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The effect of the concentration dependence of diffusivity on zeolitic sorption curves
Chemical Engineering Science, 1972, Vol. 27, pp. 417-423.
Pergamon
Press.
Printed
in Great
Britain
The effect of the concentration dependence of diffusivity on zeolitic sorption curves D. R. GARG and D. M. RUTHVEN Department of Chemical Engineering, University of New Brunswick, Fredericton, (Received
N.B., Canada
4 May 197 I)
Abstract-Solutions of the transient diffusion equation are presented for sorption in a system of spherical particles in which the diffusivity of the sorbate varies with concentration. The functional forms of the concentration dependence of diffusivity considered are commonly observed for the sorption of gases in zeolites. It is shown that, for such systems, a difference between the rates of adsorption and desorption is to be expected. Effective diffusivities are calculated by comparing the theoretical sorption curves for the concentration dependent diffusivity with the standard solution of the diffusion equation for a constant diffusivitv. The validity of the analysis is confirmed by comparison with experimental data. KINETICS of zeolitic sorption are generally studied experimentally by following the transient response curves obtained when a sample of the zeolite is subjected to a step change in sorbate pressure. The sorption curves are commonly interpreted by comparison with standard solutions of the Fickian diffusion equation. If it is assumed that the sample of zeolite crystals can be treated as an assemblage of uniform spheres of radius a, the relevant Fickian equation is:
THE
If the diffusivity is independent centration Eq. (1) becomes:
ac_
at-
D
a2c
2 ac
(aP+r*ar>
and the solution, expressed fractional uptake is [ I]:
2 =
1-
of sorbate con-
$5 f
(3)
in terms
of the
exp [-n2*Dt/a2]
(4)
?I=1 where ml = mass of sorbate adsorbed during time f. m, = mass of sorbate adsorbed as t 4 co.
where c= t= r= D =
sorbate concentration time radial coordinate diffusivity.
For a step change in sorbate partial pressure from zero to p or from p to zero at time t = 0, the appropriate initial and boundary conditions, assuming instantaneous equilibration at the crystal surface, are: c (r, 0) = 0 (adsorption) or co (desorption) c (a, f ) = c,, (adsorption) or 0 (desorption)
$(O,t)
= 0
(2)
Vol. 27 No. 2
-P
or desorbed
Since in general zeolitic diffusion coefficients are strongly concentration dependent, this analysis is a valid approximation only when the sorption curve is measured over a small differential step change in sorbate concentration. In order to interpret correctly the sorption curves measured over larger changes in concentration it is necessary to obtain the solution of the diffusion equation for the appropriate functional dependence of the diffusivity on sorbate concentration. Such solutions are not generally available although a number of special cases, mostly
417 C.E.S.
or desorbed
D. R. GARG
and D. M. RUTHVEN
Although not exact, these equations provide useful approximations for many gas-zeolite systems. It is convenient to introduce the following dimensionless variables:
for semi-infinite media, have been given by Crank [l] and by Fujita[2,3]. THEORETICAL
ANALYSIS
Recent studies of the sorption and diffusion of light hydrocarbons in type A zeolite have shown that, for these systems, the concentration dependence of the diffusivity is due to the nonlinearity of the equilibrium isotherm [4,5]:
C = c/c,, R = r/a, T = Dot/a2 f0
D1=+= 0
’ D,
- dC
,u=sR
f’DdC
*
JO
RTalna
D .-alnp O alnc
D=kx=
(5)
For the Langmuir system Eq. (1) becomes:
where
au _ @u/R.d2U
iv--
(Y= sorbate activity k = drag coefficient (constant) Do = y
where /3 = -In (1 - A) and A = co/c,.
= limiting diffusivity as c + 0.
The corresponding system is:
If the equilibrium isotherm obeys the Langmuir equation:
au
(6) a lnp -=a In c
1 1 -c/c,
(7)
D=Do
aT=
where b = adsorption equilibrium constant c, = saturation concentration. For Volmer’s equations are:
isotherm
a lnp a=
(I$
the
8=
c/c,.
{l-h(l--~lR)}~ (l--x)2
for the Volmer
a2u *aR2
(13)
u(R,O) =Oor 1 ~(1, T) = 1 or0 ~(0, T) = 0.
(14)
The solutions of these equations were obtained by the Crank-Nicholson method[6] and also by corresponding the use of an explicit finite difference scheme. Both methods gave results which were in good agreement and the theoretical sorption curves, calculated by integration of the concentration (9) profiles, are shown in Figs. 1 and 2 for several values of the parameter A. For small values of A, adsorption and desorption curves lie close to(10) gether and for A = 0 the curves become coincident with the analytical solution for the constant diffusivity case (Eq. 4). For larger values of A the (11) curves for adsorption and desorption diverge significantly and it is clear that desorption is much slower than adsorption. The initial portions of the
D= (1-?c,,, where
expression
and the modified initial and boundary conditions are:
(8)
1 - c/c,
(12)
dR2
418
The effect of the concentration dependence of diffusivity
O-6
Parameter
-
:A
Adsorption oesorptian
-0.4
0
O-I
0.2
0.3
0.4
0.5
0.6
0.7
&qz?
Fig. 1. Theoretical sorption curves for Langmuir system.
Parameter:
E1E”
0
0.I
02
03
-
Adsorption
---
ossorption
0.4
X
05
Fig. 2. Theoretical sorption curves for Volmer system.
O-7
sorption curves are linear in X6 In the case of the adsorption curves geometric similarity extends throughout and the curves obtained at high values of A do not differ significantly in shape from the constant diffusivity curve. For the Langmuir system the desorption curves show the same type of geometric similarity although a slight difference in shape becomes apparent for large values of A. The desorption curves for the Volmer system are however of a significantly different shape. Such behaviour is typical of systems in which the diffusivity increases with concentration and Crank[l] has pointed out that in such systems the shape of the desorption curves is much more sensitive than that of the adsorption curves. For the Langmuir system the geometric similarity of the sorption curves makes it possible to calculate, for any value of A, an effective diffusivity D, such that, when plotted in terms of Det/a2, the adsorption and desorption curves for all values of A reduce to a single curve which is formally identical with the curve for the constant diffusivity case. In the Volmer system the same type of representation is a good approximation for the adsorption curves but it is less satisfactory for the desorption curves due to the difference in shape. The accuracy of this type of representation may be judged from Fig. 3 in which computed points corresponding to sorp-
8
J_
JS
Fig. 3. Theoretical sorption curves plotted in terms of (D&a*).
419
D. R. GARG and D. M. RUTHVEN
tion curves for various values of A are compared with the constant diffusivity case. The corresponding plot of 0,/D, vs. h is shown in Fig. 4. The effective diffusivities differ very significantly from the mean diffusivity, defined by:
profile is significantly altered when the diffusivity is concentration dependent. The geometric simi-
which gives for the Langmuir system:
-Do D=Tln
1 l--X C-J
(16)
and for the Volmer system: D=+
D
(17)
Typical concentration profiles for adsorption and desorption with a Langmuir concentration dependent diffusivity are compared with the corresponding profiles for the constant diffusivity case in Figs. 5 and 6. The comparison is made at fixed values of concentration at the particle centre and it is evident that the shape of the 5c
,-
4c I
1
0
0.2
0.4
0.6
0.8
r/a
Fig. 5. Concentration profiles during adsorption (Langmuir system). h Dot/d : 20 10-9 0.061 O-185 3 30.9 0.0362 4 40 0.055.
-
Legend
30
I 2 3 4
Adsorption Volmer Desorption I Adsorption Desorption I Langmuir
0.6 -
0.6-
C
0.4 -
0.2 -
0
0.2
0.4
0.6
0.6
I3
r/a
I.0 0
0.2
0.4
0.6
0.6
I
A
Fig. 4. Effective diffisivity as function of parameter A.
Fig. 6. Concentration profiles during desorption (Langmuir system). x Dot/a2 : 20 10.9 0.0125 o-055 3 30.9 0.121 4 40 0.185.
420
The effect of the concentration
larity of the sorption curves is thus seen to be fortuitous in the sense that it does not involve similarity of the actual concentration profiles. These results have important practical implications since it follows that experimental sorption curves may be analyzed using the standard solutions of the Fickian diffusion equation but the diffusivity so obtained will be an effective diffusivity which will depend on the size of the step change in concentration over which the sorption curve is measured (i.e. the value of A). Furthermore, for the same step change in concentration, the value of D, calculated from the adsorption curve will be greater than the corresponding value calculated from the desorption curve. The analysis assumes an initial step change in surface concentration between the limits 0 and c,, but, by a suitable change of variable, it may be shown that the same solutions are applicable when the initial and final concentrations are both finite (cl < c,) with the parameter A = (cZ- CJC, and D,, equal to the diffusivity at the lower concentration cl. Thus, if the value of c, is known from the equilibrium isotherm, the value of DJD,, may be found from Fig. 4, and the limiting diffusivity at zero sorbate concentration (or at cl) may thus be calculated from the effective diffusivity, obtained from an experimental sorption curve measured over a relatively large step change in sorbate concentration.
dependence of diffusivity COMPARISON
WITH
EXPERIMENTAL
DATA
It is implicitly assumed in the analysis that the temperature of the zeolite remains constant during the sorption process but, when the step change in sorbate concentration is large, a significant thermal effect is to be expected as a result of the heat of sorption. Such effects may be expected to be less important in a flow system than in a closed system and the experimental data of Kondis[7] for the sorption of ethane in 4A zeolite, which were obtained in a continuous flow apparatus using ethane-helium gas mixtures, are therefore especially suitable for comparison with the theory. Such a comparison, which is shown in Figs. 7 and 8, and Tables 1 and 2 for the experimental data obtained at 25°C supports the conclusions of the theoretical analysis. A Langmuir plot of the equilibrium data showed that, over the relevant range of pressure, the isotherm may be satisfactorily fitted by a Langmuir equation with c, = 0.085 g/g. Runs 19 and 20 were carried out at low ethane concentrations and it is evident from Fig. 7 that there is no significant difference between the adsorption and desorption curves. (For the sake of clarity no distinction is drawn between the data points for runs 19 and 20 in Fig. 7.) Both curves may be approximately represented by the theoretical curve for a system of uniform spherical particles (Eq. 4) with a constant diffusivity given by
Experime&+
data
Kondis (ethone
of
on 4A)
0 Adsorption x Desorption Theoretical (Equation D,/02=2~45
25
Fig. 7. Comparison
of experimental
and theoretical sorption concentrations.
421
curves
curve 4) x IO-*set-’
30
35
for ethane in 4A zeolite at low
D. R. GARG and D. M. RUTHVEN
0
0’
0. 0
4
l
/
xcd
H~/x~perimen+aI Kandis
l’
.
(ethane
data of on 4A)
Adsorption
Run no. 86 IGA
Desarption
0
x
. a
104 D -
Theoretical D,,/a’=
0
Fig. 8. Comparison
of experimental
20
I5
IO
curves
x 10e4 SW-’
I
1
5
2.45
and theoretical sorption concentrations.
b
I
25’
30
curves for ethane in 4A zeolite at high
Table I. Details of sorption curves [7] for Fig. 7
20A
PI Mm) P, (atm) c1 Wg) c2 (g/g) x _ kc1I cs Del& Do/a2 (Set-I)
19A
20D
0
19D
0.0224 0 0.011
0.0224 0 0.011 0
0 0.0465 0 0.022
0.0465 0 0.022 0
0.129
0.129
0.258
0.258
1.15 2.45 x lo+
1.05 2.45 X lo-*
1.30 2.45 x lo-’
1.10 2.45 x 1O-4
Table 2. Details of sorption curves [7] for Fig. 8
86A
86A
103A
104D
PI (atm) pZ (atm) c1 Wg) c:!(g/g)
0 0.98 0 0.076
0.98 0 0.076 0
0 0.985 0 0.076
0.985 0 0.076 0
h = /c2-c1I c, De/D,
0.90
0.90
0.90
0.90
Do/a2 (Set-‘)
3.70 1.40 2.45 X 1O-4 2.45 X 1O-4
D/a2 = 2-45 X lop4 set-‘. The small deviation which is observed between the experimental and theoretical curves is probably due to the effect of crystal shape and size distribution [5].
3.70 2.45
x 1O-4
2.45
1.40 x 1O-4
In runs 86, 103A and 104D, the step change in concentration is much larger and it may be seen from Fig. 8 that the rate of adsorption is much faster than the rate of desorption, although both
422
The effect of the concentration
rates are significantly faster than at the low concentrations. The theoretical curves shown in Fig. 8 were calculated using the value of &,/a2 = 2.45 X 1O-4set-’ derived from the low concentration data (Fig. 7) and the value of D,/D, obtained from Fig. 4 with the value of A from the equilibrium isotherm. For the desorption curves the agreement between theory and experiment is excellent. Agreement is slightly less good for the adsorption curves but the difference is probably mainly attributable to small thermal effects. Such effects may be expected to be more significant during adsorption since the process is faster. CONCLUSION
For the sorption of light paraffins in Linde 5A zeolite Loughlin, using small differential changes in concentration, observed no significant difference between the rates of adsorption and desorp-
dependence of diffusivity
tion [5] whereas Eberley, whose measurements were carried out over rather larger changes in sorbate concentration, found the diffusivities for desorption to be much smaller than for adsorption[8]. This result is in qualitative agreement with the present analysis although it seems likely that in Eberley’s measurements the difference in sorption rates was further magnified by thermal effects. Within the limits imposed by the requirement for isothermal conditions the present analysis would appear to be quite widely applicable and, for systems in which it is difficult to make sorption measurements over small differential changes in sorbate concentration it provides a valuable method of calculating the limiting diffusivities at zero concentration. Acknowledgement-The financial support of the National Research Council of Canada is gratefully acknowledged.
REFERENCES
111CRANK J., The Mathematics ofDi’ision. Oxford University Press, (1956). PI FUJITA H., Text. Res. J. 1952 22 757. [31 FUJITA H., Text. Res. J. 1952 22 823. 197167 1661. (41 RUTHVEN D. M. and LOUGHLIN K. F., Trans. FaradaySoc. PI RUTHVEN D. M. and LOUGHLIN K. F., Nature 19712%(11)69. 161 CRANK J. and NICHOLSON P., Proc. Camb. ohil. Sot. 1950 76 634. [71 KONDIS E. F., Ph.D. Thesis, Northwestern University (1968). PI EBERLEY P. E., Jr. Ind. Engng Chem. Fundls, 1969 8 (I), 25. Resume-Les auteurs presentent des solutions a l’equation transitoire de dithtsion pour la sorption dans un systeme de particules spheriques a I’interieur duquel la ditfusivitt du produit dabsorption varie avec la concentration. On observe les formes fonctionnelles de la dependance de la concentration de la ditfusivitt pour la sorption de gaz dans des zeolites. On demontre que l’on peut s’attendre pour de tels systemes a une ditference entre les taux d’adsorption et de dtsorption. On calcule les diifusivites en comparent les courbes theoriques de sorption pour la ditfusivite dependante de la concentration a la solution standard de l’equation de diffision pour une ditfusivite constante. Les auteurs confirment la validite de l’analyse en la comparant aux don&es expCrimentales. Zusammenfassung-Es werden Losungen der momentanen Ditfusionsgleichung fur die Sorption in einem System spharischer Teilchen datgelegt, in welchem sich das Ditfusionsvermdgen des Sorbates mit der Konzentration Pndert. Die in Betracht gezogenen funktionellen Formen der Abhangigkeit des Diifusionsvermogens von der Konzentration werden gewijhnlich an der Sorption von Gasen in Zeolitehn beobachtet. Es wird gezeigt, dass bei solchen Systemen ein Unterschied zwischen Adsorptions- und Desorptionsraten zu erwarten ist. Die nutzbaren DiIfusionsvermGgen werden errechnet durch Vergleichder theoretischen Sorptionskurven filr das von der Konzentration abhdngige Ditfusionsvermiigen mit der Normallijsung der Ditfusionsgleichung fur ein konstantes Ditisionsvermiigen. Die Gtiltigkeit der Analyse wird durch einen Vergleich mit Versuchsdaten be&it&.
423
Pergamon
Press.
Printed
in Great
Britain
The effect of the concentration dependence of diffusivity on zeolitic sorption curves D. R. GARG and D. M. RUTHVEN Department of Chemical Engineering, University of New Brunswick, Fredericton, (Received
N.B., Canada
4 May 197 I)
Abstract-Solutions of the transient diffusion equation are presented for sorption in a system of spherical particles in which the diffusivity of the sorbate varies with concentration. The functional forms of the concentration dependence of diffusivity considered are commonly observed for the sorption of gases in zeolites. It is shown that, for such systems, a difference between the rates of adsorption and desorption is to be expected. Effective diffusivities are calculated by comparing the theoretical sorption curves for the concentration dependent diffusivity with the standard solution of the diffusion equation for a constant diffusivitv. The validity of the analysis is confirmed by comparison with experimental data. KINETICS of zeolitic sorption are generally studied experimentally by following the transient response curves obtained when a sample of the zeolite is subjected to a step change in sorbate pressure. The sorption curves are commonly interpreted by comparison with standard solutions of the Fickian diffusion equation. If it is assumed that the sample of zeolite crystals can be treated as an assemblage of uniform spheres of radius a, the relevant Fickian equation is:
THE
If the diffusivity is independent centration Eq. (1) becomes:
ac_
at-
D
a2c
2 ac
(aP+r*ar>
and the solution, expressed fractional uptake is [ I]:
2 =
1-
of sorbate con-
$5 f
(3)
in terms
of the
exp [-n2*Dt/a2]
(4)
?I=1 where ml = mass of sorbate adsorbed during time f. m, = mass of sorbate adsorbed as t 4 co.
where c= t= r= D =
sorbate concentration time radial coordinate diffusivity.
For a step change in sorbate partial pressure from zero to p or from p to zero at time t = 0, the appropriate initial and boundary conditions, assuming instantaneous equilibration at the crystal surface, are: c (r, 0) = 0 (adsorption) or co (desorption) c (a, f ) = c,, (adsorption) or 0 (desorption)
$(O,t)
= 0
(2)
Vol. 27 No. 2
-P
or desorbed
Since in general zeolitic diffusion coefficients are strongly concentration dependent, this analysis is a valid approximation only when the sorption curve is measured over a small differential step change in sorbate concentration. In order to interpret correctly the sorption curves measured over larger changes in concentration it is necessary to obtain the solution of the diffusion equation for the appropriate functional dependence of the diffusivity on sorbate concentration. Such solutions are not generally available although a number of special cases, mostly
417 C.E.S.
or desorbed
D. R. GARG
and D. M. RUTHVEN
Although not exact, these equations provide useful approximations for many gas-zeolite systems. It is convenient to introduce the following dimensionless variables:
for semi-infinite media, have been given by Crank [l] and by Fujita[2,3]. THEORETICAL
ANALYSIS
Recent studies of the sorption and diffusion of light hydrocarbons in type A zeolite have shown that, for these systems, the concentration dependence of the diffusivity is due to the nonlinearity of the equilibrium isotherm [4,5]:
C = c/c,, R = r/a, T = Dot/a2 f0
D1=+= 0
’ D,
- dC
,u=sR
f’DdC
*
JO
RTalna
D .-alnp O alnc
D=kx=
(5)
For the Langmuir system Eq. (1) becomes:
where
au _ @u/R.d2U
iv--
(Y= sorbate activity k = drag coefficient (constant) Do = y
where /3 = -In (1 - A) and A = co/c,.
= limiting diffusivity as c + 0.
The corresponding system is:
If the equilibrium isotherm obeys the Langmuir equation:
au
(6) a lnp -=a In c
1 1 -c/c,
(7)
D=Do
aT=
where b = adsorption equilibrium constant c, = saturation concentration. For Volmer’s equations are:
isotherm
a lnp a=
(I$
the
8=
c/c,.
{l-h(l--~lR)}~ (l--x)2
for the Volmer
a2u *aR2
(13)
u(R,O) =Oor 1 ~(1, T) = 1 or0 ~(0, T) = 0.
(14)
The solutions of these equations were obtained by the Crank-Nicholson method[6] and also by corresponding the use of an explicit finite difference scheme. Both methods gave results which were in good agreement and the theoretical sorption curves, calculated by integration of the concentration (9) profiles, are shown in Figs. 1 and 2 for several values of the parameter A. For small values of A, adsorption and desorption curves lie close to(10) gether and for A = 0 the curves become coincident with the analytical solution for the constant diffusivity case (Eq. 4). For larger values of A the (11) curves for adsorption and desorption diverge significantly and it is clear that desorption is much slower than adsorption. The initial portions of the
D= (1-?c,,, where
expression
and the modified initial and boundary conditions are:
(8)
1 - c/c,
(12)
dR2
418
The effect of the concentration dependence of diffusivity
O-6
Parameter
-
:A
Adsorption oesorptian
-0.4
0
O-I
0.2
0.3
0.4
0.5
0.6
0.7
&qz?
Fig. 1. Theoretical sorption curves for Langmuir system.
Parameter:
E1E”
0
0.I
02
03
-
Adsorption
---
ossorption
0.4
X
05
Fig. 2. Theoretical sorption curves for Volmer system.
O-7
sorption curves are linear in X6 In the case of the adsorption curves geometric similarity extends throughout and the curves obtained at high values of A do not differ significantly in shape from the constant diffusivity curve. For the Langmuir system the desorption curves show the same type of geometric similarity although a slight difference in shape becomes apparent for large values of A. The desorption curves for the Volmer system are however of a significantly different shape. Such behaviour is typical of systems in which the diffusivity increases with concentration and Crank[l] has pointed out that in such systems the shape of the desorption curves is much more sensitive than that of the adsorption curves. For the Langmuir system the geometric similarity of the sorption curves makes it possible to calculate, for any value of A, an effective diffusivity D, such that, when plotted in terms of Det/a2, the adsorption and desorption curves for all values of A reduce to a single curve which is formally identical with the curve for the constant diffusivity case. In the Volmer system the same type of representation is a good approximation for the adsorption curves but it is less satisfactory for the desorption curves due to the difference in shape. The accuracy of this type of representation may be judged from Fig. 3 in which computed points corresponding to sorp-
8
J_
JS
Fig. 3. Theoretical sorption curves plotted in terms of (D&a*).
419
D. R. GARG and D. M. RUTHVEN
tion curves for various values of A are compared with the constant diffusivity case. The corresponding plot of 0,/D, vs. h is shown in Fig. 4. The effective diffusivities differ very significantly from the mean diffusivity, defined by:
profile is significantly altered when the diffusivity is concentration dependent. The geometric simi-
which gives for the Langmuir system:
-Do D=Tln
1 l--X C-J
(16)
and for the Volmer system: D=+
D
(17)
Typical concentration profiles for adsorption and desorption with a Langmuir concentration dependent diffusivity are compared with the corresponding profiles for the constant diffusivity case in Figs. 5 and 6. The comparison is made at fixed values of concentration at the particle centre and it is evident that the shape of the 5c
,-
4c I
1
0
0.2
0.4
0.6
0.8
r/a
Fig. 5. Concentration profiles during adsorption (Langmuir system). h Dot/d : 20 10-9 0.061 O-185 3 30.9 0.0362 4 40 0.055.
-
Legend
30
I 2 3 4
Adsorption Volmer Desorption I Adsorption Desorption I Langmuir
0.6 -
0.6-
C
0.4 -
0.2 -
0
0.2
0.4
0.6
0.6
I3
r/a
I.0 0
0.2
0.4
0.6
0.6
I
A
Fig. 4. Effective diffisivity as function of parameter A.
Fig. 6. Concentration profiles during desorption (Langmuir system). x Dot/a2 : 20 10.9 0.0125 o-055 3 30.9 0.121 4 40 0.185.
420
The effect of the concentration
larity of the sorption curves is thus seen to be fortuitous in the sense that it does not involve similarity of the actual concentration profiles. These results have important practical implications since it follows that experimental sorption curves may be analyzed using the standard solutions of the Fickian diffusion equation but the diffusivity so obtained will be an effective diffusivity which will depend on the size of the step change in concentration over which the sorption curve is measured (i.e. the value of A). Furthermore, for the same step change in concentration, the value of D, calculated from the adsorption curve will be greater than the corresponding value calculated from the desorption curve. The analysis assumes an initial step change in surface concentration between the limits 0 and c,, but, by a suitable change of variable, it may be shown that the same solutions are applicable when the initial and final concentrations are both finite (cl < c,) with the parameter A = (cZ- CJC, and D,, equal to the diffusivity at the lower concentration cl. Thus, if the value of c, is known from the equilibrium isotherm, the value of DJD,, may be found from Fig. 4, and the limiting diffusivity at zero sorbate concentration (or at cl) may thus be calculated from the effective diffusivity, obtained from an experimental sorption curve measured over a relatively large step change in sorbate concentration.
dependence of diffusivity COMPARISON
WITH
EXPERIMENTAL
DATA
It is implicitly assumed in the analysis that the temperature of the zeolite remains constant during the sorption process but, when the step change in sorbate concentration is large, a significant thermal effect is to be expected as a result of the heat of sorption. Such effects may be expected to be less important in a flow system than in a closed system and the experimental data of Kondis[7] for the sorption of ethane in 4A zeolite, which were obtained in a continuous flow apparatus using ethane-helium gas mixtures, are therefore especially suitable for comparison with the theory. Such a comparison, which is shown in Figs. 7 and 8, and Tables 1 and 2 for the experimental data obtained at 25°C supports the conclusions of the theoretical analysis. A Langmuir plot of the equilibrium data showed that, over the relevant range of pressure, the isotherm may be satisfactorily fitted by a Langmuir equation with c, = 0.085 g/g. Runs 19 and 20 were carried out at low ethane concentrations and it is evident from Fig. 7 that there is no significant difference between the adsorption and desorption curves. (For the sake of clarity no distinction is drawn between the data points for runs 19 and 20 in Fig. 7.) Both curves may be approximately represented by the theoretical curve for a system of uniform spherical particles (Eq. 4) with a constant diffusivity given by
Experime&+
data
Kondis (ethone
of
on 4A)
0 Adsorption x Desorption Theoretical (Equation D,/02=2~45
25
Fig. 7. Comparison
of experimental
and theoretical sorption concentrations.
421
curves
curve 4) x IO-*set-’
30
35
for ethane in 4A zeolite at low
D. R. GARG and D. M. RUTHVEN
0
0’
0. 0
4
l
/
xcd
H~/x~perimen+aI Kandis
l’
.
(ethane
data of on 4A)
Adsorption
Run no. 86 IGA
Desarption
0
x
. a
104 D -
Theoretical D,,/a’=
0
Fig. 8. Comparison
of experimental
20
I5
IO
curves
x 10e4 SW-’
I
1
5
2.45
and theoretical sorption concentrations.
b
I
25’
30
curves for ethane in 4A zeolite at high
Table I. Details of sorption curves [7] for Fig. 7
20A
PI Mm) P, (atm) c1 Wg) c2 (g/g) x _ kc1I cs Del& Do/a2 (Set-I)
19A
20D
0
19D
0.0224 0 0.011
0.0224 0 0.011 0
0 0.0465 0 0.022
0.0465 0 0.022 0
0.129
0.129
0.258
0.258
1.15 2.45 x lo+
1.05 2.45 X lo-*
1.30 2.45 x lo-’
1.10 2.45 x 1O-4
Table 2. Details of sorption curves [7] for Fig. 8
86A
86A
103A
104D
PI (atm) pZ (atm) c1 Wg) c:!(g/g)
0 0.98 0 0.076
0.98 0 0.076 0
0 0.985 0 0.076
0.985 0 0.076 0
h = /c2-c1I c, De/D,
0.90
0.90
0.90
0.90
Do/a2 (Set-‘)
3.70 1.40 2.45 X 1O-4 2.45 X 1O-4
D/a2 = 2-45 X lop4 set-‘. The small deviation which is observed between the experimental and theoretical curves is probably due to the effect of crystal shape and size distribution [5].
3.70 2.45
x 1O-4
2.45
1.40 x 1O-4
In runs 86, 103A and 104D, the step change in concentration is much larger and it may be seen from Fig. 8 that the rate of adsorption is much faster than the rate of desorption, although both
422
The effect of the concentration
rates are significantly faster than at the low concentrations. The theoretical curves shown in Fig. 8 were calculated using the value of &,/a2 = 2.45 X 1O-4set-’ derived from the low concentration data (Fig. 7) and the value of D,/D, obtained from Fig. 4 with the value of A from the equilibrium isotherm. For the desorption curves the agreement between theory and experiment is excellent. Agreement is slightly less good for the adsorption curves but the difference is probably mainly attributable to small thermal effects. Such effects may be expected to be more significant during adsorption since the process is faster. CONCLUSION
For the sorption of light paraffins in Linde 5A zeolite Loughlin, using small differential changes in concentration, observed no significant difference between the rates of adsorption and desorp-
dependence of diffusivity
tion [5] whereas Eberley, whose measurements were carried out over rather larger changes in sorbate concentration, found the diffusivities for desorption to be much smaller than for adsorption[8]. This result is in qualitative agreement with the present analysis although it seems likely that in Eberley’s measurements the difference in sorption rates was further magnified by thermal effects. Within the limits imposed by the requirement for isothermal conditions the present analysis would appear to be quite widely applicable and, for systems in which it is difficult to make sorption measurements over small differential changes in sorbate concentration it provides a valuable method of calculating the limiting diffusivities at zero concentration. Acknowledgement-The financial support of the National Research Council of Canada is gratefully acknowledged.
REFERENCES
111CRANK J., The Mathematics ofDi’ision. Oxford University Press, (1956). PI FUJITA H., Text. Res. J. 1952 22 757. [31 FUJITA H., Text. Res. J. 1952 22 823. 197167 1661. (41 RUTHVEN D. M. and LOUGHLIN K. F., Trans. FaradaySoc. PI RUTHVEN D. M. and LOUGHLIN K. F., Nature 19712%(11)69. 161 CRANK J. and NICHOLSON P., Proc. Camb. ohil. Sot. 1950 76 634. [71 KONDIS E. F., Ph.D. Thesis, Northwestern University (1968). PI EBERLEY P. E., Jr. Ind. Engng Chem. Fundls, 1969 8 (I), 25. Resume-Les auteurs presentent des solutions a l’equation transitoire de dithtsion pour la sorption dans un systeme de particules spheriques a I’interieur duquel la ditfusivitt du produit dabsorption varie avec la concentration. On observe les formes fonctionnelles de la dependance de la concentration de la ditfusivitt pour la sorption de gaz dans des zeolites. On demontre que l’on peut s’attendre pour de tels systemes a une ditference entre les taux d’adsorption et de dtsorption. On calcule les diifusivites en comparent les courbes theoriques de sorption pour la ditfusivite dependante de la concentration a la solution standard de l’equation de diffision pour une ditfusivite constante. Les auteurs confirment la validite de l’analyse en la comparant aux don&es expCrimentales. Zusammenfassung-Es werden Losungen der momentanen Ditfusionsgleichung fur die Sorption in einem System spharischer Teilchen datgelegt, in welchem sich das Ditfusionsvermdgen des Sorbates mit der Konzentration Pndert. Die in Betracht gezogenen funktionellen Formen der Abhangigkeit des Diifusionsvermogens von der Konzentration werden gewijhnlich an der Sorption von Gasen in Zeolitehn beobachtet. Es wird gezeigt, dass bei solchen Systemen ein Unterschied zwischen Adsorptions- und Desorptionsraten zu erwarten ist. Die nutzbaren DiIfusionsvermGgen werden errechnet durch Vergleichder theoretischen Sorptionskurven filr das von der Konzentration abhdngige Ditfusionsvermiigen mit der Normallijsung der Ditfusionsgleichung fur ein konstantes Ditisionsvermiigen. Die Gtiltigkeit der Analyse wird durch einen Vergleich mit Versuchsdaten be&it&.
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