J . Phys. Chem. 1994,98, 6304-6307
6304
Coherence between the Stirring Effect in Bimolecular Reactions and the Belousov-Zhabotinskii Reaction in the Closed Batch Reactor Jana HlavfiEovfi and Peter Sevtik’ Department of Physical Chemistry, Faculty of Natural Sciences, Comenius University, 842 15 Bratislava, Slovakia Received: August 18, 1993; In Final Form: February 10, 1994’
The stirring effect in bimolecular reactions was studied to explain the coherence among turbulent mixing, concentration fluctuations, and chemical kinetics, which should lead to a stirring effect in an oscillating BelousovZhabotinskii reaction in the closed batch reactor, simulated by the changing of rate constants of some rapid radical reactions. The simulations based on a linear-eddy model of the turbulent mixing showed that the bimolecular reaction of two reactants is sped up by the increase of the stirring rate while the bimolecular reaction of one reactant is slowed in agreement with the simulations of the stirring effect in the BZ reaction. The mean deviation of concentration fluctuations from the mean concentration decreases except for the bimolecular reaction of two reactants, where it rises and then is decreased by turbulent mixing.
1. Introduction The observations of the stirring effect in the closed batch Belousov-Zhabotinskii (BZ) oscillatory reaction14 (the oxidation of malonic or oxalic acid by bromate ions catalyzed by Ce(III)/ Ce(1V) or Mn(II)/Mn(III) ions in sulfuric acid)5 have drawn attention to the question of the influence of the hydrodynamic turbulent flow on “homogeneous chemical kinetics”. This influence was studied in the continuous stirred tank flow reactors (CSTR) .&I2 Some authors try to explain the stirring effect in the closed anaerobic homogeneous reactor through the increase of the diffusion coefficient due to the turbulent transport of the reactive partners on small d i ~ t a n c e s . I ~The - ~ ~effective diffusion coefficient could be estimated as a sum of the molecular (DO)and “turbulent” (Dturb)diffusional coefficients. If the radius of the stirrer is 1.25 cm, the stirring rate is 2000 rpm and the kinematic viscosity of the solution is 10-2 cm2 8;the turbulent diffusional coefficient is 10 times higher than the molecular one (DOis cm2 s-I) when the concentration of reactants is 10-9 M. When the concentration is higher and the stirring rate is smaller, the value of Dturbdrops. They propose that this increase in the value of the diffusion coefficient would result in the proportional increase of the rate constants of diffusion rate controlled reactions. Noszticzius et al.15 simulated successfully the stirring effect in the BZ reaction with malonic acid using this hypothesis. They used the Radicalator model of the BZ reaction, according to which the oscillations in this system are controlled by the rapid reaction of Br02’ and malonyl radical BrO,’
+ MA‘
-
P, + P2
The increase in the rate constant of this reaction leads to the simulation of system behavior under higher stirring rates. According to the theory of the hydrodynamic turbulence, the overall process of the turbulent mixing comprises two main processes: turbulent convection and molecular diffusion.I6J7The smallest scale affected by turbulent convection is the Kolmogorov scale I)K = (v3/e)Il4, where v is the kinematic viscosity and e is the rate of the energy dissipation. This scale is about (v/Do)l/* times larger than the scale of molecular diffusion. The range between these two scales is called the viscousxonvective range, and concentration fluctuations can appear in this range.13J4 The value of ( v / D ) l / *is, in the BZ system, about 30. Menzinger and 0
Abstract published in Advance ACS Abstracts, June 1, 1994.
0022-3654/94/2098-6304%04.50/0
Jankowski experimentally observed concentration fluctuations in this system.’* The existence of concentration fluctuations is the basis of the other explanation of the stirring effect in our previous w0rk.1~ The whole reaction volume with concentration fluctuations can be divided into N equal subvolumes. The concentration ci of the subvolume Vi can be expressed a d 9
ci=E(l+ai) where E =
(1)
Xfv-lciisthe mean concentration and ai is the random
value higher than or equal to -1. The rate of the reaction A B
+
-
P in the subvolume Vi is
vi = kaibi
(2)
where ai and bi are the concentrations of the reactants A and B, respectively, and k is the rate constant. The mean reaction rate can be expressed using eq 1
(3) 1
zK1aij3ican be assigned as LYB and characterizes the correla-
tion between a and B. The distribution of the A and B reactants after addition to the reaction vessel is accomplished by turbulent mixing. As distributions of both reactants proceed simultaneously in the same vessel, one cannot foretell whether a and B are correlated or not. As iB can be negative, zero, or positive in this reaction, one cannot foretell the effect of the concentration fluctuations on the reaction rate in general. If it is negative, then the increase of the stirring rate should increase the reaction rate, in agreement with simulations of Noszticzius et al.15 If the reaction A + A Pis considered, the situation is simpler,
-
1
as the mean reaction rate is given by eq 4, where - ~ N is always positive.
*; ,
~=
(4)
That is why one can suppose the reaction rate of these reactions 0 1994 American Chemical Society
a
~
The Journal of Physical Chemistry, Vol. 98, No. 25, 1994 6305
Coherence between the Stirring Effect and BZ Reaction will fall with the increase of the stirring rate, as GI, which characterizes the mean deviation of the subvolume concentration from the mean, is expected to be smaller. The simulations of the stirring effect in the BZ system with oxalic acid19 are based on the decreasing rate constants of radical reactions 5 and 6 with the increase of the stirring rate. BrO,'
+ BrO,' + H,O Br'
-
+ Br'
HBrO,
-
+ BrO; + H+
(5)
-xE
Fluctuations The whole reaction volume is divided into N subvolumes with the random initial concentration. The concentration of the subvolume is changed by the chemical reaction course and by molecular diffusion. Subvolumes have a uniform size, and they represent eddies created by turbulent convection in the viscousconvective range. The location of the subvolume can be changed by turbulent convection. The algorithm of the simulation of turbulent mixing is based on the linear-eddy mode1.17v21.22 Turbulent convection is modeled by the block inversion events of the linear block. Block inversion represents the effect of turbulent convection of the linear eddy. Concentrationsof thesubvolumes incorporated in the blockwhich undergoes block inversion are inverted according to the triplet mapping rule adopted by Kerstein. The inverted block with length 1is divided into three segments. Their concentrations c(x,to) are inverted into ct(x,to) = c(3x - 2x0,t0) x,, I x I xo
+ 1/3
xo + 1/3 I x I xo + 21/3
c'(x,to) = C ( ~ X- 2x0 - 21,to)
7.5-
0
2. The Simulation of Turbulent Mixing and Concentration
+ 4x, + 21,t0)
15i
c0
8
Br,
The decrease of these rate constants lowers the number of oscillations, and under certain values, oscillations disappear,which is in agreement with the experiment.' The oscillations could be experimentally found with stirring rates from 50 to 150 rpm. The number of oscillations decrease when the stirring rate increases from 80 to 150 rpm. When the stirring rate was between 200 and 1200 rpm, no oscillations were recorded. The aim of this work is to explain the success of simulations1sJ9 of the stirring effect by the changes of the rate constant of bimolecular reactions by simulations of the reaction in the turbulent mixed system with concentration fluctuations. We have tried to verify the possibility of simulating the stirring effect by the change in the reaction rate according to eqs 3 and 4. Our theoretical assumptions of the stirring effect on the reaction rate are in agreement with work in the literature.m The origin of concentration fluctuations has not yet been well understood. They could be created by the heterogeneous processes like adsorption and catalysis on the reactor walls, on a stirrer, or on a surface of the Pt-electrode or by the nucleation of gas and liquid bubbles of reaction products.18
d(x,to) = c(-3x
W
c
XO
+ 21/3 I x I 1
c(x,to) otherwise The size of the inverted block is confined to the range q k I1 I L,where L is the integral scale (represents the largest size of the flow). The value of 1 was in this paper chosen according to the probability density functiw f(l)*' (7)
01-.0
1---
200
-7-
,--__.I
._
GO0
400
f
800
stirring rate /rpm
Figure 1. Kolmogorov scale of the turbulent flow for different lengths of stirrer as a function of stirring rate.
andp = 4/3. The location of the block inversion event is selected randomly with uniform probability along the x-axis. The frequency of block inversion events wl is selected by the Poisson process (w is the event-frequency parameter with unit (length X time)-' derived from the fluid-element diffusivity &:
-
where p is again 4/3 and (9) Molecular diffusion is treated here as a three-dimensional problem, and diffusional terms are incorporated into reactiondiffusion kinetic equations. The turbulent flow is directed along the y-axis. The number of subvolumes in the x-axis is 2 n L / q ~ ,where L is 0.5 cm and n is 3.6, and 12, so the length of the subvolume is q K / 3 , 4 6 , and QK/12, respectively. The number of subvolumes in they- and z-directions is three. The random selection of concentrations of subvolumes is based on a Gaussian distribution with certain dispersion variance u. As the concentration cannot be negative, only nonnegative concentrations are taken into account. The average concentration is kept by accepting more concentrations lower than the average concentration. As the type of distribution of concentration fluctuations is not known, this little deviation from the Gaussian distribution does not matter. This selection takes place in time t = 0 s of the chemical reaction.
3. Results and Discussion The shape of the reactor, the volume of the reaction solution, and the length and shape of the stirrer are very important for the observation of the stirring effect. It is interesting to know the effect of, e.g., the change of the stirrer length on the Kolmogorov scale (Figure 1). One can compare the change of the range of stirring rates for which Kolmogorov scales are the same as in experiments1 with that of the BZ reaction. This range can be either narrowed, so the probability of the oscillatory behavior is very small, or enlarged, but the ratio of the reaction and diffusion terms in the diffusion-reaction kinetic equations need not lead to oscillations. The Simulation of the Reaction A + A P. The simulations were carried out for initial concentrations of the reactant A of 10-4 M, the Gaussian variation was 5 X 10-5 and 2 X 10-4 M,and rate constants were 105 M-1 s-1 and 3 X 103 M-1 s-1 and n = 3. Concentrationfluctuationsare smoothedout morequicklywhen the A A P reaction proceeds, but the shape of the Gzdecay
-
+
-
H l a v l b v l and SevEk
6306 The Journal of Physical Chemistry, Vol. 98, No. 25, I994
-
TABLE 2 Relative Changes of the Reaction Rate of the A + A P Reaction: k = 105 M-'s-l, fa = 10-4 M, UA = 2x104
simulations
-3
-4
-2
log(t/s)
-
-
TABLE 1: Changes of the Half-time of the Reaction A + A P with Initial Concentration of 10-4 M A
0.041 0.044
400 A
0.047 I(tl/z)tcor
18 12 6
1.48 1.51
11.5 10
1.51 1.59
9 5
- t1/2
(t1/2)toor
with time is similar to the decay due to turbulent mixing23(Figure 2). As the half-time of this reaction varies inversely to the initial concentration, the concentration of A drops faster in the subvolumes with higher initial concentrations. That is why the equalizing of the subvolume concentrations is faster here. The reaction affects the concentration decay stronger under low stirring rates and is attenuated under higher stirring rates as the rate of the molecular diffusion is increased. When k = los M-l s-I, the timesbf i j d r o p are about 70%, 35%, 25%, and 20% shorter than in the turbulent mixing decay when the stirring rates are 50,200, 300, and 400 rpm, respectively. The decay of iiwas closer to the decay in the system where only turbulent mixing proceeded in simulations with k = 3 X lo3 M-1 s-1, as diffusional terms were higher than the reaction one. It is interesting that the value of of the reaction product P 1
Aa2m/%
Abm/%
0.03 0.06 0.09
7.5 8.5 10.0
23.1 23.5 20.8
Anm/% 10.2 15.0 16.6
Aad% 22.4 26.5 25.5
0
-1
Figure 2. &time dependence in the A + A P reaction. The initial concentration of A is 10-4 M, the Gaussian variance U A is 2 X 10-4 M, k = 105 M-I s-l, CurvesA-Care for A 50,200,and 400rpm, respectively. Curves D-F are for P 50, 200, and 400 rpm, respectively.
50 200
eq4
tls
(GI = - Eciaf,where a1 is from eq 1) is 4.6and 3.6times higher N than the aj of the reactant for U A = 2 X 1V and 5 X M, respectively, at the beginning of the reaction. This is not changed using both values of the rate constants. The resultsofthesimulations (Table 1) showed that thereaction rate is lower under higher stirring rates in agreement with the assumption from section 1 and the CSTR simulation of this reaction.23 This is more evident when a higher rate constant is used. Comparing changes in the half-time of the reaction for different values of the Gaussian variance when k = 3 X l o 3 M-' s-I, one can see that the reaction rate is higher when U A is higher because the (Yi is higher. Reaction rates calculated from &Sanda (eq 4)are about 5-10% higher than the simulated ones when the diffusion terms are not taken into account, 5% for 400 rpm and 10% in the case of 50 rpm. The relative changes of the reaction rate when the stirring rate isvaried from 50 to 200 and 400 rpm, respectively, according to simulations and according to eq 4, are in Table 2. One can see good agreement between these two approaches. The concentration of the reactant varies similarly for all stirring rates. The differences of slopes of curves expressing the time
TABLE 3: Systems Used in Simulations with the A P Reaction, R = 3 1 2 3 4 5o-c 6 7 8 9 10
10-4 10-4 10-4 10-4 10-4 10-4 10-4 1 P 1 P 10-4
"n= = 33.. b nn = = 6.
2x104 2x104 10-4 10-4 10-4 10-4 10-4
1od 10-8 10-4 C n
5x104 2x104 5X1P5 5 x 10-5 5x104 5x10-5 2x104 5x10-7 5x104 5x104
5x104 2x104 5X1e5 5 x 10-5 5x104 5X10-' 2x104 5 x w 7 5x104 5x104
+B
-
2x10' 2x105 6x10' 2 x 105 7x106
lo7
2x105 7x106 7x106 7x106
= 12.
1.2 1.2/
.9-
a' .6-
.3-
0- , -4
7 -
-2.5
-1
.5
-
Figure 3. Time dependence of I? of A in the A + B P reaction for k = 6 X l o 3 M-I s-I. The initial concentration of A and B is 10-4 M, and the Gaussian variance U A = UB = 5 X 10-5 M. Curves A-E are for 50, 100, 150, 200, and 300 rpm, respectively.
dependence of the product concentration at different stirring rates are smoothed out, and after 0.1 s of the reaction course, all curves are the same. Simulation of the A + B P Reaction. Simulations were performed for systems in Table 3. The time dependence of the of the reactants is different from the decay of the concentration fluctuations due to turbulent mixing. When k = 6 X 103 (Figure 3) and 2 X 105 M-1 s-1, the initial drop of the ai is stopped and the value of increases until the reactant A or B is consumed completely in most of the subvolumes. At this time the decay of concentratio_? fluctuations due to molecular diffusion begins. The value of a2 is increased from the beginning of the reaction when k = 7 X 106 (Figure 4) and 1 X lo7 M-I s-I, as the diffusional terms are here from 10 to 100 times lower than the reaction one (Figure 4). The of the reaction product P is a t the beginning of the reaction from 2 to 5 times higher than of reactants, depending on the value of the Gaussian variance of the reactants. The decay of concentration fluctuations is similar to the decay in turbulent mixing. The Giof the product decreases due to turbulent mixing and differences in the reaction rate in different subvolumes.
-
ai
Coherence between the Stirring Effect and BZ Reaction
1-01
-4
I
-3
,
I
-2
I
-1
-
log(t/s)
0
1
Figure 4. &time dependence of A in the A + B P reaction for k = 7 X 106 M-'61. The initial concentration of A and B is l eM,and the Gaussian variance U A = ug = 5 X 10-5 M. Curves A-E are for 50,100, 150,200,and 300 rpm, respectively. .05
-.25
-.05
+
-
Figure 5. Drop of in the A B P reaction for k = 6 X lo3 M-1 s-l. The initial concentration of A and B is M,and the Gaussian variance U A = UB = 5 X lk5M. Curves A-E are for 50,100,150,200, and 300 rpm, respectively.
-.2
-,5
-.a
-1.1 ,
rate in sim.No.5a is about 10-15% slower than in sim.No. 10 in Table 3). It is in agreement with our assumption from section 1. But when k = 6 X lo3and 2 X lo5 M-l s-I, the reaction rate is higher when the Gaussian variance of reactants is higher (the difference between sim No. 4 and sim.No.7 is from 75 to 1 1595, according to the stirring rate), as molecular diffusion terms are about 10 times higher than thereaction term. After approximately 0.05 s, the changes in reaction rate are negligible. Changes of the reaction rate under different stirring rates are much more evident when initial concentrations of reactants are lower. This is in agreement with the Noszticzius et al. ideal5 that the stirring effect would be more expressive when concentrations of reactants are lower. The simulations were carried out for the system divided into uniform scale subvolumes,though one could expect that the scales of concentration fluctuations are in real systems nonuniform, distributed somehow. The length of subvolumes was in all simulations 7K/3. The decrease of the subvolume scale to 7 ~ / 6 and ?)K/12 (sim.noSb, c) leads to results similar to those for 7K/3. The only difference -- -was that due to faster molecular diffusion the curves of a2,ab, and mean concentrations were moved to higher stirring rate curves in the simulations with 7K/3. One can say that the simulations of the stirring effects in the oscillation BZ reaction, based on the change of the value of the rate constants of some r e a c t i o n ~ , ~could ~ J ~ represent the effect of turbulent mixing and concentration fluctuations, taking into account expressions 3 and 4. The simulations in ref 19 were based on the increase of the rate constant for the reaction of A + A P type. The results of both simulations are in agreement with results of this work. Noszticzius et al.15varied the rate parameters of the bimolecular reactions of the different radicals. Agreement with experiment was obtained when the reaction rate of this reaction increased with the increase of the stirring rate. The stirring effect is in our simulations more evident when the rate constant is high. Thus one can expect that especially rapid radical reactions are important in the stirring effect, so one should use the reaction mechanism incorporating radical reactions to simulate the stirring effect. As the values of (Ys and (Yghave not yet been investigated, the change of the rate constant in simulations of the stirring effect can say about the values of (Yi and (YB according to eqs 3 and 4, taking into account that these values are calculated by omitting molecular diffusion.
-
-.55
.I
The Journal of Physical Chemistry, Vol. 98, No. 25, 1994 6307
l e
References and Notes (1) SevEfk. P.;AdamEIkovi, L. Chem. Phys. Lett.
1988, 116, 419;J . Chem. Phys. 1989, 91, 1012. (2) L6pez-Tomb, L.; Saguk, F. J. Phys. Chem. 1991,95, 701. (3) Dutt, A. K.; Menzinger, M. J . Phys. Chem. 1992, 96, 8447. (4) Ruoff, P. J . Phys. Chem. 1993, 97, 6405. (5) Field, R.J. In Oscillationsand Traveling Wavesin ChemicalSystems; Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985. (6) Curl, R. L. AIChE J . 1%3,9, 175. 171 Zweiterinn. T. N. Chem. E m . Sci. 1984. 39. 1765. (8j Villerma
