J . Phys. Chem. 1990, 94, 4867-4870
4867
Stirring and Mixing Effects on Chemical Instabilities: Bistability of the Br0,-/Br-/Ce3+
Arun K. Dutt and Michael Menzinger* Department of Chemistry, University of Toronto, Toronto, Ontario M5S 1 A I , Canada (Received: May 25, 1989; In Final Form: January 17, 1990)
The bistability hysteresis of the inorganic subsystem of the BZ reaction was studied in a CSTR as a function of stirring rate and mixing mode with the flow rate as the control parameter. We found that premixing (PM) of the feedstreams stabilizes the thermodynamic branch, while enhanced stirring in the nonpremixed (NPM) mode destabilizes the same branch. Furthermore, decreased stirring of the NPM system broadens the hysteresis and can, for sufficiently large changes of stirring rate, shift the loop to a domain that does not overlap that obtained at high stirring rate. These counterintuitive and surprising results demonstrate the different dynamical roles played by stirring and by the reactant premixing, and they show the importance of heterogeneous-noise-inducedeffects.
Introduction The fact that quantitative as well as qualitative aspects of nonlinear chemical dynamics may depend on the stirring rate S, in flowiq2and batch3 reactors alike, has demonstrated first that the standard methods of mixing and stirring may fail to establish the ideal homogeneous limit commonly assumed and second that the dynamics is a sensitive function of the reactor’s degree of heterogeneity or segregation. Experimental5and theoretical6works on coupled reactors have shown that the dynamics of spatially distributed systems may be quantitatively as well as qualitatively different from and much richer than their homogeneous counterpart. Stirring and mixing (SM) effects thus put at issue the traditional approach of interpreting experiments in stirred reactors in terms of homogeneous kinetic models which employ ordinary differential equations. Stirring- and heterogeneity-induced effects may be considered a nuisance that can be minimized by more efficient methods of stirring and mixing.“ However, heterogeneities are important in their own right in most real systems, since they occur widely in nature and in the laboratory. Their study promises to provide a missing link between the chemical instabilities in idealized “well-stirred” reactors and the patterns forming spontaneously in unstirred systems. Best studied in this respect is the C102- + I- system, where increased stirring and/or premixing of the reactant feedstreams have1 the same effects qualitatively, both stabilizing the thermodynamic branch of the bistability hysteresis. This could suggest that the process of premixing is simply equivalent to enhanced ~ t i r r i n g . ’ ~However the results presented in the present paper show that generally this need not be the case.7c The heterogeneities ( I ) (a) Roux, J . C.; DeKepper, P.; Boissonade, J. Phys. Lett. A 1983, 97, 168. (b) Boissonade, J.; Roux, J. C.; Saadaoui, H.; DeKepper, P. In Fluctuations and Sensirioity in Non-Equilibrium Systems; Horsthemke, W., Kondepudi, D. K., Eds. Springer Proc. Phys. 1984, I, 70. (c) Menzinger, M.; Boukalouch. M.; DeKepper, P.; Boissonade, J.; Roux, J. C.; Saadaoui. H. J . Phys. Chem. 1986, 90, 31 3. (2) (a) Menzinger, M.; Giraudi, A. J . Phys. Chem. 1987, 91, 4391. (b) Luo, Y.:Epstein, I. R. J. Chem. Phys. 1986, 85, 5733. (c) Boukalouch, M.; Boissonade, J.; DeKepper, P. J . Chim. Phys. 1987, 84, 1353. (3) (a) Farage. U.J.; Jancic, D. Chimia 1981.35, 289. (b) Adamcikova, I.; Sevcik, P. Chem. Phys. Lett. 1988, 146,419. (c) Li, R.; Li, J. Chem. Phys. Lett. 1988, 144,96. (d) Menzinger, M.; Jankowski, P. J. Phys. Chem. 1986, 90, 1217.
(4) Klein, J. P.; David, R.; Villermaux, J. Ind. Eng. Chem. Fundam. 1980, 19, 373. ( 5 ) (a) Marek, M.; Stuchl, I. Biophys. Chem. 1975, 3, 241. (b) Fuji, H.; Sawada, Y. J . Chem. Phys. 1978, 69, 3830. Nakajima, K.; Sawada, Y. J . Chem. Phys. 1980, 72, 223. (6) (a) Kumpinsky, E.; Epstein, I . R. J . Chem. Phys. 1985, 82, 53. (b) BarEli, K.; Noyes, R. M. J . Chem. Phys. 1986, 85, 3251. (c) Boukalouch, M.; Elezgaray, J.; Arneodo, A.; Boissonade, J.; DeKepper, P. J . Phys. Chem. 1987, 91, 5843. (d) Hocker. C. G.; Epstein, I. R. J . Chem. Phys. 1989, 90, 3017. (e) Gyorgyi, L.; Field, R. J. J . Phys. Chem. 1989, 93, 2865. (7) (a) Horsthemke, W.; Hannon, L. J . Chem. Phys. 1984,81,4363. (b) Hannon. L.; Horsthemke. W. J . Chem. Phys. 1987, 86, 140. (c) Boissonade, J.; DeKepper, P.J . Chem. Phys. 1987, 87, 210. (d) Puhl, A,; Nicolis, G. J . Chem. Phys. 1987, 87, 1070. (e) Fox, R. 0.;Villermaux, J. Chem. Eng. Sci., submitted for publication.
0022-3654/90/2094-4867$02.50/0
responsible for these SM effects ariselCprimarily from incomplete mixing of the feedstream(s) into the bulk of the CSTR. The well-known oscillations found in the C I O 2 - / T system, originally believed to represent the dynamical signature of the homogeneous system, were subsequently shown2 to arise only within a narrow window of stirring rates and to vanish when stirring (homogeneity) increased above a certain critical value. This illustrates how heterogeneities can introduce new dynamical structure that is absent from the homogeneous reference system. Different aspects of these phenomena have been simulated by models emphasizing the incomplete macromixing6and incomplete micromixing7aspects of the process. In this paper we have chosen the inorganic subsystem of the BZ reaction to study the effects of stirring rate and mixing mode on both transitions of the bistability hysteresis. This reaction was chosen for its conveniently rapid rates of both the ignition (flow-to-thermodynamic branch, also called “up” transition with reference to the electrode potentials) and extinction (thermodynamic-to-flow branch, or “down”) transitions. Its bistability, as a function of the inverse residence time ko, was previously studied by Geisler,8 and model calculations9 based on the N F T mechanism1° show gratifying agreement with experiment. We determined the hysteresis loop E(kolS, NPM/PM) as a function of ko at fixed stirring rate S, for nonpremixed (NPM) feedstreams on the one hand and for premixed (PM) streams on the other. One finds the following: 1. In contrast to the C l o y + I- system, enhanced stirring in the NPM mode destabilizes the thermodynamic branch, while premixing causes the same response as in the CIO;/I- system, namely, stabilization of the thermodynamic branch. This shows clearly that premixing is not equivalent to enhanced stirring7cand that stirring rate S and mixing mode are independent dynamical parameters. 2. For N P M streams, a sufficiently large change of stirring rate moves the hysteresis loops into nonoverlapping domains. Noise-induced transitions (through changes of S) are possible in the region between these two hystereses. 3. Decreased stirring (enhanced heterogeneity) broadens the hysteresis loop rather than contracting it, as one would expect from a consideration of the transitions near the hysteresis limits mediated by noise.3d For brevity, we refer here to the heterogeneities as “noise” and to the transitions induced by a decrease of S and the increase of noise (segregation) as noise-induced transitions (NIT).” Observations 2 and 3 agree surprisingly well with DeKepper and Horsthemke’s study’* of the effect of external noise (8) (a) Geiseler, W.; Foellner, H . Biophys. Chem. 1977, 6, 107. (b) Geiseler, W. J. Phys. Chem. 1982, 86, 4394. (9) BarEli. K.; Geiseler, W. J. Phys. Chem. 1983, 87, 3769. ( I O ) Noyes, R. M.; Field, R. J.; Thompson, R. C. J . Am. Chem. SOC.1971, 93, 7315. ( I I ) Horsthemke, W.; Lefever, R . Noise-Induced Transitions; Springer: . New York, 1984. (12) DeKepper, P.; Horsthemke. W. C. R. Seances Acad. Sci., Ser. C 1978, 278, 251.
0 1990 American Chemical Society
4868
Dutt and Menzinger
The Journal of Physical Chemistry, Vol 94, No. 12. 1990
BrO;
/ Br'/ Ce"' 5.2820
j y s tem rpm
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IO
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I 20
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25
I
V k O (m I/min )
0
5
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Vk,
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I
I
15
20
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Figure 2. Effect of mixing mode on hysteresis E(kolS, NPM/PM). Feedstream concentrations: (A) [Br-], = 8.0 X IO4 M, [Ce'+]f = 4.0 X IOd M. [H2S041f = 0.75 M; (B) [BrOC], = 1.2 X IO" M, [HzSO& = 0.75 M. S = 2820 rpm; t = 20.0 f 0.3 OC. Symbols as in Figure la. 1 0
I
I
ZCKX)
3000
I ICKX)
S irpm)
Figure 1. (a) Stirring dependence of hysteresis loop E(k,lS, NPM) for NPM feedstreams. Feedstream compositions: (A) [Br-][ = 8.32 X IO4 M, [Ce3'If = 4.24 X M, [H2SO4If= 0.75 M; (B) [BrO3-If = 1.72 X IO-' M, [H2S04jf= 0.75 M. Temperature t = 20.0 f 0.3 'C. S as indicated. Open symbols, increasing k,; full symbols, decreasing k,. (b) Stirring-induced hysteresis E(Slko, NPM) for same conditions as in panel a. The flow rate Vko = I I .3 mL/min corresponds to the arrow in panel a where the two branches are monostable a t S = 2265 and 600 rpm.
700
-.:"'hO.-
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Effect of mixing mode and stirring svstem
i i
->E
on the Briggs-Rauscher reaction, suggesting that the identification of heterogeneities as noise is appropriate and demonstrating the universality of N I T effects.
Experiments A standard plexiglass CSTR'C-2a(volume V = 3 1.5 mL; diameter 30 mm; thermostated at T = 20.0 f 0.3 "C) was used. The propeller-shaped glass stirrer (two blades, tip-to-tip diameter 17 mm; blade width 6.5 mm) was positioned 35 mm above the bottom of the cylindrical reactor and was driven at an accurately reproducible speed in the range of S = 0-2820 rpm. The state of the system was monitored by a Pt macroelectrode ( 1 5 X 0.5 mm2. protruding radially into the reactor 19 mm above its bottom) and an Ag/AgCl reference electrode attached to a Na2S04salt bridge. The feedstreams contained (Br-, Ce3+, H2S04)and (BrOq, H,S04). They entered, nonpremixed (NPM) or premixed (PM) as the case may be," through two 0.5-mm ports as the base of the reactor. The transit time from premixer to the CSTR was 0.31 s at V k , = 28 mL/min and proportionately longer at lower flow rates. The two feeding modes described here represent a subset of the modes arising from three feedstreams containing BrO,-, Br-, and Ce? Results of this complete study will be reported elsewhere. Three types of experiment were performed: ( I ) the hysteresis E(kolS)was traced out as a function of pump rate P = Vko,where ko is the inverse residence time, a t different values of the stirring rate S in the N P M mode, and (2) conversely, the hysteresis E(Slko)with S as control parameter was measured at a suitable value of k,. (3) Finally, the same type of hysteresis as in (1) was measured for PM and N P M feedstreams. In the hysteresis loops shown in Figures 1-3, the transitions, indicated by the arrows, were drawn halfway between the points just prior to and following the transitions. Results and Discussion Experiment 1, which measures the hysteresis loops E(kolS, N P M ) in the N P M mode a t high and low stirring speeds, S = 2265 and 600 rpm, is summarized in Figure l a . One finds the following:
i - i I I _I O _ _ _ _ 20L _ 25_ I -30j Vko : m [ / m r n )
Figure 3. Effect of stirring and mixing mode on hysteresis. Parameters are same as in Figure 2.
1. A decrease of S causes both up and down transitions to move to higher values of the control parameter ko. Slow stirring (enhanced noise) stabilizes the (high potential) thermodynamic branch and destabilizes the (low potential) flow branch, opposite to what is known from the CIO2-/1- system.' 2. The hysteresis broadens with decreasing S and increased noise, nearly doubling in width in going from 2265 to 600 rpm. Figure 3 summarizes similar experiments performed over the wider range S = 600-2820 rpm. Here the hysteresis width more than doubles at low S , showing that stirring is not saturated at S = 2265 rpm in Figure la or that the perfect micromixing limit has not been reached at that value. 3. For sufficiently large changes of S, as shown in Figures l a and 3, the hysteresis loops become entirely disjointed. This gives rise to a range of control parameter Vko = 9.8-12.6 cm3/min in Figure la), flanked by the down transition a t high S and the up transition at low S, for which the thermodynamic and flow branches are monostable at low and high S , respectively. Consequently, noise-induced transitions between both branches may be realized in this range of ko by simply changing S. This is done in experiment 2. The corresponding stirring-induced hysteresis E ( S J k o )a t the intermediate value Vko = 11.3 mL/min is summarized in Figure Ib. It is evident that the transitions near S = 750 and 2500 rpm are induced by the Sdependent attributes of the incomplete-mixing-induced noise. Experiment 3, which probes the influence of the mixing mode, is summarized in Figure 2. In the P M mode, the hysteresis loop is shifted to much higher values of the control parameter ko and its width is considerably broadened relative to the N P M case. The pronounced stabilization of the thermodynamic branch is similar to that found in the chlorite/iodide system' and has probably a
Bistability of the Br03-/Br-/Ce3+ System
The Journal of Physical Chemistry, Vol. 94, No. 12, 1990 4869
similar i n t e r p r e t a t i ~ n .However, ~~ this is contrary to what one would expect from the naive assumption that premixing is simply equivalent to enhanced stirring. Comparison of Figures l a and 2 shows that the former stabilizes while the latter destabilizes the thermodynamic branch. This means that rapid stirring and premixing are not equivalent and that premixing initiates a specific chemical response which could not be achieved in the N P M mode by stirring at arbitrary high rates. This conclusion is supported by comparing the potentials E of the steady states in the P M and N P M modes: the thermodynamic branch lies 30-80 mV higher in PM than in N P M mode, much more than the difference of a few millivolts achieved by changing S in the NPM mode (Figures 1 and 3). In Figure 3, for Vko > 3 mL/min in the N P M mode, enhanced stirring even lowers the potential of the thermodynamic branch, in agreement with its destabilization and opposite to the situation under PM conditions. Figure 3 summarizes the effects of stirring in both mixing modes. One notes for P M feedstreams that the down transition is independent of stirring within the resolution of the experiment, while the up transition, on the other hand, is shifted to higher values of the control parameter by decreasing S, similar qualitatively and even quantitatively to the N P M mode. Thus, the principal difference between the two mixing modes lies in the facts that in PM the down transition is relatively insensitive to S while in NPM it is shifted to higher ko values by decreased stirring. In the PM mode the high-noise (low-S) hysteresis is entirely contained within the low-noise loop, or noise contracts the hysteresis as may be expected intuitively by considering the effect of noise-induced transitions near the hysteresis boundary,3d while the NPM mode, surprisingly, causes the above-noted broadening of the loop and shift to nonoverlapping positions. With regard to the NPM mode it was noted above that stirring is not saturated, Le., the ideal micromixing limit is not reached, at S = 2265 rpm (Figure la), as the further shift and broadening of the hysteresis show upon further increase of the stirring speed to its maximum value of S = 2820 rpm (Figure 3). Obviously, the ideal micromixing limit is not reached at S = 2265 rpm. Indeed, considering the distribution20 of the turbulent intensity in the vicinity of the stirrer and in the weakly stirred regions in the far corners of the CSTR, it appears likely that this is true even under conditions of reactant premixing and more efficient stirring4
Interpretation Turbulent mixing of two or more dissimilar fluids is known to give rise to complicated objectsI3 (“turbulent eddies”) distributed in Euclidian and concentration space. The energy of the turbulent flow field is dissipated by successively stretching and folding of the fluid parcels, which assume a complex fractal geometry14down to a length scale LK,the Kolmogorov limit,16 below which viscous forces dominate the shear forces acting on the liquid, and molecular diffusion completes the mixing process. Under the present conditions, LK is of the orderI3 of 0.1 mm. The intensity of stirring (expressed by the stirring rate S or the rate c of energy dissipation per unit volume) determines the time tM = t K + t D required for perfect mdecular micr~mixing.’~ ( 1 3) (a) Tennekes, H.; Lumley, J. L. A First Course in Turbulence: MIT Press: Cambridge, MA, 1972. (b) Villermaux, J. In Encyclopedia of Fluid Mechanics; Cheremisinoff, N., Ed.; Gulf Publishing: Houston, 1985; Vol. 2, Chapter 33. (14) (a) Sreenivasan, K. R.; Ramshankar, R.; Menevau, C. Proc. R . Soc., London 1989, A421.79. (b) Argoul, F.; Arneodo, A,; Grasseau, G.: Gagne, Y.; Hopfinger, E. J.; Frisch, U. Nature 1989, 338, 51. ( I 5) (a) Dewel, G.; Borckmans, P.; Walgraef, D. J . Phys. Chem. 1984,88, 5422; 1985,89, 4670. (b) Laplante, J. P.; Borckmans, P.; Dewel, G.; Gimenez, M.; Micheau, J. C. J . Phys. Chem. 1987, 91, 3401. (16) Kolmogorov, A. N. J . Fluid Mech. 1962, 13, 82. (17) BarEli, K. J . Phys. Chem. 1984.88, 6174. (18) Field, R. J.; Koros, E.; Noyes, R. M. J . Am. Chem. SOC.1972, 94, 8649. (19) (a) Villermaux, J. Genie de la Reaction Chimique; Edition Tech & Doc: Lavoisier, Paris, 1972. (b) Westerterp, K. R.; VanSawaaij, W. P. M.;
Beenackers, A. A. C. M. Chemical Reactor Design and Operation; Wiley: New York, 1983. (c) Sorensen, P. G.Disc. Meeting Deutsche Bunsenges. Phys. Chem., Aachen, 1979; preprints, p 41.
It is controlled by the time tK spent by a fluid element in traversing the Kolmogorov cascade and by the time to = L K 2 / D= ( V / S ) ~ / * / D required for the dissipation of the residual droplets by diffusion, where D is the diffusion coefficient and v the kinematic visc~sity.’~ t M is estimated to be of the order of 1 s for mixing of nonreactive aqueous The mixing mode, Le., the composition and the entry points of the feedstream(s), on the other hand, determines which species are contiguous and which are segregated from each other in the early stage of the process. Since the chemical environment of a molecule together with its reactive possibilities, embodied by the chemical mechanism, determines its probability and mode of reaction, it is clear that the gross kinetic behavior can be modified by stirring rate and mixing mode. Stirring effects will be prominent whenever micromixing time t M becomes comparable to or longer than the shortest chemical relaxation time tCH. While the overall reaction rate is determined by the slowest step, relevant in the present context are only the fast switching steps. Reactions like the present one that are relatively fast but not too fast are therefore good candidates for S M studies. The theoretical approaches taken to simulate S M effects fall into two categories. The first emphasizes the aspect of imperfect macromixing by representing the nonideal reactor as two or more ideal CSTR’s that are coupled through a stirring-dependent term6 The second class of models attempts to mimic the imperfect micromixing process described above, e g , by a n a l y t i ~ a land ~~.~ numerical7c realizations of Curl’s coalescence-dispersion model, by a m ~ d i f i c a t i o nof~ ~the “back-mixing” model proposed by Zwietering, or by the mean-field IEM model.7e In the past, experimental results on the C l 0 +~ I- system have ~ ~ , ~ ~the simulations of stird i r e ~ t l y ~ ~ - ’or~ *i ,n~d i r e ~ t l y guided ring6a*b,7a.bie and mixing7w effects. However, mainly the thermodynamic-to-flow branch transition has been documented experimentally, due to the sluggish kinetics of the reverse transit i 0 1 - 1 , and ~ ~ ~ the stirring and mixing effects were found to be “normal” compared to the present ones in the sense that premixing as well as enhanced stirring have the same stabilizing effect on the thermodynamic branch. This can be readily e ~ p l a i n e d ’in~ terms of the chemical and mixing mechanisms. In the present paper, in contrast, both down and up transitions are examined and it is found that the stirring effect in the N P M mode is counterintuitive and opposite to what was found in the chloriteiodide system. In fact, the following discussion does not succeed to rationalize all of the observations. The NFT mechanismlo of the BrO
