I n d . Eng. Chem. Res. 1987, 26, 945-947
945
Production Rate Improvement in Plug-Flow Reactors with Concentration Forcing Jan Thullie Institute of Chemical Engineering, Polish Academy of Science, Gliwice 44100, Poland
Leroy Chiao and Robert G. Rinker* Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, California 93106
A comparison of two kinetic models, under forced concentration oscillations, for the same chemical process is presented. It is shown that, for a plug-flow reactor with a two-step adsorption-desorption model, it is possible to improve the reaction rate and the production rate by periodic concentration forcing.
It is common practice to use kinetic expressions obtained from steady-state analyses for the purpose of modeling chemical reactor dynamics. Usually this results from the scarcity of kinetic data under transient conditions. In this paper, we compare the predictions of a dynamic reactor model which accounts for the transient behavior of the catalyst surface (model I) with a model using steady-state kinetics (model 11). Both models are based on a two-step, adsorption-desorption reaction sequence with Eley-Rideal kinetics. Model I represents real-system behavior more closely under dynamic conditions, but model I1 is often used and is also much simpler to use. The differences in the resulting calculated behavior between the two models for forced periodic oscillations of feed composition (concentration forcing) in a fixed-bed reactor are significant and cannot generally be ignored. Improvements in time-averaged production rates resulting from concentration forcing have been observed experimentally by a number of researchers including, for example, Bailey et ai. (1971), Barshad and Gulari (1985), Chiao et al. (19861, Cutlip (1979), Denis and Kabel (19701, Graham and Lynch (1984), Jain et al. (1982, 1983), Li et al. (1984), Lynch and Wanke (1981), Wandrey and Renken (1977), and Wilson and Rinker (1982). Several comprehensive reviews on concentration forcing are available (Bailey, 1973, 1977; Renken, 1984).
Mathematical Models
--
Consider the irreversible reaction A 2B 3C which consists of two phenomenological steps A S* AS* AS* + 2B 3C S* where AS* is a chemisorbed species on the catalytic surface. Thus, gaseous species A adsorbs on the catalytic surface in the first step and reacts with gaseous species B via Eley-Rideal kinetics in the second step. For a fixedbed reactor in which these reactions are taking place under isothermal plug-flow conditions, we can describe the two models as model I
+
+
where eq l a and l b are pseudohomogeneous species balances for A and B, respectively, and eq ICis a transient site balance for the kinetic model chosen. The variables, YA and YB, are the mole fractions of A and B in the gas phase; [ is dimensionlesstime; x is the dimensionlessaxial distance; K~ and K~ are dimensionless rate constants; and 0 is the fraction of sites covered by species A. The appropriate boundary and initial conditions for this system of equations are given for concentration forcing in a later paragraph. To obtain model 11, we first reduce eq 1 to the steadystate forms which are
with boundary conditions YA(0) = y& YB(0) = 1 - YAo (3) In eq 2, the steady-state reaction rate, r, is given by (4) for this particular reaction scheme, without assuming a rate-limiting step. If the steady-state reaction rate is arbitrarily substituted into eq l a and l b to replace the kinetic expressions, this gives the commonly used form below; i.e., model I1
+
Note that the time-dependent site-balance equation is nonexistent in model 11. To reiterate, a comparison of models I and I1 in forced concentration (pulsed) reactor behavior is the basis of this paper. The boundary conditions for both models are expressed in the form yA(C;,O)= y b + A for E 5 n;i + y7 (6) yA(C;,O)= y b - A for C; > + 77 YB(E,O) 1 - YA([,O) (7) where T is the dimensionless cycle time, n is the number of cycles, and A is the amplitude of the square pulses. Also, for eq IC,the initial condition for the site balance is given by
* Author to whom correspondence should
be addressed.
0888-5885/87/2626-0945$01.50/0
e(o,x) = 0 1987 American Chemical Society
o
(8)
946 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 0901
=
I
.I.....
I
I
I
I
I
0001 000
I
I
Steady-state Solution y B 2 =1-yA O + A Steady-state Solution yB '1-yA - A 0
1
0.20
0 10
I
I
K1=22O K2= 2 2 M = 02 "
020
=
-
020
....... Steady-state
Solution yB1:1 - y A o - A
Y=O5 ?=0122
"
"
"
040
060
080
'
1 I
1
100
000
020
040
Reactor Length
080
060
100
Reactor Length
Figure 1. Invariant cycling results for model I1 for typical values of parameters K ~ K, ~ and , M and an arbitrary value of cycle time, ?. With a cycle split of y = 0.5, the amplitude, A , is symmetric a t an arbitrary value of 0.2. The average feed composition is stoichiometric with yBo = 0.67.
We do not specify a particular form of the initial conditions here because we are interested in results for time 5 >> 1, when invariant cycling is obtained (Renken, 1984). In the results presented below, the invariant cycling achieved after several cycles of periodic operation for both models I and 11does not include the "relaxed" steady state as defined by Lee and Bailey (1974), regardless of the magnitude of the frequency (w = 27r/r). In other words, the reaction conditions on the catalyst surface are permitted to follow the fastest oscillations in the gas phase. This means that l/w, which is the time constant for the oscillations, is large compared to the relaxation times for composition changes on the catalyst surface.
Results To make the comparison more meaningful, we introduce mean reaction rates, with the first based on the reactor length, L, (9) where 5 is treated as a parameter. The second mean value is obtained by integration over the cycle period, i.e.,
The results obtained for model I1 give a wave form of concentration distribution along the reactor length as shown in Figure 1. It is noteworthy that the upper and lower bounds of this wave are determined by the solution of only the steady-state equations. The consequence of this is that the average global reaction rate, R , for model I1 is given by R = YR, + (1- r ) R , (11) where y is the cycle split, R2 is the average global reaction rate at the steady state having the same inlet concentration as the second part of the cycle which is rich in reactant B, and R1 is the average global reaction rate at the steady state having the same inlet concentration as the first part of the cycle which is rich in reactant A. According to eq 11, there is no possibility for model I1 to achieve an average global reaction rate greater than the
Figure 2. Invariant cycling results for model I using the same parameter values as in Figure l. 120-
A=O 3
0
K,
5
090
K2=
0
20
220 22
40
M= 02
yBo= 067 60
80
100
120
Cycle Frequency, w
Figure 3. Cyclic reaction rate improvement for model I relative to the stoichiometric reaction rate at steady state. The improvement is better for increasing amplitude, A .
mean reaction rate between the two steady states which limit the oscillations. Consequently, no improvement is achieved over steady-state operation. Furthermore, eq 11 applies to model I1 regardless of the complexity of the kinetic expression for the reaction rate, r = r(YA,yB), so that there is no need to solve eq 5 for other rate expressions. The situation is quite different for model I. Here, as shown in Figure 2, the resulting wave concentration profile along the reactor no longer reaches the steady-state extremes, and eq 11 does not apply. Thus, eq 1 was solved by an appropriate integration scheme (the method of characteristics). Results are shown in Figure 3. It is seen that there is an improvement in the average global reaction rate at high frequency in comparison with stoichiometric steady-state operation. There is also enhancement in the production rate of C compared to optimal steady-state operation as shown in Figure 4. The question of a cycle time corresponding to a maximum enhancement in production rate immediately arises. It is clear from Figure 4 that there is a frequency above which the production enhancement, $, remains unchanged at its maximum value, and this can be predicted from the model. It is also clear that the amplitude has an influence on the results. The larger it is, the better the improvement in reaction rate. Conclusions This work shows the possibility of improvement in average global reaction rate or enhancement in production rate by unsteady operation of an isothermal fixed-bed
Ind. Eng. Chem. Res., Vol. 26, No. 5 , 1987 947 K1
= 22.0
y~~ = 0.825
K2
= 2.2
A =0.165
M
= 0.2
Ro = stoichiometric global reaction rate at steady state t = time, s i = L/v, s u = velocity, m/s 3c = z / L , dimensionless axial distance in the reactor y = mole fraction yA,= average mole fraction of species A in the feed, 1- yBo yBo = average mole fraction of species B in the feed, YYB, + (1 - Y)YB = mole rraction of reactant B in the feed in the first part
YB,
0.98
of the cycle which is rich in reactant A mole fraction of reactant B in the feed in the second part of the cycle which is rich in reactant B z = axial distance in the reactor, m
yB2 =
t 0
20
40
60
80
100
Cycle Frequency, o
Figure 4. Production rate enhancement for model I, using nonstoichiometric average feed composition.
reactor using model I, which accounts for the transient behavior of the catalyst surface. More detailed kinetic expressions should be tested in that model. Model 11, which makes use of steady-state kinetics, shows no improvement in reaction or production rate relative to steady-state operation and is therefore misleading. It is important to note that these comparisons are based on the reactor operating under plug-flow transient conditions but not including the relaxed steady state. The adequacy of model I in describing real systems which follow the same kinetic model has not been addressed here. To achieve good agreement with experiments, it may be necessary to include other features in the model such as catalyst deactivation, reactant or product storage on the catalyst, and deviations from plug flow.
Acknowledgment Support of J. T. through the National Academy of Sciences is gratefully acknowledged. The work was also supported by the Division of Chemical Sciences (Office of Basic Energy Sciences) of the Depmtment of Energy under Grant DE-FG03-84ER13300.
Nomenclature u p = specific surface area of the catalyst, m2/m3 A = amplitude of feed composition oscillation given by yBp - YBo or YBo - YB1 L = reactor length, m M = dimensionless constant resulting from nondimensionalizing the species balance equations n = number of cycles r = instantaneous global reaction rate R = average global reaction rate
Greek Symbols y = cycle split, which is the fraction of the total cycle that the stream rich in species B flows to the reactor 0 = dimensionless surface concentration K~ K~
= dimensionless rate constant
= dimensionless rate constant
6 = t/f, dimensionless time T
= cycle time, s
7=
./it dimensionless cycle time
+ = production enhancement, yc(cyclic)/yc(optimal steady state)
w = 27r/t,
dimensionless cycle frequency
Literature Cited Bailey, J. E. Chem. Eng. Commun. 1973,1 , 111. Bailey, J. E. In Chemical Reactor Theory. A Reuiew; Lapidus, L., Ammundson, N. R., Eds.; Prentice Hall: Englewood Cliffs, NJ, 1977;pp 758-813. Bailey, J. E.; Horn, F. J. M.; Lin, R. C. AIChE J . 1971, 17, 818. Barshad, Y.;Gulari, E. AIChE J. 1985,31,649. Chiao, L.; Zack, F.; Rinker, R. G.;Thullie, J. Chem. Eng. Commun. 1986,in press. Cutlip, M. B. AIChE J. 1979,25,502. Denis, G. H.; Kabel, R. L. Chem. Eng. Sci. 1970,25,1054. Graham, W. R. C.; Lynch, D. T. In Catalysis on the Energy Scene; Kaliaguine, S., Mahay, E., Eds.; Elsevier Science: Amsterdam, 1984;pp 197-204. Jain, A. K.; Hudgins, R. R.; Silveston, P. L. ACS Symp. Ser. 1982, 196,97. Jain, A. K.; Hudgins, R. R.; Silveston, P. L. Can. J. Chem. Eng. 1983, 61,824. Lee, C. K.; Bailey, J. E. Chem. Eng. Sci. 1974,29, 1157. Li, Chengyne; Hudgins, R. R.; Silveston, P. L. In Catalysis on the Energy Scene; Kaliaguina, S., Mahay, A., Eds.; Elsevier Science: Amsterdam, 1984;pp 229-233. Lynch, D. T.; Wanke, S. E. Can. J . Chem. Eng. 1981,59,766. Renken, A. Int. Chem. Eng. 1984,24,202. Wandrey, C.; Renken, A. Chem. Eng. Sci. 1977,32,448. Wilson, H. D.; Rinker, R. G. Chem. Eng. Sci. 1982,37,343.
Receiued for review October 7, 1985 Accepted December 12, 1986
