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Experimental observations of isolated surface steady-state branches
Surface Science 146 (19S4) L569-L575 North Holland, Amsterdam
L569
SURFACE SCIENCE LETTERS EXPERIMENTAL OBSERVATIONS OF ISOLATED SURFACE STEADY-STATE BRANCHES T.H. L I N D S T R O M
and T.T. T S O T S I S
Department of Chemical Engineering, Unit,ersiO' of Southern California, Los Angeles. California 90089-1211, USA Received 9 March 1984; accepted for publication 19 June 1984
In this paper we report infrared spectroscopic observations of isolated surface steady-state branches during CO oxidation over Pt/y-Al~O 3 catalysts, These are branches that cannot be reached by continuous smooth changes of the experimentally controlled variables, but only through global perturbations of the catalytic reaction system.
Several theoretical investigations [1-4] have p r e d i c t e d the existence of isolated s t e a d y - s t a t e b r a n c h e s (isolas) for catalytic reaction systems. Until recently, however, no e x p e r i m e n t a l evidence of such b r a n c h e s was available [5,6]. Luss a n d c o w o r k e r s were the first to r e p o r t e x p e r i m e n t a l o b s e r v a t i o n s of isolated s t e a d y - s t a t e b r a n c h e s for the C O o x i d a t i o n reaction system on P t - t y p e c a t a l y s t s [6]. In their e x p e r i m e n t s they m o n i t o r e d the s t e a d y - s t a t e t e m p e r a t u r e of a single catalyst pellet s u s p e n d e d in a t u b u l a r reactor as a function of inlet s t r e a m c o m p o s i t i o n . In this work we present a d d i t i o n a l evidence for the existence of such isolated s t e a d y - s t a t e branches. D u r i n g our experiments, carried out in a C S T R , we have s i m u l t a n e o u s l y m o n i t o r e d the catalyst surface state, by I R T r a n s m i s s i o n S p e c t r o s c o p y , as well as the reaction rate a n d c a t a l y s t t e m p e r a t u r e as a function of inlet stream c o m p o s i t i o n and t e m p e r a ture. O u r e x p e r i m e n t a l setup and p r o c e d u r e s have been d e s c r i b e d in detail elsewhere [7,8]. Special p r e c a u t i o n s have been taken d u r i n g our studies to e l i m i n a t e gas p h a s e i m p u r i t i e s in the inlet r e a c t a n t s t r e a m b y utilizing a series of gas p u r i f i c a t i o n steps and cold traps. N o d e t e c t a b l e gas p h a s e i m p u r i t i e s ( < 1 p p m ) were p r e s e n t d u r i n g o u r experiments. T h e reactor cell was m a d e from Pyrex glass with C a F 2 w i n d o w s sealed to the o u t s i d e surface with Viton o-rings: The r e a c t a n t gases d o not c o m e in direct c o n t a c t with a n y o t h e r reactor m a t e r i a l besides the Pyrex glass wails a n d C a F z windows. Sonic orifice meters were utilized to achieve c o n s t a n t r e a c t a n t flows. T h e r e a c t o r cell has been shown to b e h a v e as a C S T R for the flow c o n d i t i o n s 0 0 3 9 - 6 0 2 8 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
T.H. Lindstrom, T.T. 7~9otsis / Isolated surface steady-state branches
L570
utilized in this investigation [8]. The catalyst wafer (1 in. diameter, 30 m g / c m : thick) was prepared by impregnation of "/-alumina (Degussa type C. BET surface area 100 m2/g, average particle size 200 A) with H2PtCIe,. 6 H 2 0 solution up to a total Pt content of 3.2%. (Further details of our catalyst preparation techniques can be found elsewhere [8].) Before the series of experiments reported here was initiated, the catalyst was treated for 24 h in a 10% CO in air mixture. This treatment eliminates the catalyst's ability to form the 2120 cm ] band reported earlier [7]. The experimental results presented here are part of an ongoing investigation of the steady-state and the dynamic behavior of this and other catalytic oxidation reaction systems. Fig. 1 shows the "bifurcation set" of this reaction system in the Tg (gas phase temperature), %CO in the inlet reactant stream parameter space for two distinct flowrates (40 and 50 cm3/s). This figure was constructed by running a series of experiments, where the inlet gas phase temperature was kept constant (within + 1 °C) and the composition was varied (0.65%-8.0% CO in synthetic air, i.e., a 20.93% O 2, 79.07% N 2 mixture). In the figure we present the concentration values (within 0.1% CO) for which (self-induced) reaction ignition and extinction p h e n o m e n a were observed. A similar figure for the same catalyst has been obtained by a series of experiments,
Tg °C '
,,~~~ ~o~/s
'E80 t60
lily
t4O 120
r
'I00 80 60 4C 2C
t.0
2.0
3.0
4.0
5.0 %C0
Fig. 1. "Bifurcation set" for the CO oxidation reaction ~ystem. T~ versus %CO in inlet stream.
T.H. Lindstrom, T.T. Tsotsis /lsolated surface steady-state branches
L571
w h e r e the gas p h a s e c o m p o s i t i o n was k e p t c o n s t a n t a n d the t e m p e r a t u r e was varied. E a c h e x p e r i m e n t in the series was p r e c e d e d b y a p r e t r e a t r n e n t o f the c a t a l y s t , w h i c h c o n s i s t e d of a 2 h e x p o s u r e to f l o w i n g N 2 f o l l o w e d b y a 2 h
A/•,s t,0 0.9
I
0.8 0.7 0.6
I
Extinction
I
0.5 0.4 0.3
t
0.2 0.!
,
,
t.0 2.0 3.0 4.0 5.0 6,0 7.0 %C0
J
A/A s
1.0 O.9 O,8 0.7 0.6 O.5 0.4 0.3 0.2
It jl
I Forced ]I~nitioo
II
t
It t
0.~ I
I
I -
J.
2
f
•
I
~t
4.0 2.0 3,0 4.0 5.0 6,0 7.0 8.0%C 0 b Fig. 2. "Bifurcation diagram" for the CO oxidation. Normalized absorbance versus ~CO in inlet stream (50 cm3/s): (a) Tg = 180°C; (b) Tg = 120°C.
72 H. Lindstrom, 71 T. Tsotsis / Isolated surface steadv- state branches
L572
e x p o s u r e to f l o w i n g s y n t h e t i c air at c a t a l y s t t e m p e r a t u r e s e x c e e d i n g 2 0 0 ° C . T h i s p r e t r e a t m e n t p r o d u c e d a s p e c t r o s c o p i c a l l y c l e a n c a t a l y s t surface. E a c h s t e a d y - s t a t e s h o w n in figs. 2 a n d 3 was m o n i t o r e d for m o r e t h a n 2 h f o l l o w i n g initial s t a b i l i z a t i o n .
A/A s 4.0
x
0.9 0.8
,
t
I
0.7 0,6 0.5
Forced lqnition
_ ~
,
I
- II
I
0.4 0.3
II
0.2 0,1 t
I
,.~
•
~.
1
•
~.0 2.0 3,0 4.0 5~'0 6.0 7]0 %C0 O
A/A s i.O
i.
. . . .
}-(
0,9
Z
0.8-
I
0.70.6-
Forced Ignilion
0.5-
o.4~ 0.3 0.2
×t
O,i
e~
>_ 4,0 2,0 3.0 4,0 5.0 6,0 7,0 */,,CO b Fig. 3. "Bifurcation diagram" for the CO oxidation. Normalized absorbance versus %CO in inlet stream (50 cm3/s): (a) Tg = ll0~C; (b) Tg = 100°C. I
t
l
t
1
1
7£H. Lindstrom, T. T. Tsotsis / Isolated surface steady-state branches
L573
Figs. 2 and 3 contain the four different types of "bifurcation diagrams" observed for this reaction system at 50 cm~/s. What we present here is the normalized (in terms of maximum) absorbance of the linearly adsorbed CO species (around 2088 cm-a). It should be noted that in agreement with other recent studies [9] no frequency shift greater than our instrumental resolution ( + 4 c m - t) was observed in any of these experiments. In experiment 1 (fig. 2a) the catalyst was at an ignited state for the whole region of concentrations starting from 0.65% up to 7% CO, at which point the catalyst was forcefully extinguished by momentarily turning off the 02 stream ( - 1 rain). After :he 02 flow was resumed the catalyst remained at an extinguished state for concentrations down to 3.3% CO, where an ignition occurred. Experiment 1 does not exhibit an isolated ignited branch, but an isolated extinguished branch. In experiment 2 (fig. 2b) the catalyst was ignited between 0.65% and 1% CO at which point an extinction occured. In the same region surface state oscillations were observed. The catalyst remained extinguished up to 8% CO, where the catalyst was forcefully ignited by momentarily turning off the CO flow ( < 15 s). The catalyst then remained ignited down to 2.35% CO, at which point an extinction occurred. The catalyst remained extinguished from 2.35% CO to 0.8% CO, where an ignition was again observed. It is apparent from fig. 2b that in this experiment an isolated ignited branch occurs. In experiment 3 (fig. 3a) the catalyst was ignited between 0.65% and 0.70% CO, after which point an extinction occurred. In this region surface state oscillations were again observed. The catalyst remained extinguished up to 7% CO, where it was forcefully ignited, using the previously described technique. The catalyst then remained ignited down to 2.75% CO, at which point a self-induced extinction occurred. The catalyst subsequently remained extinguished from 2.75% CO to 0.65% CO, where again an ignition was observed. Once more it is apparent that an isolated ignited branch occurs. In experiment 4 (fig. 3b) the catalyst was extinguished for the entire region of concentrations starting from 0.65% up to 7% CO, at which point we forcefully ignited the catalyst as in experiment 1. The catalyst remained ignited down to 3% CO, where a self-induced extinction occurred. In this experiment no ignition at lower concentrations was observed. Reaction rate data shows qualitatively similar behavior with that of the presented surface coverage in figs. 2 and 3 (high reaction rates, however, correspond to low surface coverages and vice versa). To the best of our knowledge this is the first time that surface state data has been presented corroborating thermal data showing the existence of isolated steady-state branches in catalytic reaction systems. Furthermore the experimental information presented here clearly points out the difficulties one could encounter in spectroscopic investigations of coadsorbed species at atmospheric conditions.
L574
T.H. Lindstrorn, T.T. Tsotsis / Isolated surface steady-state branches
The (bifurcation) behavior observed here, also shown schematically in fig. 4, can be explained in terms of mathematical models which exhibit codimension 2 singularities (pitch-fork). Our experiments are equivalent to changing only one of the two parameters of the universal unfolding of this singularity, i.e. x -~- ~,x + c~ +/~x 2. It is, therefore, not surprising that only three of the four possible bifurcation diagrams are observed, experimentally. It is also conceivable that the experimental behavior of this system can be described by mathematical models exhibiting singularities of higher codimension, such as of the winged-cusp type. Furthermore, a n u m b e r of complex mechanistic reaction schemes can be postulated that will result in mathematical models exhibiting such singularities, of codimension 2 or higher. The value however of such a modeling effort is questionable since the quantitative verification of these models is limited by the nonisothermal effects and transport limitations ever present in studies at atmospheric conditions. We do not wish, of course, to discount the importance of studies at "realistic" atmospheric conditions. We believe, however, that .
AIAs J I
A/As~
~/--
ir
y
%CO
,7
I
Fig. 4. A schematic representation of the results in fig. 1 through fig. 3.
T.H. Lindstrom, T. 72 Tsotsis / Isolated surface steady-state branches
L575
t h e s e s t u d i e s c a n o n l y r e a l i z e t h e i r full p o t e n t i a l if c o u p l e d w i t h s t u d i e s at lower pressures, where nonisothermal effects and transport limitations are eliminated. This work was supported by a Grant from the National Science Foundation ( C P E 8 1 - 0 6 5 7 1 ) . W e a r e t h a n k f u l to P r o f e s s o r s V. B a l a k o t a i a h , M . S h e i n t u c h a n d D. L u s s f o r m a n y h e l p f u l d i s c u s s i o n s .
References [1] [2] [3] [4} [5J [6]
V. Hlavacek, M. Kubicek and J.Jelinek, Chem. Eng. Sci. 25 (1970) 1441. M. Chang and R.A. Schmitz, Chem. Eng. Sci. 30 (1975) 21. A. Uppal, W.H. Ray and A.B. Poore, Chem. Eng. Sci. 31 (1976) 205. V. Balakotaiah and D. Euss, Chem. Eng. Commun. 13 (1981) 111. M. Sheintuch and D. Luss, Ind. Eng. Chem. Fundamentals 22 (1983) 209. M. Harold and D. Luss, paper presented at the AIChE Annual Meeting, Los Angeles, CA, Nov. 1982. [71 A.E. Elhaderi and T.T. Tsotsis, ACS Symp. Ser. 196 (1982) 77. [8] A.E. Elhaderi, PhD Thesis, University of Southern California, Los Angeles, CA (1982). [9] D.M. Haaland and F.L. Williams, J. Catalysis 76 (1982) 450.
L569
SURFACE SCIENCE LETTERS EXPERIMENTAL OBSERVATIONS OF ISOLATED SURFACE STEADY-STATE BRANCHES T.H. L I N D S T R O M
and T.T. T S O T S I S
Department of Chemical Engineering, Unit,ersiO' of Southern California, Los Angeles. California 90089-1211, USA Received 9 March 1984; accepted for publication 19 June 1984
In this paper we report infrared spectroscopic observations of isolated surface steady-state branches during CO oxidation over Pt/y-Al~O 3 catalysts, These are branches that cannot be reached by continuous smooth changes of the experimentally controlled variables, but only through global perturbations of the catalytic reaction system.
Several theoretical investigations [1-4] have p r e d i c t e d the existence of isolated s t e a d y - s t a t e b r a n c h e s (isolas) for catalytic reaction systems. Until recently, however, no e x p e r i m e n t a l evidence of such b r a n c h e s was available [5,6]. Luss a n d c o w o r k e r s were the first to r e p o r t e x p e r i m e n t a l o b s e r v a t i o n s of isolated s t e a d y - s t a t e b r a n c h e s for the C O o x i d a t i o n reaction system on P t - t y p e c a t a l y s t s [6]. In their e x p e r i m e n t s they m o n i t o r e d the s t e a d y - s t a t e t e m p e r a t u r e of a single catalyst pellet s u s p e n d e d in a t u b u l a r reactor as a function of inlet s t r e a m c o m p o s i t i o n . In this work we present a d d i t i o n a l evidence for the existence of such isolated s t e a d y - s t a t e branches. D u r i n g our experiments, carried out in a C S T R , we have s i m u l t a n e o u s l y m o n i t o r e d the catalyst surface state, by I R T r a n s m i s s i o n S p e c t r o s c o p y , as well as the reaction rate a n d c a t a l y s t t e m p e r a t u r e as a function of inlet stream c o m p o s i t i o n and t e m p e r a ture. O u r e x p e r i m e n t a l setup and p r o c e d u r e s have been d e s c r i b e d in detail elsewhere [7,8]. Special p r e c a u t i o n s have been taken d u r i n g our studies to e l i m i n a t e gas p h a s e i m p u r i t i e s in the inlet r e a c t a n t s t r e a m b y utilizing a series of gas p u r i f i c a t i o n steps and cold traps. N o d e t e c t a b l e gas p h a s e i m p u r i t i e s ( < 1 p p m ) were p r e s e n t d u r i n g o u r experiments. T h e reactor cell was m a d e from Pyrex glass with C a F 2 w i n d o w s sealed to the o u t s i d e surface with Viton o-rings: The r e a c t a n t gases d o not c o m e in direct c o n t a c t with a n y o t h e r reactor m a t e r i a l besides the Pyrex glass wails a n d C a F z windows. Sonic orifice meters were utilized to achieve c o n s t a n t r e a c t a n t flows. T h e r e a c t o r cell has been shown to b e h a v e as a C S T R for the flow c o n d i t i o n s 0 0 3 9 - 6 0 2 8 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
T.H. Lindstrom, T.T. 7~9otsis / Isolated surface steady-state branches
L570
utilized in this investigation [8]. The catalyst wafer (1 in. diameter, 30 m g / c m : thick) was prepared by impregnation of "/-alumina (Degussa type C. BET surface area 100 m2/g, average particle size 200 A) with H2PtCIe,. 6 H 2 0 solution up to a total Pt content of 3.2%. (Further details of our catalyst preparation techniques can be found elsewhere [8].) Before the series of experiments reported here was initiated, the catalyst was treated for 24 h in a 10% CO in air mixture. This treatment eliminates the catalyst's ability to form the 2120 cm ] band reported earlier [7]. The experimental results presented here are part of an ongoing investigation of the steady-state and the dynamic behavior of this and other catalytic oxidation reaction systems. Fig. 1 shows the "bifurcation set" of this reaction system in the Tg (gas phase temperature), %CO in the inlet reactant stream parameter space for two distinct flowrates (40 and 50 cm3/s). This figure was constructed by running a series of experiments, where the inlet gas phase temperature was kept constant (within + 1 °C) and the composition was varied (0.65%-8.0% CO in synthetic air, i.e., a 20.93% O 2, 79.07% N 2 mixture). In the figure we present the concentration values (within 0.1% CO) for which (self-induced) reaction ignition and extinction p h e n o m e n a were observed. A similar figure for the same catalyst has been obtained by a series of experiments,
Tg °C '
,,~~~ ~o~/s
'E80 t60
lily
t4O 120
r
'I00 80 60 4C 2C
t.0
2.0
3.0
4.0
5.0 %C0
Fig. 1. "Bifurcation set" for the CO oxidation reaction ~ystem. T~ versus %CO in inlet stream.
T.H. Lindstrom, T.T. Tsotsis /lsolated surface steady-state branches
L571
w h e r e the gas p h a s e c o m p o s i t i o n was k e p t c o n s t a n t a n d the t e m p e r a t u r e was varied. E a c h e x p e r i m e n t in the series was p r e c e d e d b y a p r e t r e a t r n e n t o f the c a t a l y s t , w h i c h c o n s i s t e d of a 2 h e x p o s u r e to f l o w i n g N 2 f o l l o w e d b y a 2 h
A/•,s t,0 0.9
I
0.8 0.7 0.6
I
Extinction
I
0.5 0.4 0.3
t
0.2 0.!
,
,
t.0 2.0 3.0 4.0 5.0 6,0 7.0 %C0
J
A/A s
1.0 O.9 O,8 0.7 0.6 O.5 0.4 0.3 0.2
It jl
I Forced ]I~nitioo
II
t
It t
0.~ I
I
I -
J.
2
f
•
I
~t
4.0 2.0 3,0 4.0 5.0 6,0 7.0 8.0%C 0 b Fig. 2. "Bifurcation diagram" for the CO oxidation. Normalized absorbance versus ~CO in inlet stream (50 cm3/s): (a) Tg = 180°C; (b) Tg = 120°C.
72 H. Lindstrom, 71 T. Tsotsis / Isolated surface steadv- state branches
L572
e x p o s u r e to f l o w i n g s y n t h e t i c air at c a t a l y s t t e m p e r a t u r e s e x c e e d i n g 2 0 0 ° C . T h i s p r e t r e a t m e n t p r o d u c e d a s p e c t r o s c o p i c a l l y c l e a n c a t a l y s t surface. E a c h s t e a d y - s t a t e s h o w n in figs. 2 a n d 3 was m o n i t o r e d for m o r e t h a n 2 h f o l l o w i n g initial s t a b i l i z a t i o n .
A/A s 4.0
x
0.9 0.8
,
t
I
0.7 0,6 0.5
Forced lqnition
_ ~
,
I
- II
I
0.4 0.3
II
0.2 0,1 t
I
,.~
•
~.
1
•
~.0 2.0 3,0 4.0 5~'0 6.0 7]0 %C0 O
A/A s i.O
i.
. . . .
}-(
0,9
Z
0.8-
I
0.70.6-
Forced Ignilion
0.5-
o.4~ 0.3 0.2
×t
O,i
e~
>_ 4,0 2,0 3.0 4,0 5.0 6,0 7,0 */,,CO b Fig. 3. "Bifurcation diagram" for the CO oxidation. Normalized absorbance versus %CO in inlet stream (50 cm3/s): (a) Tg = ll0~C; (b) Tg = 100°C. I
t
l
t
1
1
7£H. Lindstrom, T. T. Tsotsis / Isolated surface steady-state branches
L573
Figs. 2 and 3 contain the four different types of "bifurcation diagrams" observed for this reaction system at 50 cm~/s. What we present here is the normalized (in terms of maximum) absorbance of the linearly adsorbed CO species (around 2088 cm-a). It should be noted that in agreement with other recent studies [9] no frequency shift greater than our instrumental resolution ( + 4 c m - t) was observed in any of these experiments. In experiment 1 (fig. 2a) the catalyst was at an ignited state for the whole region of concentrations starting from 0.65% up to 7% CO, at which point the catalyst was forcefully extinguished by momentarily turning off the 02 stream ( - 1 rain). After :he 02 flow was resumed the catalyst remained at an extinguished state for concentrations down to 3.3% CO, where an ignition occurred. Experiment 1 does not exhibit an isolated ignited branch, but an isolated extinguished branch. In experiment 2 (fig. 2b) the catalyst was ignited between 0.65% and 1% CO at which point an extinction occured. In the same region surface state oscillations were observed. The catalyst remained extinguished up to 8% CO, where the catalyst was forcefully ignited by momentarily turning off the CO flow ( < 15 s). The catalyst then remained ignited down to 2.35% CO, at which point an extinction occurred. The catalyst remained extinguished from 2.35% CO to 0.8% CO, where an ignition was again observed. It is apparent from fig. 2b that in this experiment an isolated ignited branch occurs. In experiment 3 (fig. 3a) the catalyst was ignited between 0.65% and 0.70% CO, after which point an extinction occurred. In this region surface state oscillations were again observed. The catalyst remained extinguished up to 7% CO, where it was forcefully ignited, using the previously described technique. The catalyst then remained ignited down to 2.75% CO, at which point a self-induced extinction occurred. The catalyst subsequently remained extinguished from 2.75% CO to 0.65% CO, where again an ignition was observed. Once more it is apparent that an isolated ignited branch occurs. In experiment 4 (fig. 3b) the catalyst was extinguished for the entire region of concentrations starting from 0.65% up to 7% CO, at which point we forcefully ignited the catalyst as in experiment 1. The catalyst remained ignited down to 3% CO, where a self-induced extinction occurred. In this experiment no ignition at lower concentrations was observed. Reaction rate data shows qualitatively similar behavior with that of the presented surface coverage in figs. 2 and 3 (high reaction rates, however, correspond to low surface coverages and vice versa). To the best of our knowledge this is the first time that surface state data has been presented corroborating thermal data showing the existence of isolated steady-state branches in catalytic reaction systems. Furthermore the experimental information presented here clearly points out the difficulties one could encounter in spectroscopic investigations of coadsorbed species at atmospheric conditions.
L574
T.H. Lindstrorn, T.T. Tsotsis / Isolated surface steady-state branches
The (bifurcation) behavior observed here, also shown schematically in fig. 4, can be explained in terms of mathematical models which exhibit codimension 2 singularities (pitch-fork). Our experiments are equivalent to changing only one of the two parameters of the universal unfolding of this singularity, i.e. x -~- ~,x + c~ +/~x 2. It is, therefore, not surprising that only three of the four possible bifurcation diagrams are observed, experimentally. It is also conceivable that the experimental behavior of this system can be described by mathematical models exhibiting singularities of higher codimension, such as of the winged-cusp type. Furthermore, a n u m b e r of complex mechanistic reaction schemes can be postulated that will result in mathematical models exhibiting such singularities, of codimension 2 or higher. The value however of such a modeling effort is questionable since the quantitative verification of these models is limited by the nonisothermal effects and transport limitations ever present in studies at atmospheric conditions. We do not wish, of course, to discount the importance of studies at "realistic" atmospheric conditions. We believe, however, that .
AIAs J I
A/As~
~/--
ir
y
%CO
,7
I
Fig. 4. A schematic representation of the results in fig. 1 through fig. 3.
T.H. Lindstrom, T. 72 Tsotsis / Isolated surface steady-state branches
L575
t h e s e s t u d i e s c a n o n l y r e a l i z e t h e i r full p o t e n t i a l if c o u p l e d w i t h s t u d i e s at lower pressures, where nonisothermal effects and transport limitations are eliminated. This work was supported by a Grant from the National Science Foundation ( C P E 8 1 - 0 6 5 7 1 ) . W e a r e t h a n k f u l to P r o f e s s o r s V. B a l a k o t a i a h , M . S h e i n t u c h a n d D. L u s s f o r m a n y h e l p f u l d i s c u s s i o n s .
References [1] [2] [3] [4} [5J [6]
V. Hlavacek, M. Kubicek and J.Jelinek, Chem. Eng. Sci. 25 (1970) 1441. M. Chang and R.A. Schmitz, Chem. Eng. Sci. 30 (1975) 21. A. Uppal, W.H. Ray and A.B. Poore, Chem. Eng. Sci. 31 (1976) 205. V. Balakotaiah and D. Euss, Chem. Eng. Commun. 13 (1981) 111. M. Sheintuch and D. Luss, Ind. Eng. Chem. Fundamentals 22 (1983) 209. M. Harold and D. Luss, paper presented at the AIChE Annual Meeting, Los Angeles, CA, Nov. 1982. [71 A.E. Elhaderi and T.T. Tsotsis, ACS Symp. Ser. 196 (1982) 77. [8] A.E. Elhaderi, PhD Thesis, University of Southern California, Los Angeles, CA (1982). [9] D.M. Haaland and F.L. Williams, J. Catalysis 76 (1982) 450.