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NaX zeolites
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Applied Catalysis A: General 151 (1997) 267-287
A PA LE IY D CP AT L SS I &GENERAL
Mathematical modeling of transient diffusion and adsorption of cyclopropane in NaX, Ni/NaX and Eu/NaX zeolites Venkatesan V. Krishnan a, Carroll O. Bennett a, Steven L. Suib a,b,* a U-222, Department of Chemical Engineering, University of Connecticut, Storrs, CT 06269-3222, USA b U-60, Department of Chemistry, Unic'ersity of Connecticut, Storrs, CT 06269-3060, USA
Abstract
The transient uptake of cyclopropane gas in NaX zeolite under isothermal conditions has been simulated by a mathematical model. The model has been curve-fit to the experimental data obtained in a stainless steel microreactor with a small bed of catalyst (NaX zeolite). Intra-crystalline diffusion of cyclopropane gas was assumed to play a significant role in the uptake of the adsorbate gas in the zeolite. Based upon this assumption, the response to a switch from a stream of pure non-adsorbing argon to one of 0.5% cyclopropane/argon (a step function) was simulated by assuming an effective intracrystalline diffusivity of the cyclopropane/argon mixture and also taking into account the CSTR conditions in the microreactor. The Langmuir adsorption isotherm was also used to explain the adsorption of the cyclopropane gas in the active sites of the zeolite catalyst. From the simulation results and subsequent curve-fit of the experimental data, the effective diffusivity of the cyclopropane/argon was estimated to be about 2 × 10 ~ cm2/s. The diffusivity of the cyclopropane in Ni/NaX was estimated to be about 2 x 1 0 - 1 2 cm2/s and 3 X 10 -12 cm2/s for the Eu/NaX system. These values of the diffusion coefficient seem reasonable in comparison to the results of diffusion coefficients obtained by similar methods such as Zero Length Chromatography and gravimetric techniques for other organic components in different zeolites. A non-isothermal model taking into account the heat of adsorption of cyclopropane and the activation energy of diffusion of the gas mixture was also formulated to observe any possible temperature rise in the catalyst bed. The temperature of the bed rose not more than 2°C, causing very little changes in the effective diffusivity of the gas mixture, thereby justifying the assumption of isothermality of the uptake process, according to the modeling data. Keywords: Cyclopropaneadsorption;Mathematical modeling;Zeolites; Europium;Sodium
* Correspondingauthor. 0926-860X/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0926- 860X(96)00269-4
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V. V. Krishnan et al. // Applied Catalysis A: General 151 (1997) 267-287
I. Introduction Transient techniques by the use of pulse and step functions have been used before to observe adsorption processes and kinetics of heterogeneous catalytic reactions. The use of such switching techniques in conjunction with high resolution mass spectrometry has been the focus of research in our laboratory over a period of time [ 1-4]. Originally, the sorption of cyclopropane in NaX and ion-exchanged forms of zeolite X was studied by observing the response of the step function generated by switching from a non-adsorbing argon gas to an adsorbate mixture of cyclopropane/argon. [5-8,11] The gases were passed over a small bed of catalyst (about 15-20 mg) in a stainless steel microreactor. The response of the cyclopropane was measured by a high resolution mass spectrometer. The time period over which the response of the cyclopropane reached steady state was found to be much more in zeolites (NaX, ion-exchanged forms of zeolite X) than for supports like alumina (BET surface areas of about 200 m2/g). This behavior of the transient response hinted at the importance of diffusion at the intra-crystalline level as opposed to mere molecular diffusion which one may see in macroporous or mesoporous catalysts. Intra-crystalline diffusion has been described as an activated process with an Arrhenius type temperature dependence. [9,10] Some of the prior work involving the uptake of cyclopropane in NaX and its ion-exchanged forms indicated that the nature of the cartier gas could also play a role in the behavior of the uptake curve for cyclopropane. This "carrier gas" effect was demonstrated in the transient uptake curve of cyclopropane on NaX and its ion-exchanged forms, by Efstathiou et al. [5] Upon switching the gas from pure argon to 0.5% cyclopropane/H 2 a sigmoidal uptake curve was observed which was in contrast with the uptake curve obtained upon a switch from pure argon to 0.5% cyclopropane/Ar. [5] In addition to simulating the transient curve for the Ar ~ 0.5% cyclopropane/Ar switch, we have tried to explain the effect of the cartier gas in terms of the "effective diffusivity", inside the crystallites of the zeolite. The details of the model are discussed in later sections. For the sorption process occurring in a microporous material such as a zeolite, deconvolution of the equilibrium adsorption process from the intra-crystalline transport is desirable. However, in situations where both the processes play a vital role, or when experimental techniques are not able to resolve either effect, mathematical modeling with curve-fitting of the uptake curves provides an opportunity for evaluating diffusion coefficients and equilibrium constants for the sorption process. The mathematical model considered in this work is based upon a simple CSTR coupled with the unsteady state diffusion process occurring in the crystallite (Fick's Law, with a concentration independent "effective diffusivity"). The possibility of intrapellet diffusion resistance or external mass transfer resistance was already considered in the work Efstathiou et al. [5] and proved
V.V. Krishnan et al./Applied Catalysis A: General 151 (1997) 267-287
269
experimentally to be insignificant in the behavior of the isothermal uptake curves. Considerable work has been done, regarding the measurement of diffusivities by Ruthven, Karger and co-workers. [9,12-14,16] The techniques used in these references have been primarily: Zero Length Chromatography (ZLC), gravimetric techniques (with microbalance), IR measurements and self diffusion experiments by Pulsed Field Gradient (PFG) NMR. FTIR spectroscopy has also been used to determine diffusivities in zeolites by Niessen and Karge. [15] The Fickian diffusivity obtained by ZLC and gravimetric techniques is considerably different from that observed by PFG methods. The reason for this discrepancy is still unclear. Some of our predictions for the Fickian intracrystalline diffusivity have been compared with other organic-zeolite systems available in the referenced literature.
2. Mathematical model 2.1. Isothermal model
The reactor used in the experiments done by Efstathiou et al., [5-8,11] was verified to be a CSTR, based on prior work. It was also assumed that the crystallites in the zeolite were spherical and the Fick's Law for unsteady state diffusion in a spherical geometry was thus applied. The adsorption of cyclopropane in NaX is an exothermic process but considering the extremely small quantity of catalyst used (18 mg), isothermality was assumed. Keeping these assumptions in mind, the following material balance equations were postulated: 1. Unsteady state material balance of cyclopropane in the CSTR microreactor: dt
-- 1 -- Cg(}) - ~, ra [ O?Jr=,
The variables may be defined as follows: Cg = f ( - ~ ) /
Cinle t .
(l(ii))
t=t/r. r= (V-
(l(i))
W)/(
? = r / r c. r d = r2c/Oe .
PsQ).
(l(iii)) (l(iv)) (l(v))
ce~ = 3 w / ( a p s ) .
(l(vi))
C r = C s a t / f i n l e t.
(1 (vii))
2. Unsteady state material balance of the "sorbed" phase inside the crystal:
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D e, the effective diffusivity is assumed to be independent of the intracrystalline concentration because, the crystal size is very small (0.6 txm, radius) and the levels of concentration may be sufficiently low to ignore this dependence. ~0(?,})
-~7
[ 320(?,t)
--0/2[
2 30(?,t) ]
~'~"~ -'{- r
~
]'
(2)
where 0[. 2 =
T/
(2(i))
Td .
Initial condition: t=0,
Cg(0) = 0 ,
t=0,
0(?,0) =0.
(2(ii))
Boundary conditions: r=0,
?=0,
3?=0
'
r = r c,
?=1,
O(1,-t)=Cg/(Cg+l/K).
(2(iii)) (2(iv))
The last boundary condition indicates that at the entrance of the crystal, the adsorption of cyclopropane occurs instantaneously by a simple Langmuir type of isotherm. [5,11] The transport of the adsorbed phase in the crystals is what has been described by the "effective diffusivity" in the Fick's Law equation. The values of the various constants have been taken from the experimental work of Efstathiou et al. [5].
K=bCiHRT, O=bP/(1 + bP).
(2(v)) (2(vi))
The constant b is the original Langmuir constant represented in the Langmuir equation, above. The parameter K has been postulated to represent the Lmagmuir adsorption using dimensionless gas phase concentration instead of partial pressure as in Eq. (2)(vi). O=C(r,t)/Csa
t.
(2(vii))
The variable 0 represents the dimensionless "coverage" of the adsorbed species as a function of radial distance in the crystallite and time. Csat is the saturation value of the adsorbed phase concentration at the temperature of sorption. In zeolites, Csat is observed to be a function of temperature [9], but we have considered the maximum value of the saturation concentration, which occurs in the room temperature. Lower temperature experiments may give more saturation uptake, but that could include some physisorption which is undesirable. From the work of Efstathiou et al. [5], the value of C~at is taken as 0.0043 m o l / c m 3. The radius of the crystallite (re), obtained by SEM micrographs [5], is about 0.6 ta,m. The porosity of the bed, used in the work of Efstathiou et al. [5], was
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V.V. Krishnan et al./Applied Catalysis A: General 151 (1997) 267-287
estimated to be about 0.4. The inlet concentration (Cin) of cyclopropane used was 0.5%, i.e. 1.95 × 10 -7 m o l / c m 3. The volume of the reactor has been estimated to be about 0.785 cm 3 and the weight of the adsorbent bed used was 18 mg. The density of the solid phase of the zeolite NaX was found to be about 1.46 g / c m 3.
2.2. Non-isothermal model The non-isothermal model is derived based on the presence of heat transfer resistance external to the pellets. It is assumed that the pellet heats up uniformly and the temperature gradient occurs entirely in the film external to the pellet. To take into account the non-isothermality of the process, the diffusivity and the equilibrium constants are expressed in terms of their Arrhenius dependence with temperature: D = Doexp[- EJ(RT)],
(3(i))
K = A e x p [ - A H / ( R T ) + AS~R],
(3(ii))
where D O is the pre-exponential factor and E a and A H are the activation energy of diffusion and the enthalpy of adsorption, respectively. Assuming that the change in entropy, AS, is invariant with temperature, the value of K can be approximated as follows:
K = K o e x p [ - A H / ( RT)].
(3(iii))
For the non-isothermal model, Eqs. (1) and (2) remain the same, with modifications in the constants to accommodate the temperature changes, but the energy balance needs to be carried out also, and may be written as follows: d~/(t) dt
O0(1,t) 0~30/2 ~
O~4T('t ~ -- 1).
(3)
Initial conditions: Cg(0) = 0,
0(?,0) =0,
7 ( 0 ) - - 1.
(3(iv))
Boundao' conditions:
00(o,?) - -
O?
=0,
(3(v))
Cg 0(1,}) = Cg + l / K ( r / ) "
(3(vi))
The dimensionless constants have been modified and re-written as follows: ~ ( t ) -= Tpellet(})/Tb,
(3(vii))
rc
rd = Ooexp[_Ed/(RTb~?) ] .
(3(viii))
V.V. Krishnan et aL / Applied Catalysis A: General 151 (1997) 267-287
272
This modification leads to a temperature dependence of the constants such as a 2. It also introduces two new constants which are indirect functions of temperature, a 3 and a n, defined as follows, in Eqs. 3(ix) and 3(x): 3(- A/~)C~ a3=
ps Cp~Tb
,
(3(ix))
3h O~4= PsCps rp(1 -- ep)"
(3(x))
These constants are related to the density and specific heat capacity of the solid catalyst (denoted by the subscript " s " . The subscript " p " denotes "pellet", because the experiments had been done by using small pellets of average diameter of about 0.5 mm). The heat transfer film coefficient, h, has been predicted according to the simplest case, i.e. a spherical particle of radius rp, in a stream of near stagnant gas with Reynolds number approaching zero. The Nusselt number approaches 2 in this case.
h = 2k/de.
(3(xi))
The heat transfer coefficient is a linear function of the thermal conductivity of the adsorbate gas mixture according to this assumption. This considers the worst possible case where the rate of heat removal from the catalyst pellet is the least.
2.3. Solution of the combined system of differential equations The solution of a combined system of ordinary and partial differential equations has to be obtained by using numerical techniques because the two equations are interlinked by the diffusive flux term at the intracrystalline boundary. This problem has been solved by using a program called differential systems solver (DSS), based upon a technique known as the numerical method of lines, developed by Prof. W.E. Schiesser of Lehigh University. A subroutine specifying the problem with its initial and boundary conditions was written, compiled and linked with a Microsoft Fortran compiler.
3. Results
3.1. Isothermal model The results presented in our simulation indicate the trends obtained by variation of parameters such as diffusivity and equilibrium constant at different conditions of flow rate. Fig. 1 shows the variation of the uptake curve at various flow regimes. Fig. 2a and Fig. 2b show the simulated gas phase response with
V. V. Krishnan et al. / A p p l i e d Catalysis A: General 151 (1997) 267-287
273
1.0 5 2 0.8 r,,,)=
i
0.6
~e
0.4
1 - 0.34cc/s
"~
2 - 0 . 5 0 cc/s 3 - 1.50 cc/s 0.2
4 - 2 . 5 0 cc/s 5 - 6 . 0 0 cc/s
0.0 0
q
r
~
,
200
400
600
800
1000
Real time, s Fig. 1. V a r i a t i o n o f u p t a k e c u r v e s w i t h f l o w rate D e = 2 . 0 E -
11 c m 2 / s ;
K = 0.65; N a X .
dimensionless time at different values of diffusivity. Fig. 2b shows the increase in the resolution in the uptake response with diffusivity at a higher flow rate of 0.75 cm3//s. The diffusivity of 0.5% cyclopropane/argon in the NaX zeolite is approximately in the range 10-11-10 -~2 c m 2 / s and such values have been chosen in the simulation. At the low flow rate of 0.34 cm3/s, the effect of diffusivity does not appear to change the uptake curve significantly. Varying the equilibrium constant seems to have a stronger effect, as shown in Fig. 3a. Figs. 4 and 5 show the effect of intracrystalline diffusivity on the concentration profile of the adsorbed phase inside the crystal at various times. The spatial concentration gradients are lower at a diffusivity of about 2 X 10 -11 c m 2 / s than those at a diffusivity of 4 X 10 -12 c m e / s , which would be expected. For a system with an effective diffusivity of 2 × 10 -1~ cm2/s, the response time to achieve a near zero concentration gradient of the intracrystal phase is less than 200 s (Fig. 5) for a flow rate of 0.34 cm3/s, whereas the total time taken for the gas phase concentration to equilibrate at the same flow rate (Fig. 2a) is over 1000 s. The implications of this observation will be discussed in the next section. Fig. 6 shows the comparison of the simulated gas phase response with the experimental data from the work of Efstathiou et al. [5]. The values of the diffusivity have been arrived at by trial and error and they indicate an excellent fit with the experimental data. The diffusivity was estimated to be about 2.0 X 10-11 c m 2 / s and the equilibrium constant about 0.7, at a temperature of 40°C, at which the original experiments were performed [5].
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V. V. Krishnan et al. /Applied Catalysis A: G e n e r a l 151 (1997) 267-287
1.0 0.8
=
/
0.6
D --9.0E-12cm/s
0.4 De=2.0E-I1cm2/s 0.2 0.0 0
100
200
300
400
500
600
Dimensionless time
1.0
b)
0.8
/
i 0.6
De=4.0E-12c m ' / s
~
De=9"0E-12cm2/s
"~ 0.4 ///
0.2
0.0
0
D =2.0E-11cm2/s
i
i
i
i
i
100
200
300
400
500
600
Dimensionless time
2. (a) Variation o f diffusivities at a l o w f l o w rate: 0.34cm3/s. Equilibriumconstant = 0.65; NaX.(b) Variationo f diffusivities at a f l o w rate o f 0.75 cm3/s. Equilibrium constant = 0.65;NaX. Fig.
Figs. 7 - 1 0 are similar to the simulations shown from Figs. 1-4, but these are done at high flow rates: 1.5 cm3/s. Unfortunately, the experimental data at 1.5 c m 3 / s are not available for us to curve-fit with the aid of our simulation. It can be seen that the variation in diffusivity caused a more significant change in the uptake curve, as the flow rates were increased from 0.34 cm3/s to 0.75 c m 3 / s
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
1.0
275
(a)
0.8
0.6 ~
K--0.5]4
0.4
0.2
0.0 0
I
I
l
7
T
100
2~
3~
4~
5~
6~
Dimensionless time
0.8
o.6
°i///
"~ 0.4
0.2
0.0
] . . . . . . . 0
100
200
i 300
400
500
Dimensionless time
Fig. 3. (a) Variation of equilibrium constants at low flow rate: 0.34 cm3/s. Diffusivity = 9.0E- 12 cm2/s; NaX. (b) Variation of equilibrium constants. Intermediate flow rates: q = 0.75 cm3/s; diffusivity = 9.0E- 12 cm2/s.
(Fig. 2b) and all the way up to 1.5 cm3/s. The adsorbed phase concentration profile at a diffusivity of 2.0 × 10 -11 cm2/s as shown in Fig. 8 may be compared to Fig. 5 (low flow rate). The effect of the flow rate on the concentration gradient can be seen by a mere visual comparison of the two
276
V. V. Krishnan et al. / A p p l i e d Catalysis A." General 151 (1997) 267-287 0.5
t = 500 s
0.4
8 -8 o.o t = 200 s
0.2 0.1
==
5 o.o
t=lOs
-0.1
0.0
I
I
F
I
0.2
0.4
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 4. Adsorbed phase concentration vs. intracrystal distance; NaX. D e = 4 . 0 E - 12 c m 2 / s , Q = 0.34 c m 3 / s , K = 0.65.
figures. Fig. 9 shows the immense difference in the adsorbed phase concentration profile due to a drop in the diffusivity from 2 . 0 X 10 - u c m 2 / s to 4.0 x 10 -12 cm2/s. A strong concentration gradient exists even after 200
0.5
e~
0.4
0.3
8
t = 200 s
0.2 t=50s
o
•~ 0.1 t=30s t=lOs
0.0
-0.1
I
0.0
0.2
-
~
0.4
I
7
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 5. Adsorbed phase concentration vs. intracrystal distance; NaX. D~ = 2 . 0 E - 11 c m 2 / s , Q = 0.34 c m 3 / s , K = 0,65.
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
277
1.0
0.8
0.6
"~
Simulated data
0.4
N 0.2
0.0 0
I
r
100
200
7
.....
300
I
I
I
I
400
500
600
700
800
Real time, sec Fig. 6. Comparison of experimental data vs. simulated response; NaX. Switch: Ar ~ cm3/s; De = 2 . 0 E - 11 cm2/s; K = 0.7.
0.5% CP/Ar. Q = 0.5
seconds (Fig. 8) and this causes a significant difference in the gas phase uptake curve at a diffusivity of 4.0 X 10- ~2 cm2/s, in comparison to the uptake curve at a diffusivity of 2.0 X 10 -~1 cm2/s. Fig. 10 seems to be identical to Fig. 3, which indicates that changes in the uptake curve due to variation in the equilibrium constant are similar in nature regardless of the flow rates. 1.0
~
0.8
0.6
~
0.4
0.2
! 0.0 0
100
200
300
400
500
600
Dimensionless time Fig. 7. Variation of diffusivities at a high flow rate: 1.5 cm3/s. Equilibrium constant: K = 0.65; NaX.
278
V. V. Krishnan et al. / A p p l i e d Catalysis A: General 151 (1997) 267-287 0.5
t = 800 s
0.4
0.3 t = 200 s
= 0.2
= o.1
E o.o
t=10s
-0.1 0.0
0.2
0.4
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 8. A d s o r b e d p h a s e concentration vs. intracrystal distance; NaX. D r = 2 . 0 E - 11 c m 2 / s , Q = 1.5 c m 3 / s , K = 0.65.
Figs. 12-14 show how the diffusivity and the equilibrium constant have been changed to curve-fit the transient responses for the argon to cyclopropane/argon switch in the case of systems like N i / N a X , E u / N a X and NaX (at 80°C). The 0.5
ee~
t = 800 s
0.4
0.3 0.2
8 0.l .o
o.0
t= 10s
-0.1 0.0
I
I
I
I
0.2
0.4
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 9. A d s o r b e d p h a s e concentration vs. intracrystal distance; NaX. D e = 4 . 0 E - 12 c m 2 / s , Q = 1.5 c m 3 / s , K = 0.65.
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
279
1.0
K
0.8
=
0
.
5
1
~
K=0.65 0.6 5 •~ 0.4
0.2
0.0 0
I
t
I
I
I
100
200
300
400
500
600
Dimensionless time
Fig. 10. Variation of equilibriumconstants at a high flow rate: 1.5 cm3/s. Diffusivity = 9.0E-12 cm2/s: NaX.
diffusivity of cyclopropane in Ni-exchanged NaX (Fig. 12), is evaluated to be about 2 × 10-12 cm2/s, with an equilibrium constant ( K ) of 1.3 (corresponding to 0.33 Torr -1, in dimensional terms). For the E u / N a X system (Fig. 13), a diffusivity of about 3 × 10-12 c m 2 / s and an equilibrium constant of about 0.58 (b = 0.15 Torr -1) gave us the best curve fit with the experimental data. In both the cases, the diffusivity has clearly dropped by almost an order of magnitude in comparison with NaX at 40°C (Fig. 6). The uptake curve for the NaX system at 80°C with a 0.3% cyclopropane/argon mixture has been simulated in Fig. 14. The diffusivity of this system remains at about 1.0 × 10 -11 c m 2 / s but the equilibrium constant has decreased considerably to about 0.15 (b = 0.039); understandably, because adsorption is an exothermic process. The implications of these values are discussed in the next subsection. 3.2. Non-isothermal model
The results of the non-isothermal model are shown in Fig. 11. The maximum rise in temperature accompanying the adsorption and diffusion of the gas mixture is only about 1.3°C for argon as a carrier gas under the conditions of the experiment (0.5 cm3/s). The interesting thing is the increase in the rise in temperature at a higher volumetric flow rate of the adsorbate gas mixture. The simulations run at a flow rate of 2.5 cm3/s, show a maximum temperature increase of about 2.4°C. This aspect will be discussed in the next section. For hydrogen as a carrier gas, the maximum possible temperature is even smaller,
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V. V. Krishnan et al. / Applied Catalysis A." General 151 (1997) 267-287 3.0
2.5 ;,,)
2.0 =
,~ 1.5
.
[,. .~ 1.0
0.5
0.0 I
I
I
I
I
I
0
100
200
300
400
500
600
Dimensionless time Fig. 11. Rise in temperature (°C) in the catalyst bed. Non-isothermal model (heat transfer film resistances included). H = - 8.0 K C a l / m o l ; E d = 8.9 K C a l / m o l ; NaX.
only about 0.3-0.4°C. Due to the extremely small values of the temperature increase, inter-pellet heat transfer resistance is not a factor. The assumption of isothermality is therefore justified.
4. Discussion 4.1. Isothermal model
The low flow rate of 0.34 c m 3 / s does not seem to be a good condition for doing uptake experiments because the intracrystalline adsorbed phase concentration gradient, as shown from our simulation (Figs. 4 and 5) diminishes drastically in less than 200 seconds in both cases. The sensitivity of the ~0/07 term at the crystal boundary, in response to changes in diffusivity, is the primary element causing a change in the uptake curve. If however, the system behaves like the NaX system, i.e., a diffusivity of about 2.0 × 10 -11 cm2/s, then any experiment done in the range of about 0.34 c m 3 / s will be ineffective in resolving small changes in diffusivity, because of the lack of concentration gradients inside the crystal (Fig. 5). The effect of intracrystalline concentration gradients is strong up to 200 seconds or so, which is in the early phase of the adsorption uptake curve. The
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
281
diffusivity is therefore an important parameter at the early stages of the uptake experiment. At higher times, the concentration gradients will approach zero anyway as the uptake curve approaches equilibrium feed concentration. Based upon the same reasoning, simulations at a high flow rate of 1.5 cm3/s can now be analyzed. The differences in the uptake curve with changes in diffusivity are more discernible (Fig. 7) and can be easily explained on the basis of the intracrystalline concentration gradient; compare Figs. 8 and 9. A change of diffusivity from 2 . 0 × 10 -1! c m 2 / s to 4 × 10 -12 cm2/s shows drastic differences in the concentration profiles of the adsorbed phase. It is clear that in order to do uptake experiments, it is essential to go to higher flow rates of over 1 cm3/s, in order for diffusivity to play a significant role in the uptake curve. The only limitation could be due to the time at which equilibrium is achieved. If the transient time is too small, small changes in uptake curves may not be easily detected upon changing parameters. The advantage of working with lower flow rates is the large transient time of the uptake curve, but this may not indicate a diffusion controlled regime. Fig. 1 shows a comparison of simulated uptake curves upon changes in flow rates. This gives an idea of the time scale of the transient curves in the cyclopropane/argon - NaX system. The equilibrium constant seems to be a very important criterion because of the strong chemisorptive interaction of cyclopropane with the NaX zeolite [5]. That is the reason why the adsorption of cyclopropane has been used as the boundary condition. An equilibrium constant of 0.7 which has been obtained by curve-fitting, corresponds to a value of 0.18 Torr- 1, on a dimensional scale. Evaluation of the enthalpy of adsorption has also been done with the aid of curve-fits by the non-isothermal model. A value of A H = - 8 . 0 kCal/mol is arrived at based on this exercise. The activation energy of diffusion, Ej, is estimated to be about 8.9 kCal/mol. These are approximate values, because of the non-availability of uptake data at higher temperatures. The above mentioned values correspond to the diffusivity and equilibrium constant found by curve-fitting the uptake curves at 40°C. The N i / N a X system [7] shows a long time for equilibration (Fig. 12), though the initial rise in the uptake curve is rapid. As discussed before, for a system with a diffusivity of about 2 × 10 -12 cm2/s, the initial part of the uptake curve is more dependent on the diffusivity of the system rather than the equilibrium constant. At higher times, the transient response is unable to equilibrate within a reasonable amount of time, which can be explained by a strong adsorptive interaction of cyclopropane, as shown by a high value of K (1.3). The E u / N a X system [6] shows a smaller equilibrium time than NaX, but also weaker adsorptive interaction with cyclopropane gas. This is clearly seen by the diffusivity of 3 X 10-~2 c m 2 / s (nearly an order of magnitude smaller than that obtained for NaX). The simulation of the cyclopropane/argon transient curve for NaX at 80°C is shown in Fig. 14. The equilibrium constant is drastically lowered to about 0.15
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V. K Krishnan et al. /Applied Catalysis A: General 151 (1997) 267-287 1.0
0.8
0.6
8 o 0.4
0.2
0.0 0
r
i
i
500
1000
1500
2000
Time, s Fig. 12. C o m p a r i s o n - m o d e l vs. experimental data: N i / N a X . Switch: A r ~ D e = 2 . 0 E - 12 c m 2 / s ; K = 1.3.
0.5% C P / A r . Q = 0.5 c m 3 / s ;
(b = 0.064 Torr -~) from 0.7 for NaX at 40°C and it contributes directly to the smaller response time. The diffusivity is still about 1.0 × 10 - l l cm2/s. This is because of our reasoning that at values of diffusivity greater than 10-~1 cme/s,
1.0
0.8 a~
.~ 0.6
"
d data
0.4 ..~ 0.2
0.0 0
~
f
500
1000
1500
Time, s Fig. 13. C o m p a r i s o n - m o d e l vs. experimental data, E u / N a X . Switch: A r ~ De = 3 . 0 E - 12 c m Z / s ; K = 0.58.
0.5% C P / A r . Q = 0.5 cm3/s;
V. V. Krishnan et al. / A p p l i e d Catalysis A: General 151 (1997) 267-287 1.0
283
yaW¢*,-
0.8
0.6
0.4 .I
0.2
I
0.0 0
i
F
~
i
100
200
300
400
500
Time, s Fig. 14. Comparison - model vs. experimental data: NaX at 80°C. Switch: Ar ~ c m 3 / s ; De = 1.0E-11 c m 2 / s ; K = 0.15.
0.3% C P / A r . Q = 0.5
the intracrystalline concentration gradients become very small. Diffusivities in this regime cannot be resolved at this flow rate (0.5 cm3/s). However, the flow rate of 0.5 cm3/s may be adequate to resolve diffusivities for both E u / N a X and N i / N a X because their diffusivity is as low as 2-3 x 10-12 c m 2 / s at 40°C, which means that intracrystalline concentration gradients play a significant role in the uptake process. Ruthven et al. [10] have used Zero Length Chromatography (ZLC) extensively to measure the diffusivity of o-xylene in NaX. Upon extrapolating the diffusivities measured in this work to about 40°C, we obtain a value of 2.0 X 10 -l° cm2/s for the o-xylene/NaX system. Gravimetric techniques (Ruthven et al. [12]) and ZLC yield consistent results for the benzene/NaX system: about 4.0 X 10 -9 cmZ/s, extrapolated to 40°C. Using gravimetric techniques, Ruthven et al. [13] have estimated "corrected diffusivities" to be about 2 x 10 -11 cmZ/s at 150°C for the cyclohexane/silicalite system. Upon extrapolation of this data to about 40°C, the diffusivity of this system may be as low as 2 - 3 X 10 -13 cmZ/s. FTIR experiments by Niessen et al. [15] have been used to determine the Fickian diffusivity of p-xylene in H-ZSM-5, which is about 8 X 10-10 cmZ/s at about 100°C. Extrapolation of these values to 40°C yields a diffusivity of about 8 X 10 -11 cmZ/s by using adsorption data. Karger, Ruthven and co-workers [14] have used IR, PFG-NMR and ZLC techniques for measurement of diffusiv-
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V.V. Krishnan et aL /Applied Catalysis A: General 151 (1997) 267-287
ity of methanol in NaX crystals. The IR data in reference 14 give a value of about 2.6 x 10 -8 c m 2 / s for the "corrected" diffusivity for methanol in NaX at 27°C. Ruthven and Karger [14] have obtained a value of 0.8 X 10 -8 cmZ/s from the ZLC data at 40°C and about 1.5 X 10 -8 c m 2 / s for the same system, by using PFG-NMR methods. Earlier measurements by Karger and Ruthven [16] in the benzene/NaX system, using the PFG-NMR technique, lead to a value of self-diffusivity of about 10 -6 cm2/s, considerably higher than with other techniques mentioned in the above references. This discrepancy has been commented upon in several references [9,10,12,13] but is something that has not yet been resolved. 4.2. Non-isothermal model
To explain the increase in rise in temperature of the system (AT) with higher flow rates, assuming film heat transfer resistance, we will have to review the dimensionless constants in the equations defined earlier. One would expect under normal conditions that an increase in heat transfer coefficient will occur with increase in flow rate, thereby causing heat removal at a faster rate, leading to a smaller temperature buildup in the catalyst pellets. According to our simulation, the opposite result is observed (Fig. 11). An increase in flow rate results in a higher d C g / d t , ot2 in Eq. (2) decreases because of the decrease in residence time in the reactor. This decreases the temporal derivatives (Eq. (2)) of the interior points in the crystal, i.e 00/Ot. The higher value of the gas phase concentration results in a higher value of the adsorbed phase concentration at the crystal boundary (from the Langmuir boundary condition). The end result is an increase in the concentration gradient at the crystal boundary (term 1, Eq. (3)). If this term dominates, the trend shown in Fig. 1 will be observed. Perhaps, under conditions of very high flow rate, the second term of Eq. (3) (dimensionless constant, a 4) may be dominant and that might result in a reversed trend. Under very high flow rates, however, the heat transfer coefficient cannot be calculated from near stagnant fluid assumptions (Nusselt number = 2) because it is now a function of the Reynolds and Prandtl numbers. This analysis is beyond the scope of this work. The simulations involving a switch in the gas from argon to 0.5% cyclopropane/hydrogen have not been able to reproduce the sigmoidal signal obtained in Ref. [5]. It is suspected that this carrier gas "effect", shown in the work of Efstathiou et al. [5], may be due to the inlet system in the mass spectrometer, because changes in two gases with diverse molecular weight cause a lag time at the inlet. The step function therefore may not be exact. An in-depth analysis of the inlet system is being done to observe, by fluid mechanics, how the pressure at the inlet changes by changing the molecular weight of the feed. This will be part of an ensuing paper.
V. V. Krishnan et al. /Applied Catalysis A: General 151 (1997) 267-287
285
5. Conclusion
The value of diffusivity arrived at for the cyclopropane/argon-NaX system is about 2.0 × 10 -ll c m 2 / s and seems to be in the 10-9-10 -12 c m e / s range for similar organics in other zeolites, measured by other techniques in the literature. The prerequisites for modeling no doubt include accurate physical data, including the density of the solid, the crystal size and void fraction of the bed a n d / o r the pellet. Upon estimation of these quantities, the number of adjustable parameters reduces to just two - the equilibrium constant and the diffusivity. The value of the equilibrium constant from the Langmuir isotherm is very close to that obtained by previous work [5]. A simple Henry's Law equation for predicting equilibrium is not applicable in this case, because the partial pressure of the sorbate gas is too high. The values of diffusivity obtained by simulating the response for E u / N a X and N i / N a X are almost one order of magnitude smaller than that for NaX at 40°C. The low values of the intracrystalline diffusivities for the Eu and Ni exchanged NaX systems indicate the presence of concentration gradients at the intracrystalline level even at a flow rate of 0.5 cm3/s. The model we have proposed is, therefore, the simplest model which can be used to simulate the sorption uptake process using a laboratory scale flow system. There are no mathematical simplifications needed, since numerical techniques have been used. It is further possible to increase the complexity of the diffusivity and the equilibrium terms and still use numerical analysis without having to worry about non-linearity. The modeling work also shows us the correct regime in which the experiments need to be performed, in order to obtain maximum information on the diffusivity and equilibrium constants.
6. List of notations /-
r~ t
c(~) C~at Cinlet
Cr
Radial distance in the spherical crystallite. Radius of the crystallite. Real time (s). Residence time in the microreactor (s). Intracrystalline diffusional time constant (s). Dimensionless distance in the crystallite. Dimensionless time. Bulk gas phase concentration in the reactor (mol/cm3). Saturation concentration of the adsorbed phase in the crystallite (mol/cm3). Inlet gas phase concentration (mol/cm3). Ratio of the saturation adsorbed phase concentration to the inlet gas concentration.
286
V.V. Krishnan et al./Applied Catalysis A: General 151 (1997) 267-287
0 W V P~
g De K Do AH AS Tpellet
C,s
rp h k
%
Dimensionless gas phase concentration. Adsorbed phase concentration (mol/cm3). Dimensionless adsorbed phase concentration. Weight of the catalyst bed (g). Volume of the microreactor (cm3). Density of the solid phase of the catalyst (g/cm3). Volumetric flow rate of the adsorbate gas mixture (cm3/s). Effective Fickian diffusivity of the sorbate gas (cm2/s). Langmuir equilibrium constant (dimensionless). Pre-exponential factor in the estimation of diffusivity (cmZ/s). Activation energy of diffusion (Kcal/mol). Enthalpy of adsorption (Kcal/mol). Entropy of adsorption (Kcal/mol • K). Temperature of the pellet (°C). Temperature of the bulk fluid (°C). Dimensionless temperature of the pellet. Specific heat of the solid pellet (Kcal/mol • K). Radius of the catalyst pellet (cm). Heat transfer coefficient of the fluid to the particle (Kcal/cm 2 • s • K). Thermal conductivity of the gas (Kcal/cm • s • K). Porosity of the pellet.
Acknowledgements We acknowledge the support of the Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences for this research. The authors also wish to thank Prof. M.B. Cutlip (Department of Chemical Engineering, University of Connecticut) for allowing us to use his DSS software and a copy of his Microsoft Fortran compiler, for this research.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
C.O. Bennett, Catal. Rev., 13 (1976) 121. C.O. Bennett, ACS Symp. Ser., 178 (1982) 1. D.M. Stockwell, J.S. Chung and C.O. Bennett, J. Catal., 112 (1988) 135. A. Efstathiou and C.O. Bennett, J. Catal., 120 (1989) 137. A.M. Efstathiou, S.L. Suib and C.O. Bennett, J. Catal., 131 (1991) 94. A.M. Efstathiou, S.L. Suib and C.O. Bennett, J. Catal., 135 (1992) 223. A.M. Efstathiou, S.L. Suib and C.O. Bennett, J. Catal., 135 (1992) 236. M.W. Simon, A.M. Efstathiou and C.O. Bennett, J. Catal., 138 (1992) 1. D.M. Ruthven, Principles of adsorption and adsorption processes, Wiley, New York. M. Eic and D.M. Ruthven, Zeolites, 8 (1988) 40.
V.K Krishnan et al. /Applied Catalysis A: General 151 (1997) 267-287
[11] [12] [13] [14] [15] [16]
C.O. Bennett and S.L. Suib, Catal. Today., 15 (1992) 503. M. Eic, M. Goddard and D.M. Ruthven, Zeolites, 8 (1988) 327. C.L. Cavalcante, Jr. and D.M. Ruthven, Ind. Eng. Chem. Res., 34 (1995) 185. Ph. Grenier, F. Meunier, P.G. Gray, J. Karger, Z. Xu and D.M. Ruthven, Zeolites, 14 (1994) 242. W. Niessen and H.G. Karge, Microporous Materials, 1 (1993) 1. J. Karger and D.M. Ruthven, J. Chem. Soc., Faraday Trans. I, 77 (1981) 1485.
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A PA LE IY D CP AT L SS I &GENERAL
Mathematical modeling of transient diffusion and adsorption of cyclopropane in NaX, Ni/NaX and Eu/NaX zeolites Venkatesan V. Krishnan a, Carroll O. Bennett a, Steven L. Suib a,b,* a U-222, Department of Chemical Engineering, University of Connecticut, Storrs, CT 06269-3222, USA b U-60, Department of Chemistry, Unic'ersity of Connecticut, Storrs, CT 06269-3060, USA
Abstract
The transient uptake of cyclopropane gas in NaX zeolite under isothermal conditions has been simulated by a mathematical model. The model has been curve-fit to the experimental data obtained in a stainless steel microreactor with a small bed of catalyst (NaX zeolite). Intra-crystalline diffusion of cyclopropane gas was assumed to play a significant role in the uptake of the adsorbate gas in the zeolite. Based upon this assumption, the response to a switch from a stream of pure non-adsorbing argon to one of 0.5% cyclopropane/argon (a step function) was simulated by assuming an effective intracrystalline diffusivity of the cyclopropane/argon mixture and also taking into account the CSTR conditions in the microreactor. The Langmuir adsorption isotherm was also used to explain the adsorption of the cyclopropane gas in the active sites of the zeolite catalyst. From the simulation results and subsequent curve-fit of the experimental data, the effective diffusivity of the cyclopropane/argon was estimated to be about 2 × 10 ~ cm2/s. The diffusivity of the cyclopropane in Ni/NaX was estimated to be about 2 x 1 0 - 1 2 cm2/s and 3 X 10 -12 cm2/s for the Eu/NaX system. These values of the diffusion coefficient seem reasonable in comparison to the results of diffusion coefficients obtained by similar methods such as Zero Length Chromatography and gravimetric techniques for other organic components in different zeolites. A non-isothermal model taking into account the heat of adsorption of cyclopropane and the activation energy of diffusion of the gas mixture was also formulated to observe any possible temperature rise in the catalyst bed. The temperature of the bed rose not more than 2°C, causing very little changes in the effective diffusivity of the gas mixture, thereby justifying the assumption of isothermality of the uptake process, according to the modeling data. Keywords: Cyclopropaneadsorption;Mathematical modeling;Zeolites; Europium;Sodium
* Correspondingauthor. 0926-860X/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0926- 860X(96)00269-4
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V. V. Krishnan et al. // Applied Catalysis A: General 151 (1997) 267-287
I. Introduction Transient techniques by the use of pulse and step functions have been used before to observe adsorption processes and kinetics of heterogeneous catalytic reactions. The use of such switching techniques in conjunction with high resolution mass spectrometry has been the focus of research in our laboratory over a period of time [ 1-4]. Originally, the sorption of cyclopropane in NaX and ion-exchanged forms of zeolite X was studied by observing the response of the step function generated by switching from a non-adsorbing argon gas to an adsorbate mixture of cyclopropane/argon. [5-8,11] The gases were passed over a small bed of catalyst (about 15-20 mg) in a stainless steel microreactor. The response of the cyclopropane was measured by a high resolution mass spectrometer. The time period over which the response of the cyclopropane reached steady state was found to be much more in zeolites (NaX, ion-exchanged forms of zeolite X) than for supports like alumina (BET surface areas of about 200 m2/g). This behavior of the transient response hinted at the importance of diffusion at the intra-crystalline level as opposed to mere molecular diffusion which one may see in macroporous or mesoporous catalysts. Intra-crystalline diffusion has been described as an activated process with an Arrhenius type temperature dependence. [9,10] Some of the prior work involving the uptake of cyclopropane in NaX and its ion-exchanged forms indicated that the nature of the cartier gas could also play a role in the behavior of the uptake curve for cyclopropane. This "carrier gas" effect was demonstrated in the transient uptake curve of cyclopropane on NaX and its ion-exchanged forms, by Efstathiou et al. [5] Upon switching the gas from pure argon to 0.5% cyclopropane/H 2 a sigmoidal uptake curve was observed which was in contrast with the uptake curve obtained upon a switch from pure argon to 0.5% cyclopropane/Ar. [5] In addition to simulating the transient curve for the Ar ~ 0.5% cyclopropane/Ar switch, we have tried to explain the effect of the cartier gas in terms of the "effective diffusivity", inside the crystallites of the zeolite. The details of the model are discussed in later sections. For the sorption process occurring in a microporous material such as a zeolite, deconvolution of the equilibrium adsorption process from the intra-crystalline transport is desirable. However, in situations where both the processes play a vital role, or when experimental techniques are not able to resolve either effect, mathematical modeling with curve-fitting of the uptake curves provides an opportunity for evaluating diffusion coefficients and equilibrium constants for the sorption process. The mathematical model considered in this work is based upon a simple CSTR coupled with the unsteady state diffusion process occurring in the crystallite (Fick's Law, with a concentration independent "effective diffusivity"). The possibility of intrapellet diffusion resistance or external mass transfer resistance was already considered in the work Efstathiou et al. [5] and proved
V.V. Krishnan et al./Applied Catalysis A: General 151 (1997) 267-287
269
experimentally to be insignificant in the behavior of the isothermal uptake curves. Considerable work has been done, regarding the measurement of diffusivities by Ruthven, Karger and co-workers. [9,12-14,16] The techniques used in these references have been primarily: Zero Length Chromatography (ZLC), gravimetric techniques (with microbalance), IR measurements and self diffusion experiments by Pulsed Field Gradient (PFG) NMR. FTIR spectroscopy has also been used to determine diffusivities in zeolites by Niessen and Karge. [15] The Fickian diffusivity obtained by ZLC and gravimetric techniques is considerably different from that observed by PFG methods. The reason for this discrepancy is still unclear. Some of our predictions for the Fickian intracrystalline diffusivity have been compared with other organic-zeolite systems available in the referenced literature.
2. Mathematical model 2.1. Isothermal model
The reactor used in the experiments done by Efstathiou et al., [5-8,11] was verified to be a CSTR, based on prior work. It was also assumed that the crystallites in the zeolite were spherical and the Fick's Law for unsteady state diffusion in a spherical geometry was thus applied. The adsorption of cyclopropane in NaX is an exothermic process but considering the extremely small quantity of catalyst used (18 mg), isothermality was assumed. Keeping these assumptions in mind, the following material balance equations were postulated: 1. Unsteady state material balance of cyclopropane in the CSTR microreactor: dt
-- 1 -- Cg(}) - ~, ra [ O?Jr=,
The variables may be defined as follows: Cg = f ( - ~ ) /
Cinle t .
(l(ii))
t=t/r. r= (V-
(l(i))
W)/(
? = r / r c. r d = r2c/Oe .
PsQ).
(l(iii)) (l(iv)) (l(v))
ce~ = 3 w / ( a p s ) .
(l(vi))
C r = C s a t / f i n l e t.
(1 (vii))
2. Unsteady state material balance of the "sorbed" phase inside the crystal:
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V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
D e, the effective diffusivity is assumed to be independent of the intracrystalline concentration because, the crystal size is very small (0.6 txm, radius) and the levels of concentration may be sufficiently low to ignore this dependence. ~0(?,})
-~7
[ 320(?,t)
--0/2[
2 30(?,t) ]
~'~"~ -'{- r
~
]'
(2)
where 0[. 2 =
T/
(2(i))
Td .
Initial condition: t=0,
Cg(0) = 0 ,
t=0,
0(?,0) =0.
(2(ii))
Boundary conditions: r=0,
?=0,
3?=0
'
r = r c,
?=1,
O(1,-t)=Cg/(Cg+l/K).
(2(iii)) (2(iv))
The last boundary condition indicates that at the entrance of the crystal, the adsorption of cyclopropane occurs instantaneously by a simple Langmuir type of isotherm. [5,11] The transport of the adsorbed phase in the crystals is what has been described by the "effective diffusivity" in the Fick's Law equation. The values of the various constants have been taken from the experimental work of Efstathiou et al. [5].
K=bCiHRT, O=bP/(1 + bP).
(2(v)) (2(vi))
The constant b is the original Langmuir constant represented in the Langmuir equation, above. The parameter K has been postulated to represent the Lmagmuir adsorption using dimensionless gas phase concentration instead of partial pressure as in Eq. (2)(vi). O=C(r,t)/Csa
t.
(2(vii))
The variable 0 represents the dimensionless "coverage" of the adsorbed species as a function of radial distance in the crystallite and time. Csat is the saturation value of the adsorbed phase concentration at the temperature of sorption. In zeolites, Csat is observed to be a function of temperature [9], but we have considered the maximum value of the saturation concentration, which occurs in the room temperature. Lower temperature experiments may give more saturation uptake, but that could include some physisorption which is undesirable. From the work of Efstathiou et al. [5], the value of C~at is taken as 0.0043 m o l / c m 3. The radius of the crystallite (re), obtained by SEM micrographs [5], is about 0.6 ta,m. The porosity of the bed, used in the work of Efstathiou et al. [5], was
271
V.V. Krishnan et al./Applied Catalysis A: General 151 (1997) 267-287
estimated to be about 0.4. The inlet concentration (Cin) of cyclopropane used was 0.5%, i.e. 1.95 × 10 -7 m o l / c m 3. The volume of the reactor has been estimated to be about 0.785 cm 3 and the weight of the adsorbent bed used was 18 mg. The density of the solid phase of the zeolite NaX was found to be about 1.46 g / c m 3.
2.2. Non-isothermal model The non-isothermal model is derived based on the presence of heat transfer resistance external to the pellets. It is assumed that the pellet heats up uniformly and the temperature gradient occurs entirely in the film external to the pellet. To take into account the non-isothermality of the process, the diffusivity and the equilibrium constants are expressed in terms of their Arrhenius dependence with temperature: D = Doexp[- EJ(RT)],
(3(i))
K = A e x p [ - A H / ( R T ) + AS~R],
(3(ii))
where D O is the pre-exponential factor and E a and A H are the activation energy of diffusion and the enthalpy of adsorption, respectively. Assuming that the change in entropy, AS, is invariant with temperature, the value of K can be approximated as follows:
K = K o e x p [ - A H / ( RT)].
(3(iii))
For the non-isothermal model, Eqs. (1) and (2) remain the same, with modifications in the constants to accommodate the temperature changes, but the energy balance needs to be carried out also, and may be written as follows: d~/(t) dt
O0(1,t) 0~30/2 ~
O~4T('t ~ -- 1).
(3)
Initial conditions: Cg(0) = 0,
0(?,0) =0,
7 ( 0 ) - - 1.
(3(iv))
Boundao' conditions:
00(o,?) - -
O?
=0,
(3(v))
Cg 0(1,}) = Cg + l / K ( r / ) "
(3(vi))
The dimensionless constants have been modified and re-written as follows: ~ ( t ) -= Tpellet(})/Tb,
(3(vii))
rc
rd = Ooexp[_Ed/(RTb~?) ] .
(3(viii))
V.V. Krishnan et aL / Applied Catalysis A: General 151 (1997) 267-287
272
This modification leads to a temperature dependence of the constants such as a 2. It also introduces two new constants which are indirect functions of temperature, a 3 and a n, defined as follows, in Eqs. 3(ix) and 3(x): 3(- A/~)C~ a3=
ps Cp~Tb
,
(3(ix))
3h O~4= PsCps rp(1 -- ep)"
(3(x))
These constants are related to the density and specific heat capacity of the solid catalyst (denoted by the subscript " s " . The subscript " p " denotes "pellet", because the experiments had been done by using small pellets of average diameter of about 0.5 mm). The heat transfer film coefficient, h, has been predicted according to the simplest case, i.e. a spherical particle of radius rp, in a stream of near stagnant gas with Reynolds number approaching zero. The Nusselt number approaches 2 in this case.
h = 2k/de.
(3(xi))
The heat transfer coefficient is a linear function of the thermal conductivity of the adsorbate gas mixture according to this assumption. This considers the worst possible case where the rate of heat removal from the catalyst pellet is the least.
2.3. Solution of the combined system of differential equations The solution of a combined system of ordinary and partial differential equations has to be obtained by using numerical techniques because the two equations are interlinked by the diffusive flux term at the intracrystalline boundary. This problem has been solved by using a program called differential systems solver (DSS), based upon a technique known as the numerical method of lines, developed by Prof. W.E. Schiesser of Lehigh University. A subroutine specifying the problem with its initial and boundary conditions was written, compiled and linked with a Microsoft Fortran compiler.
3. Results
3.1. Isothermal model The results presented in our simulation indicate the trends obtained by variation of parameters such as diffusivity and equilibrium constant at different conditions of flow rate. Fig. 1 shows the variation of the uptake curve at various flow regimes. Fig. 2a and Fig. 2b show the simulated gas phase response with
V. V. Krishnan et al. / A p p l i e d Catalysis A: General 151 (1997) 267-287
273
1.0 5 2 0.8 r,,,)=
i
0.6
~e
0.4
1 - 0.34cc/s
"~
2 - 0 . 5 0 cc/s 3 - 1.50 cc/s 0.2
4 - 2 . 5 0 cc/s 5 - 6 . 0 0 cc/s
0.0 0
q
r
~
,
200
400
600
800
1000
Real time, s Fig. 1. V a r i a t i o n o f u p t a k e c u r v e s w i t h f l o w rate D e = 2 . 0 E -
11 c m 2 / s ;
K = 0.65; N a X .
dimensionless time at different values of diffusivity. Fig. 2b shows the increase in the resolution in the uptake response with diffusivity at a higher flow rate of 0.75 cm3//s. The diffusivity of 0.5% cyclopropane/argon in the NaX zeolite is approximately in the range 10-11-10 -~2 c m 2 / s and such values have been chosen in the simulation. At the low flow rate of 0.34 cm3/s, the effect of diffusivity does not appear to change the uptake curve significantly. Varying the equilibrium constant seems to have a stronger effect, as shown in Fig. 3a. Figs. 4 and 5 show the effect of intracrystalline diffusivity on the concentration profile of the adsorbed phase inside the crystal at various times. The spatial concentration gradients are lower at a diffusivity of about 2 X 10 -11 c m 2 / s than those at a diffusivity of 4 X 10 -12 c m e / s , which would be expected. For a system with an effective diffusivity of 2 × 10 -1~ cm2/s, the response time to achieve a near zero concentration gradient of the intracrystal phase is less than 200 s (Fig. 5) for a flow rate of 0.34 cm3/s, whereas the total time taken for the gas phase concentration to equilibrate at the same flow rate (Fig. 2a) is over 1000 s. The implications of this observation will be discussed in the next section. Fig. 6 shows the comparison of the simulated gas phase response with the experimental data from the work of Efstathiou et al. [5]. The values of the diffusivity have been arrived at by trial and error and they indicate an excellent fit with the experimental data. The diffusivity was estimated to be about 2.0 X 10-11 c m 2 / s and the equilibrium constant about 0.7, at a temperature of 40°C, at which the original experiments were performed [5].
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V. V. Krishnan et al. /Applied Catalysis A: G e n e r a l 151 (1997) 267-287
1.0 0.8
=
/
0.6
D --9.0E-12cm/s
0.4 De=2.0E-I1cm2/s 0.2 0.0 0
100
200
300
400
500
600
Dimensionless time
1.0
b)
0.8
/
i 0.6
De=4.0E-12c m ' / s
~
De=9"0E-12cm2/s
"~ 0.4 ///
0.2
0.0
0
D =2.0E-11cm2/s
i
i
i
i
i
100
200
300
400
500
600
Dimensionless time
2. (a) Variation o f diffusivities at a l o w f l o w rate: 0.34cm3/s. Equilibriumconstant = 0.65; NaX.(b) Variationo f diffusivities at a f l o w rate o f 0.75 cm3/s. Equilibrium constant = 0.65;NaX. Fig.
Figs. 7 - 1 0 are similar to the simulations shown from Figs. 1-4, but these are done at high flow rates: 1.5 cm3/s. Unfortunately, the experimental data at 1.5 c m 3 / s are not available for us to curve-fit with the aid of our simulation. It can be seen that the variation in diffusivity caused a more significant change in the uptake curve, as the flow rates were increased from 0.34 cm3/s to 0.75 c m 3 / s
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
1.0
275
(a)
0.8
0.6 ~
K--0.5]4
0.4
0.2
0.0 0
I
I
l
7
T
100
2~
3~
4~
5~
6~
Dimensionless time
0.8
o.6
°i///
"~ 0.4
0.2
0.0
] . . . . . . . 0
100
200
i 300
400
500
Dimensionless time
Fig. 3. (a) Variation of equilibrium constants at low flow rate: 0.34 cm3/s. Diffusivity = 9.0E- 12 cm2/s; NaX. (b) Variation of equilibrium constants. Intermediate flow rates: q = 0.75 cm3/s; diffusivity = 9.0E- 12 cm2/s.
(Fig. 2b) and all the way up to 1.5 cm3/s. The adsorbed phase concentration profile at a diffusivity of 2.0 × 10 -11 cm2/s as shown in Fig. 8 may be compared to Fig. 5 (low flow rate). The effect of the flow rate on the concentration gradient can be seen by a mere visual comparison of the two
276
V. V. Krishnan et al. / A p p l i e d Catalysis A." General 151 (1997) 267-287 0.5
t = 500 s
0.4
8 -8 o.o t = 200 s
0.2 0.1
==
5 o.o
t=lOs
-0.1
0.0
I
I
F
I
0.2
0.4
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 4. Adsorbed phase concentration vs. intracrystal distance; NaX. D e = 4 . 0 E - 12 c m 2 / s , Q = 0.34 c m 3 / s , K = 0.65.
figures. Fig. 9 shows the immense difference in the adsorbed phase concentration profile due to a drop in the diffusivity from 2 . 0 X 10 - u c m 2 / s to 4.0 x 10 -12 cm2/s. A strong concentration gradient exists even after 200
0.5
e~
0.4
0.3
8
t = 200 s
0.2 t=50s
o
•~ 0.1 t=30s t=lOs
0.0
-0.1
I
0.0
0.2
-
~
0.4
I
7
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 5. Adsorbed phase concentration vs. intracrystal distance; NaX. D~ = 2 . 0 E - 11 c m 2 / s , Q = 0.34 c m 3 / s , K = 0,65.
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
277
1.0
0.8
0.6
"~
Simulated data
0.4
N 0.2
0.0 0
I
r
100
200
7
.....
300
I
I
I
I
400
500
600
700
800
Real time, sec Fig. 6. Comparison of experimental data vs. simulated response; NaX. Switch: Ar ~ cm3/s; De = 2 . 0 E - 11 cm2/s; K = 0.7.
0.5% CP/Ar. Q = 0.5
seconds (Fig. 8) and this causes a significant difference in the gas phase uptake curve at a diffusivity of 4.0 X 10- ~2 cm2/s, in comparison to the uptake curve at a diffusivity of 2.0 X 10 -~1 cm2/s. Fig. 10 seems to be identical to Fig. 3, which indicates that changes in the uptake curve due to variation in the equilibrium constant are similar in nature regardless of the flow rates. 1.0
~
0.8
0.6
~
0.4
0.2
! 0.0 0
100
200
300
400
500
600
Dimensionless time Fig. 7. Variation of diffusivities at a high flow rate: 1.5 cm3/s. Equilibrium constant: K = 0.65; NaX.
278
V. V. Krishnan et al. / A p p l i e d Catalysis A: General 151 (1997) 267-287 0.5
t = 800 s
0.4
0.3 t = 200 s
= 0.2
= o.1
E o.o
t=10s
-0.1 0.0
0.2
0.4
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 8. A d s o r b e d p h a s e concentration vs. intracrystal distance; NaX. D r = 2 . 0 E - 11 c m 2 / s , Q = 1.5 c m 3 / s , K = 0.65.
Figs. 12-14 show how the diffusivity and the equilibrium constant have been changed to curve-fit the transient responses for the argon to cyclopropane/argon switch in the case of systems like N i / N a X , E u / N a X and NaX (at 80°C). The 0.5
ee~
t = 800 s
0.4
0.3 0.2
8 0.l .o
o.0
t= 10s
-0.1 0.0
I
I
I
I
0.2
0.4
0.6
0.8
1.0
Dimensionless intracrystalline distance Fig. 9. A d s o r b e d p h a s e concentration vs. intracrystal distance; NaX. D e = 4 . 0 E - 12 c m 2 / s , Q = 1.5 c m 3 / s , K = 0.65.
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
279
1.0
K
0.8
=
0
.
5
1
~
K=0.65 0.6 5 •~ 0.4
0.2
0.0 0
I
t
I
I
I
100
200
300
400
500
600
Dimensionless time
Fig. 10. Variation of equilibriumconstants at a high flow rate: 1.5 cm3/s. Diffusivity = 9.0E-12 cm2/s: NaX.
diffusivity of cyclopropane in Ni-exchanged NaX (Fig. 12), is evaluated to be about 2 × 10-12 cm2/s, with an equilibrium constant ( K ) of 1.3 (corresponding to 0.33 Torr -1, in dimensional terms). For the E u / N a X system (Fig. 13), a diffusivity of about 3 × 10-12 c m 2 / s and an equilibrium constant of about 0.58 (b = 0.15 Torr -1) gave us the best curve fit with the experimental data. In both the cases, the diffusivity has clearly dropped by almost an order of magnitude in comparison with NaX at 40°C (Fig. 6). The uptake curve for the NaX system at 80°C with a 0.3% cyclopropane/argon mixture has been simulated in Fig. 14. The diffusivity of this system remains at about 1.0 × 10 -11 c m 2 / s but the equilibrium constant has decreased considerably to about 0.15 (b = 0.039); understandably, because adsorption is an exothermic process. The implications of these values are discussed in the next subsection. 3.2. Non-isothermal model
The results of the non-isothermal model are shown in Fig. 11. The maximum rise in temperature accompanying the adsorption and diffusion of the gas mixture is only about 1.3°C for argon as a carrier gas under the conditions of the experiment (0.5 cm3/s). The interesting thing is the increase in the rise in temperature at a higher volumetric flow rate of the adsorbate gas mixture. The simulations run at a flow rate of 2.5 cm3/s, show a maximum temperature increase of about 2.4°C. This aspect will be discussed in the next section. For hydrogen as a carrier gas, the maximum possible temperature is even smaller,
280
V. V. Krishnan et al. / Applied Catalysis A." General 151 (1997) 267-287 3.0
2.5 ;,,)
2.0 =
,~ 1.5
.
[,. .~ 1.0
0.5
0.0 I
I
I
I
I
I
0
100
200
300
400
500
600
Dimensionless time Fig. 11. Rise in temperature (°C) in the catalyst bed. Non-isothermal model (heat transfer film resistances included). H = - 8.0 K C a l / m o l ; E d = 8.9 K C a l / m o l ; NaX.
only about 0.3-0.4°C. Due to the extremely small values of the temperature increase, inter-pellet heat transfer resistance is not a factor. The assumption of isothermality is therefore justified.
4. Discussion 4.1. Isothermal model
The low flow rate of 0.34 c m 3 / s does not seem to be a good condition for doing uptake experiments because the intracrystalline adsorbed phase concentration gradient, as shown from our simulation (Figs. 4 and 5) diminishes drastically in less than 200 seconds in both cases. The sensitivity of the ~0/07 term at the crystal boundary, in response to changes in diffusivity, is the primary element causing a change in the uptake curve. If however, the system behaves like the NaX system, i.e., a diffusivity of about 2.0 × 10 -11 cm2/s, then any experiment done in the range of about 0.34 c m 3 / s will be ineffective in resolving small changes in diffusivity, because of the lack of concentration gradients inside the crystal (Fig. 5). The effect of intracrystalline concentration gradients is strong up to 200 seconds or so, which is in the early phase of the adsorption uptake curve. The
V. V. Krishnan et al. / Applied Catalysis A: General 151 (1997) 267-287
281
diffusivity is therefore an important parameter at the early stages of the uptake experiment. At higher times, the concentration gradients will approach zero anyway as the uptake curve approaches equilibrium feed concentration. Based upon the same reasoning, simulations at a high flow rate of 1.5 cm3/s can now be analyzed. The differences in the uptake curve with changes in diffusivity are more discernible (Fig. 7) and can be easily explained on the basis of the intracrystalline concentration gradient; compare Figs. 8 and 9. A change of diffusivity from 2 . 0 × 10 -1! c m 2 / s to 4 × 10 -12 cm2/s shows drastic differences in the concentration profiles of the adsorbed phase. It is clear that in order to do uptake experiments, it is essential to go to higher flow rates of over 1 cm3/s, in order for diffusivity to play a significant role in the uptake curve. The only limitation could be due to the time at which equilibrium is achieved. If the transient time is too small, small changes in uptake curves may not be easily detected upon changing parameters. The advantage of working with lower flow rates is the large transient time of the uptake curve, but this may not indicate a diffusion controlled regime. Fig. 1 shows a comparison of simulated uptake curves upon changes in flow rates. This gives an idea of the time scale of the transient curves in the cyclopropane/argon - NaX system. The equilibrium constant seems to be a very important criterion because of the strong chemisorptive interaction of cyclopropane with the NaX zeolite [5]. That is the reason why the adsorption of cyclopropane has been used as the boundary condition. An equilibrium constant of 0.7 which has been obtained by curve-fitting, corresponds to a value of 0.18 Torr- 1, on a dimensional scale. Evaluation of the enthalpy of adsorption has also been done with the aid of curve-fits by the non-isothermal model. A value of A H = - 8 . 0 kCal/mol is arrived at based on this exercise. The activation energy of diffusion, Ej, is estimated to be about 8.9 kCal/mol. These are approximate values, because of the non-availability of uptake data at higher temperatures. The above mentioned values correspond to the diffusivity and equilibrium constant found by curve-fitting the uptake curves at 40°C. The N i / N a X system [7] shows a long time for equilibration (Fig. 12), though the initial rise in the uptake curve is rapid. As discussed before, for a system with a diffusivity of about 2 × 10 -12 cm2/s, the initial part of the uptake curve is more dependent on the diffusivity of the system rather than the equilibrium constant. At higher times, the transient response is unable to equilibrate within a reasonable amount of time, which can be explained by a strong adsorptive interaction of cyclopropane, as shown by a high value of K (1.3). The E u / N a X system [6] shows a smaller equilibrium time than NaX, but also weaker adsorptive interaction with cyclopropane gas. This is clearly seen by the diffusivity of 3 X 10-~2 c m 2 / s (nearly an order of magnitude smaller than that obtained for NaX). The simulation of the cyclopropane/argon transient curve for NaX at 80°C is shown in Fig. 14. The equilibrium constant is drastically lowered to about 0.15
282
V. K Krishnan et al. /Applied Catalysis A: General 151 (1997) 267-287 1.0
0.8
0.6
8 o 0.4
0.2
0.0 0
r
i
i
500
1000
1500
2000
Time, s Fig. 12. C o m p a r i s o n - m o d e l vs. experimental data: N i / N a X . Switch: A r ~ D e = 2 . 0 E - 12 c m 2 / s ; K = 1.3.
0.5% C P / A r . Q = 0.5 c m 3 / s ;
(b = 0.064 Torr -~) from 0.7 for NaX at 40°C and it contributes directly to the smaller response time. The diffusivity is still about 1.0 × 10 - l l cm2/s. This is because of our reasoning that at values of diffusivity greater than 10-~1 cme/s,
1.0
0.8 a~
.~ 0.6
"
d data
0.4 ..~ 0.2
0.0 0
~
f
500
1000
1500
Time, s Fig. 13. C o m p a r i s o n - m o d e l vs. experimental data, E u / N a X . Switch: A r ~ De = 3 . 0 E - 12 c m Z / s ; K = 0.58.
0.5% C P / A r . Q = 0.5 cm3/s;
V. V. Krishnan et al. / A p p l i e d Catalysis A: General 151 (1997) 267-287 1.0
283
yaW¢*,-
0.8
0.6
0.4 .I
0.2
I
0.0 0
i
F
~
i
100
200
300
400
500
Time, s Fig. 14. Comparison - model vs. experimental data: NaX at 80°C. Switch: Ar ~ c m 3 / s ; De = 1.0E-11 c m 2 / s ; K = 0.15.
0.3% C P / A r . Q = 0.5
the intracrystalline concentration gradients become very small. Diffusivities in this regime cannot be resolved at this flow rate (0.5 cm3/s). However, the flow rate of 0.5 cm3/s may be adequate to resolve diffusivities for both E u / N a X and N i / N a X because their diffusivity is as low as 2-3 x 10-12 c m 2 / s at 40°C, which means that intracrystalline concentration gradients play a significant role in the uptake process. Ruthven et al. [10] have used Zero Length Chromatography (ZLC) extensively to measure the diffusivity of o-xylene in NaX. Upon extrapolating the diffusivities measured in this work to about 40°C, we obtain a value of 2.0 X 10 -l° cm2/s for the o-xylene/NaX system. Gravimetric techniques (Ruthven et al. [12]) and ZLC yield consistent results for the benzene/NaX system: about 4.0 X 10 -9 cmZ/s, extrapolated to 40°C. Using gravimetric techniques, Ruthven et al. [13] have estimated "corrected diffusivities" to be about 2 x 10 -11 cmZ/s at 150°C for the cyclohexane/silicalite system. Upon extrapolation of this data to about 40°C, the diffusivity of this system may be as low as 2 - 3 X 10 -13 cmZ/s. FTIR experiments by Niessen et al. [15] have been used to determine the Fickian diffusivity of p-xylene in H-ZSM-5, which is about 8 X 10-10 cmZ/s at about 100°C. Extrapolation of these values to 40°C yields a diffusivity of about 8 X 10 -11 cmZ/s by using adsorption data. Karger, Ruthven and co-workers [14] have used IR, PFG-NMR and ZLC techniques for measurement of diffusiv-
284
V.V. Krishnan et aL /Applied Catalysis A: General 151 (1997) 267-287
ity of methanol in NaX crystals. The IR data in reference 14 give a value of about 2.6 x 10 -8 c m 2 / s for the "corrected" diffusivity for methanol in NaX at 27°C. Ruthven and Karger [14] have obtained a value of 0.8 X 10 -8 cmZ/s from the ZLC data at 40°C and about 1.5 X 10 -8 c m 2 / s for the same system, by using PFG-NMR methods. Earlier measurements by Karger and Ruthven [16] in the benzene/NaX system, using the PFG-NMR technique, lead to a value of self-diffusivity of about 10 -6 cm2/s, considerably higher than with other techniques mentioned in the above references. This discrepancy has been commented upon in several references [9,10,12,13] but is something that has not yet been resolved. 4.2. Non-isothermal model
To explain the increase in rise in temperature of the system (AT) with higher flow rates, assuming film heat transfer resistance, we will have to review the dimensionless constants in the equations defined earlier. One would expect under normal conditions that an increase in heat transfer coefficient will occur with increase in flow rate, thereby causing heat removal at a faster rate, leading to a smaller temperature buildup in the catalyst pellets. According to our simulation, the opposite result is observed (Fig. 11). An increase in flow rate results in a higher d C g / d t , ot2 in Eq. (2) decreases because of the decrease in residence time in the reactor. This decreases the temporal derivatives (Eq. (2)) of the interior points in the crystal, i.e 00/Ot. The higher value of the gas phase concentration results in a higher value of the adsorbed phase concentration at the crystal boundary (from the Langmuir boundary condition). The end result is an increase in the concentration gradient at the crystal boundary (term 1, Eq. (3)). If this term dominates, the trend shown in Fig. 1 will be observed. Perhaps, under conditions of very high flow rate, the second term of Eq. (3) (dimensionless constant, a 4) may be dominant and that might result in a reversed trend. Under very high flow rates, however, the heat transfer coefficient cannot be calculated from near stagnant fluid assumptions (Nusselt number = 2) because it is now a function of the Reynolds and Prandtl numbers. This analysis is beyond the scope of this work. The simulations involving a switch in the gas from argon to 0.5% cyclopropane/hydrogen have not been able to reproduce the sigmoidal signal obtained in Ref. [5]. It is suspected that this carrier gas "effect", shown in the work of Efstathiou et al. [5], may be due to the inlet system in the mass spectrometer, because changes in two gases with diverse molecular weight cause a lag time at the inlet. The step function therefore may not be exact. An in-depth analysis of the inlet system is being done to observe, by fluid mechanics, how the pressure at the inlet changes by changing the molecular weight of the feed. This will be part of an ensuing paper.
V. V. Krishnan et al. /Applied Catalysis A: General 151 (1997) 267-287
285
5. Conclusion
The value of diffusivity arrived at for the cyclopropane/argon-NaX system is about 2.0 × 10 -ll c m 2 / s and seems to be in the 10-9-10 -12 c m e / s range for similar organics in other zeolites, measured by other techniques in the literature. The prerequisites for modeling no doubt include accurate physical data, including the density of the solid, the crystal size and void fraction of the bed a n d / o r the pellet. Upon estimation of these quantities, the number of adjustable parameters reduces to just two - the equilibrium constant and the diffusivity. The value of the equilibrium constant from the Langmuir isotherm is very close to that obtained by previous work [5]. A simple Henry's Law equation for predicting equilibrium is not applicable in this case, because the partial pressure of the sorbate gas is too high. The values of diffusivity obtained by simulating the response for E u / N a X and N i / N a X are almost one order of magnitude smaller than that for NaX at 40°C. The low values of the intracrystalline diffusivities for the Eu and Ni exchanged NaX systems indicate the presence of concentration gradients at the intracrystalline level even at a flow rate of 0.5 cm3/s. The model we have proposed is, therefore, the simplest model which can be used to simulate the sorption uptake process using a laboratory scale flow system. There are no mathematical simplifications needed, since numerical techniques have been used. It is further possible to increase the complexity of the diffusivity and the equilibrium terms and still use numerical analysis without having to worry about non-linearity. The modeling work also shows us the correct regime in which the experiments need to be performed, in order to obtain maximum information on the diffusivity and equilibrium constants.
6. List of notations /-
r~ t
c(~) C~at Cinlet
Cr
Radial distance in the spherical crystallite. Radius of the crystallite. Real time (s). Residence time in the microreactor (s). Intracrystalline diffusional time constant (s). Dimensionless distance in the crystallite. Dimensionless time. Bulk gas phase concentration in the reactor (mol/cm3). Saturation concentration of the adsorbed phase in the crystallite (mol/cm3). Inlet gas phase concentration (mol/cm3). Ratio of the saturation adsorbed phase concentration to the inlet gas concentration.
286
V.V. Krishnan et al./Applied Catalysis A: General 151 (1997) 267-287
0 W V P~
g De K Do AH AS Tpellet
C,s
rp h k
%
Dimensionless gas phase concentration. Adsorbed phase concentration (mol/cm3). Dimensionless adsorbed phase concentration. Weight of the catalyst bed (g). Volume of the microreactor (cm3). Density of the solid phase of the catalyst (g/cm3). Volumetric flow rate of the adsorbate gas mixture (cm3/s). Effective Fickian diffusivity of the sorbate gas (cm2/s). Langmuir equilibrium constant (dimensionless). Pre-exponential factor in the estimation of diffusivity (cmZ/s). Activation energy of diffusion (Kcal/mol). Enthalpy of adsorption (Kcal/mol). Entropy of adsorption (Kcal/mol • K). Temperature of the pellet (°C). Temperature of the bulk fluid (°C). Dimensionless temperature of the pellet. Specific heat of the solid pellet (Kcal/mol • K). Radius of the catalyst pellet (cm). Heat transfer coefficient of the fluid to the particle (Kcal/cm 2 • s • K). Thermal conductivity of the gas (Kcal/cm • s • K). Porosity of the pellet.
Acknowledgements We acknowledge the support of the Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences for this research. The authors also wish to thank Prof. M.B. Cutlip (Department of Chemical Engineering, University of Connecticut) for allowing us to use his DSS software and a copy of his Microsoft Fortran compiler, for this research.
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C.O. Bennett, Catal. Rev., 13 (1976) 121. C.O. Bennett, ACS Symp. Ser., 178 (1982) 1. D.M. Stockwell, J.S. Chung and C.O. Bennett, J. Catal., 112 (1988) 135. A. Efstathiou and C.O. Bennett, J. Catal., 120 (1989) 137. A.M. Efstathiou, S.L. Suib and C.O. Bennett, J. Catal., 131 (1991) 94. A.M. Efstathiou, S.L. Suib and C.O. Bennett, J. Catal., 135 (1992) 223. A.M. Efstathiou, S.L. Suib and C.O. Bennett, J. Catal., 135 (1992) 236. M.W. Simon, A.M. Efstathiou and C.O. Bennett, J. Catal., 138 (1992) 1. D.M. Ruthven, Principles of adsorption and adsorption processes, Wiley, New York. M. Eic and D.M. Ruthven, Zeolites, 8 (1988) 40.
V.K Krishnan et al. /Applied Catalysis A: General 151 (1997) 267-287
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