ n)1
6(1 + R)
2
9R(1 + R) + un
}
e-un βt
2
(37)
This equation is equivalent to the formula based on the assumption of an ideal step change of the gas-phase concentration.8,16 It is furthermore interesting to analyze the concentration profile during the period of gas expansion, i.e., the second range of time. If the volume of the gas phase is infinite, the sorption process at constant amount of (16) Crank, J. Mathematics of Diffusion; Clarendon Press: Oxford, 1975; Chapter 6, pp 85ff.
{
K(c0 - cv) R
c(3)(t) ) cv + c 0 - cv
( )
6R(3R + un2)
- 1]e-un βt
2βt δ
(38)
Since sin(un) can be transformed according to eq 24 this leads to
The mean concentration of sorbate averaged over the sorbent volume is given by
q j (t) ) qv +
sorbate becomes equivalent to the sorption process at constant external concentration. Mathematically, this is described as the limit R f 0. This transition does not contain any difficulties except for the terms of the series. A single term of the series in eq 23 is given by
r2
1
6 6R 2 r 2 2 (-1)n sin 2π e-n π βt R n3π3 + βt -
2
}
( )
∑n
(40)
This is equivalent to the respective formula published by Crank.16 4.2. Influence of the Valve Effect on the Uptake Curves. It is interesting to examine how the valve effect influences experimental uptake curves. Figures 2-4 show calculations for the development of the sorbate concentration in the gas phase and the mass uptake in the sorbent. In each case, tδ is assumed to be 0.1 s; the time constant for diffusion-limited sorption was varied as indicated in the figure captions. The results can be applied to experiments with sorbents of different diameter, if the curve with the same value for β is chosen according to eq 12. Three examples are studied for the cases of strong, intermediate, and weak adsorption, the parameter R equals 10, 1, and 0.1, respectively. All curves of a diagram approach identical equilibrium values. Obviously, the shape of the uptake curves and the maximum concentration in the gas phase are severely affected. Especially the drop of the maximum gas-phase concentration proves that in some cases the experimental conditions may deviate significantly from the ideal conditions. 4.3. Influence on the Shape of the Observed Uptake Curves. Especially for barometric devices, the influence of the valve effect on the shape of the experimental curves in the third time range is important. It can be studied if the appropriate dependency is plotted in reduced coordinates. ∞
∑ an[e-u n)1
2βt δ
n
(3)
Ω)
c (t) - c(∞) c(2)(tδ) - c(∞)
)
∞
∑ ane
n)1
-un2βtδ
- 1]e-un β(t-tδ) 2
(41) -
1 15(1 + R)
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Langmuir, Vol. 15, No. 11, 1999 3969
Figure 2. Sorption kinetic curves for strong adsorption (R ) 10), all curves are approaching identical equilibrium values: (a) normalized concentration in the gas phase, D/R2)10n s-1, n varying between -6 and 1 from the highest to the lowest curve; (b) normalized uptake, n varying between 1 and -6 from the highest to the lowest curve.
with an and un as defined above and
Ω ) Ω(R,β,tδ,t)
Figure 3. Same as Figure 2 for intermediate adsorption (R ) 1): (a) n varying between -5 and 1 from the highest to the lowest curve; (b) n varying between 1 and -5 from the highest to the lowest curve.
reduced form,
Θ) (42)
This relation is plotted in Figure 5 as a function of (t tδ), where R ) 1, β ) 0.01, and tδ varies in the range tδ ) 1.0-10-3. It is obvious that the shape of the curve is changed by the variation of tδ. If tδ is getting shorter, the initial slope of the uptake curves becomes steeper. This effect can be explained phenomenologically as a consequence of sorbate preloading during gas expansion. According to Fick’s law, the sorption rate is directly proportional to the concentration gradient at the outer boundary of the sorbent. If the amount of sorbate loaded during gas expansion is large, the concentration gradient at the boundary of the crystal will be less steep and the sorbate flux will be lower. This has consequences for practical measurements. If fitting of the conventional analytical solution8,16 is chosen for the determination of diffusion coefficients and the valve effect is not taken into account, then the evaluation of uptake curves will yield diffusivities smaller than their true value. 4.4. Applicability of Sorption Uptake Methods for the Determination of Diffusion Coefficients. The influence of the valve effect on the experimental uptake curve can be determined quantitatively by the relative amount of sorbate sorbed at time tδ, when the sorbate expansion is finished. It is useful to study the average sorbate concentration as given from eqs 25 and 34 in a
q j (t)tδ) - qv q j (t)∞) - qv
1-
1
{
)
1
βtδ 15(1 + R)
∞
-
∑ n)1
6(1 + R)
4
9R(1 + R)un + un
}
e-un βtδ 2
2
(43)
Θ will be within 0 and 1, giving the fraction of sorbate sorbed during gas expansion. If it is close to 0, the valve effect is negligible and the sorption kinetics is controlled by diffusion. If it is close to 1, the uptake rate is mainly controlled by the kinetics of gas expansion, whereas diffusion is too fast to be rate-determining. In that case, any evaluation of the experimental data with respect to a diffusion coefficient is impossible. Θ is a function of D, R, tδ, and R. As three of the parameters occur in combination, it is possible to reduce the number of variables to two.
Θ ) Θ(D,R,tδ,R) ) Θ(f,R); f )
Dtδ R2
(44)
The function Θ(f) is plotted in Figure 6 for various values of R. The Θ(f) curves converge when R approaches zero. Apparently, there is a methodical limit for the experimental determination of diffusion coefficients. Evaluation is practically impossible if f > 0.1, which provides a general upper limit for the diffusion coefficients to be
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Schumacher et al.
Figure 6. Fraction of sorbate within the sorbent after full equilibration of the gas phase; R ) 100, 50, 20, 10, 5, 2, 1, 0.5, 0.2, 0.1, 0.05, 0.02, and 0.01 from left to right.
completion of gas expansion might be considered to be negligible, while experiments with uptakes between 5 and 50% during gas expansion might still be evaluated, if the valve effect is taken into account. From Figure 6 it can be concluded that in this case the range in which sorption uptake curves can be reliably evaluated is extended by about 2 orders of magnitude. 5. Conclusions
Figure 4. Same as Figure 2 for weak adsorption (R ) 0.1): (a) n varying between -3 and 2 from the highest to the lowest curve; (b) n varying between 2 and -4 from the highest to the lowest curve.
Figure 5. Normalized gas-phase concentration curves for tδ ) 1, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, and 0.001 from the highest to the lowest curve; all curves approaching identical equilibrium values.
determined with a sorbent with a given radius. As has been pointed out above, the limit R ) 0 yields the theoretical description of sorption uptake at constant boundary conditions so that f must be smaller than 0.1 also in the case of dynamic adsorption methods as, for example, the FTIR method.17 It is a necessary condition, however, especially in case of strongly adsorbing samples (large R), the practical limit can be considerably lower. It is now in the hands of the experimentalist to decide whether the influence of the valve effect is negligible or whether it should be taken into account when evaluating the experimental data. Uptakes of less than 5% prior to (17) Karge, H. G., Niessen, W. Catal. Today 1991, 8, 451.
The results provide insight into the way sorption kinetic measurements are affected by the finite rate of gas expansion. If compared to the case of an ideal concentration step, sorption uptake rates decrease due to the valve effect. No determination of diffusivities is possible if Dtδ/R2 > 0.1, which has to be regarded as an overall limit for the applicability of sorption kinetic devices. However, the limit is shifted to much lower values for strongly adsorbing samples (see above). The equations developed above are suitable for the determination of diffusion coefficients from sorption kinetic experiments of the constant volume/variable gasphase concentration type even if the valve effect is not negligible any more. Experimental results based on the proposed method of evaluation have been published elsewhere.18 It can be estimated that the applicability of sorption uptake experiments is extended by roughly 2 orders of magnitude. Apart from classical gravimetric or barometric devices, the model is also suitable to describe, for example, experiments from the single-step frequency response type. Glossary A(s) an c(t) cv c1
parameter of the solution in the Laplacedomain; see eq 19 set of parameters of the time-domain solution; see eq 27 gas-phase concentration gas-phase concentration before the beginning of the experimental run initial gas-phase concentration at the beginning of the third time range (after gas expansion is completed)
(18) Schumacher, R.; Lorenz, P.; Karge, H. G. In Progress in Zeolite and Microporous Materials, Proc. 11th Intern. Conf. Zeolites, Seoul, Korea, August 12-17 1996; Chon, H., Ihm, S. K., Uh, Y. S., Eds.; Elsevier: Amsterdam, 1997. Schumacher, R.; Lorenz, P.; Karge, H. G. Stud. Surf. Sci. Catal. 1997, 105, 1747-1754.
Diffusion Coefficients from Kinetic Measurements c(2)(t), c(3)(t) gas-phase concentration during and after expansion, respectively, see eq 8 c0 theoretical maximum value of gas-phase concentration in absence of valve effect D diffusivity f dimensionless parameter describing the uptake kinetics; see eq 44 H(s) Laplace-transformed of the reduced gas-phase concentration h(y,s) Laplace-transformed of the reduced sorbate concentration in the adsorbed state K differential adsorption constant P(t), P ˜ (t) reduced gas-phase concentration during and after gas expansion Q(y,t), Q ˜ (r,t) reduced sorbate concentration in the sorbent during and after gas expansion q(r,t) sorbate concentration in the adsorbed state q1(r) initial sorbate concentration profile in the adsorbed state for the third time range sorbate concentration in the sorbent before the qv start of the experimental run sorbate concentration in the adsorbed state q(2)(r,t), during and after gas expansion, respectively q(3)(r,t) R radius of sorbent particles r radial coordinate s coordinate in the Laplace domain
Langmuir, Vol. 15, No. 11, 1999 3971 sn t tδ un Vgas Vdos, Vsorpt Vsorbent y R β φtotal φ0 φv τ Θ Ω LA9816454
poles of the solution in the Laplace domain time valve opening time parameter of the time domain solution; see eq 24 gas-phase volume volumes of the dosing and sorption chamber volume of the adsorbent reduced space coordinate distribution parameter, ratio of absolute amounts of sorbate in the sorbent and in the gas phase diffusional time constant, see eq 12 total amount of sorbate within the apparatus, see eq 4 amount of sorbate added during a single experimental run amount of sorbate within the apparatus before the start of the experimental run reduced time coordinate; see eq 11 fraction of sorbate adsorbed during gas expansion; see eq 43 reduced curve shape; see eq 41
