15810
J. Phys. Chem. 1996, 100, 15810-15814
Differential Flow Instability in Tubular Flow Reactor: Its Convective Nature Xiao-Guang Wu,† Satoshi Nakata,‡ Michael Menzinger,*,† and Arkady Rovinsky† Department of Chemistry, UniVersity of Toronto, Toronto, Ontario, Canada M5S 1A1, and Nara UniVersity of Education, Nara 630 Japan ReceiVed: February 1, 1996X
Experiments and stochastic simulations are presented as evidence for the convective nature of the differential flow instability, observed in a tubular flow reactor, using the Belousov-Zhabotinsky system as the active medium. The irregular spacing and amplitudes of the traveling wave fronts is characteristic of the noiseinduced structures that arise through the convective instability. When recycling of the reactive flow is introduced in the simulations in the form of periodic boundary conditions, the waves become periodic and the instability absolute. The average concentrations of both activator and inhibitor are enhanced in the patterned state, and the enhancements are greater under conditions of absolute than under convective instability.
Introduction A differential flow1 between the counteracting species of a dynamical activator-inhibitor system may destabilize its homogeneous reference state and cause the medium to selforganize into a pattern of traveling waves through the differential flow instability2-4 (DIFI). Differential flow induced waves and stationary Turing patterns share the same mechanism of pattern formation:2,4 both are caused by the spatial uncoupling of the activator and inhibitor species by their differential transportsflow in the former and diffusion in the latter case. This uncoupling locally releases the inherent potential of the activator to grow, while spreading of the activation over the whole space is prevented by lateral inhibition. The DIFI was first predicted theoretically2 and verified experimentally3 in the BelousovZhabotinsky (BZ) system using a tubular flow reactor (Figure 1) in which the inhibitor was immobilized. In spatially extended dynamical systems whose reflection symmetry is broken by a flow, instabilities are either “convective” or “absolute”.10-12 These two types of instability can be distinguished physically by considering the evolution of an isolated, spatially localized perturbation of the unstable reference state in the presence of a flow.11 The instability is called convective if the disturbance spreads at a velocity less than the flow rate. It will be carried away from its point of origin and its amplitude grows in the course of its downstream propagation: as a result, the system acts as a tuned amplifier of noise.8 In a finite, bounded system, the resulting wave front is eventually washed out by the flow and the system returns to its steady state. This type of system is called convectively unstable and absolutely stable. In practice, there is always internal or external noise, which acts as a persistent source of perturbations that are selectively amplified and leave the system in an aperiodic, noise-sustained spatiotemporal pattern.12 The absolute instability, on the other hand, arises if the perturbation spreads faster than the flow. In this case, when the perturbation reaches a point in space, it continues to grow in time at that point. A convectively unstable system may be made absolutely unstable by introducing a spatial feedback loop,11 since reinjection of an amplified perturbation gives rise to its continued growth. As a result, the system settles in a state of selfsustained, periodic waves that circulate through the loop. More formally, the convective and the absolute instabilities may be †
University of Toronto. Nara University of Education. X Abstract published in AdVance ACS Abstracts, September 1, 1996. ‡
S0022-3654(96)00354-1 CCC: $12.00
Figure 1. Schematic diagram of the flow reactor (inner diameter, 0.35 cm; length 40 cm). The tube is packed with cation exchanger on which ferroin is immobilized.3 Key: RV, regulating valve; P, manometer; FD, fritted disk; S, stopcock. M, mask; PD, photodiode. M and PD are attached to the video monitor, and not to the flow tube, for image analysis.
distinguished in terms of the roots of a dispersion relation of the waves in the complex plane.11 Experimentally, the two types of instability may be distinguished then by the aperiodic nature of the waves and their spatial or temporal growth characteristics on the one hand and by their periodicity on the other. On the level of linear stability analysis they are characterized by the eigenvalues: a finite, bounded system is convectively unstable if all its eigenvalues have negative real parts while it nevertheless amplifies the perturbations in the course of their propagation.8,10-12 An absolutely unstable system possesses at least one eigenvalue with positive real part.11 The DIFI waves that were predicted2 and experimentally observed3 in the BZ system were initially interpreted2,3 in terms of an absolute instability that is the direct consequence of the periodic boundary conditions used in these early works. Subsequently we described the DIFI-induced waves that arise in the nonisothermal, tubular packed bed reactor in the presence of an exothermic reaction.7-9 In this setting, DIFI may occur due to the autocatalysis nature of the released heat and the different rates of flow of heat and matter.7 The instability was © 1996 American Chemical Society
Differential Flow Instability shown to be convective when it occurred in the conventional, tubular packed bed reactor8 with appropriate flow boundary conditions. When a part of the product stream or of the reaction heat is recycled, the wave fronts are reinjected and the instability takes an absolute character.9 The resulting periodic wave circulate around the reactor loop.9 Physically the isothermal flow reactor (Figure 1) and the tubular packed bed reactor are closely related. By analogy, one expects then the DIFI to be convective in the isothermal reactor as it is in the nonisothermal reactor. The main aim of this paper is to present experimental evidence for the convective instability in the BZ reaction in the flow reactor shown in Figure 1. Together with stochastic simulations, these observations confirm directly, without elaborate stability analysis, the convective nature of the DIFI in this experimental setup. The observations focus on the aperiodic nature of the waves and on their spatial growth characteristic in the upstream portion of the reactor. These experiments are compared with stochastic simulations in which an external white noise process, meant to represent persistent perturbations arising within the granular medium, acts on the governing reactionflow-diffusion equations.2-4 The resulting DIFI waves possess the same stochastic characteristics of a noise-induced pattern as those found experimentally. Additional simulations are reproted to illustrate how the introduction of a recycle loop converts the convective into absolute instability: the irregular waves becomes periodic through feedback, even in the continued presence of noise. We showed elsewhere13 that the productivity (average concentrations and production rates) of the spatially uniform reference state without flow may change dramatically as the systems transits to a DIFI-induced patterned state. Observed enhancements13 of the space- or time-averaged ferriin concentrations in the BZ system of the order of 100% may be understood as the consequence of the spatial decoupling of activator and inhibitor. This uncoupling releases the tendency of the activator to grow, gives rise to spatial gradients, and enhances thereby the average concentration and formation rate of both activator and inhibitor. In real flow systems, there is always a mechanism that tends to counteract the spatial uncoupling by randomizing the pattern to a certain extent, be it through hydrodynamic mixing or through a local noise process of the kind considered in the present simulations. As a consequence, the productivity of this stochastically perturbed system is expected to be closer to that of the homogeneous reference state which is reached in the limit of rapid mixing. Since the differential flow induced control of productivity may be practically useful,13 it is important to know to which extent this control is limited by fluctuations. To address this problem we finally compare the productivity enhancement in the tubular, convectively unstable flow reactor with that of the absolutely unstable feedback reactor with periodic boundary conditions, both in the presence of noise. As expected, the productivity is reduced in the convectively unstable state. Experiments The experiments were done in the tubular reactor (0.35 cm × 40 cm) shown in Figure 1 and are described in greter detail elsewhere.3 Briefly, the BZ solution, which contains NaBrO3, bromomalonic acid, and H2SO4 flowed under nitrogen pressure through a packed bed of cation exchanger on which ferroin was immobilized. Ferriin, its oxidized, blue form, in the inhibitor, and the activator is the intermediate HBrO2. The differential flow, in this case equal to the fluid flow rate, was controlled through the applied pressure. In the absence of flow, the system
J. Phys. Chem., Vol. 100, No. 39, 1996 15811
Figure 2. (a) Snapshots of experimentally observed oxidation fronts (bright ) ferriin) traveling from left to right in a 5 cm long segment near the center of the flow tube at four successive times 20 s apart (bottom to top). Initial concentrations: [NaBrO3] ) 0.8 M, [bromomalonic acid] ) 0.4 M, [H2SO4] ) 0.02 M; [ferroin] ) 1 × 10-4 M (see ref 3). Nitrogen pressure is 0.34 bar. The bright lines near the ends of the tube segments are 5 cm apart. (b) Snapshots of simulated 1D waves of inhibitor Y. The concentration is plotted as gray shades at four equally spaced time intervals of ∆t ) 108 s (which corresponds to 5000 integration steps). Length of the displayed section, L ) 9 cm. The dashed line crosses the peak value of a traveling wave. Parameters: ) 2.45, q ) 0.5, R ) 0.48, β ) 1.45 × 10-3, µ ) 5 × 10-4, V ) 0.015, Dx ) 3 × 10-3, Dy ) 0, h ) 0.002, ξx0 ) 10-4, and ξy0 ) 0. A ) 0.125, B ) 0.2, C ) 3 × 10-4, and h0 ) 0.03.
was initially in the reduced (red) state. When the flow was started by applying pressure and opening the stopcock, blue oxidation fronts evolved in the upstream portion gradually and grew in amplitude as they moved down the tube. They were recorded on video tape. A simple image analysis of the video images was performed by attaching a photodiode and a mask to the screen of the video monitor and recording the light signal from a 0.3-cm section of the flow tube as a function of time. Figure 2a shows four sequential images of a central portion of the flow tube, taken at 20-s intervals. They show four oxidation wavefronts (bright ) ferriin) that travel along the tube at a rate of 0.016 ( 0.002 cm/s. Attention is drawn to the irregular spacing of the waves and the jagged appearance of the wave fronts. Figure 3a shows a time series of the photodiode signal which is a measure of the relative ferriin concentration at that fixed, central location on the flow rate as a function of time. It reproduces the irregular spacing of the wave fronts seen in Figure 2a and shows in addition a nonconstant peak height. The spatial growth of the waves was measured by freezing an image of the upstream portion of the flow tube on the video screen and scanning it with the photodiode detector. The zero of position corresponds to the top of the narrow part of the flow tube. In Figure 4a, the relative peak amplitudes (the peak signal, normalized by the reference signal from the homogeneous state at zero flow) are plotted as a function of their position along the flow tube. The sigmoid curve through the points is drawn as a guide to the eye. Clearly, the waves grow in the course of
15812 J. Phys. Chem., Vol. 100, No. 39, 1996
Wu et al.
Figure 3. (a) Experimental time series of the brightness (a measure of the ferriin concentration) of the video image Figure 2a, recorded near the center of the flow tube. Conditions are as in Figure 2a. (b) Calculated time series X(L/2,t). Parameters are the same as in Figure 2b, except V ) 0.015. T ) 5 × 104 time steps ∼18.13 min.
their downstream propagation. The wide scatter of wave amplitudes around the average line is further evidence for the stochastic nature of the growth process. The simulated results, Figure 3b, confirm that this scatter of wave amplitudes is not due to measurement errors but that it is an intrinsic stochastic property of the system. Computer Simulations The governing stochastic reaction-flow-diffusion equations are2,3
∂X(r,t) ) F(X(r,t),Y(r,t)) + Dx∇2X(r,t) + ∂t v‚∇X(r,t) + ξx(r,t) ∂Y(r,t) ) G(X(r,t),Y(r,t)) + Dy∇2Y(r,t) + ξy(r,t) ∂t
(1)
where the reaction terms are given by the Puschinator model2,6 of the BZ reaction:
X-µ 1 Y F(X,Y) ) X(1 - X) - 2qR +β 1-Y x+µ
[
(
)
]
Y G(X,Y) ) X - R 1-Y and X(r,t) and Y(r,t) are the scaled concentrations of bromous acid (activator) and ferriin (inhibitor) at the spatial point r and time t. The additive noise terms ξx,y(r,t) describe the perturbations of the BZ medium. Physically these persistent perturbations may be thought to arise from the granular nature of the reactor packing and the CO2 bubbles formed during the later stages of the reaction. Here the noise process is taken to affect
Figure 4. (a) Spatial dependence of the normalized photodiode peak signal (a measure of the ferriin concentration; see text) at a fixed time. Experimental conditions: [NaBrO3] ) 0.45 M, [bromomalonic acid] ) 0.1 M, [H2SO4] ) 0.02 M; [ferroin] ) 2 × 10-4 M (see ref 3). (b) Simulation: Spatial growth of Xm. Concentration maxima at five successive times are superimposed. Smooth curve is a hyperbolic fit. Same system parameters as in Figure 2b but V ) 0.02. L ) 1000l ∼18.7 cm. (c) Simulation: Temporal growth of four individual peaks in X concentration. System parameters as in (b). T ) 8 × 105 time steps ∼29.01 min.
only the X variable by setting ξy ) 0. The model and the parameters , q, R, β, and µ are defined elsewhere.2,3,6 The parameter values were chosen to agree with the values of the experiments. They are given in the caption of Figure 2. Since the instability occurs along the flow direction z when the flow velocity ν exceeds a threshold value,2,3 we may simplify eqs 1 to a set of one-dimensional PDEs. Our computer simulations are based on a coupled-map lattice:
X(i,t+1) ) X(i,t) + ∫t
t+h
dt′F(X(x,t′),Y(i,t′)) +
Dx[X(i+1,t) + X(i-1,t) - 2X(i,t)] + V[X(i+1,t) X(i-1,t)] + ξx(t) Y(i,t+1) ) Y(i,t) + ∫t
t+h
dt′G(x(i,t′),Y(i,t′))
+Dy[Y(i+1,t) + Y(i-1,t) - 2Y(i,t)]
(2)
obtained by discretizing the one-dimensional set of reactionflow-diffusion equations. The scaled flow rate and diffusion coefficients are defined as
V ) hV/2l, Dx,y ) hDx,y/l2
Differential Flow Instability
J. Phys. Chem., Vol. 100, No. 39, 1996 15813
where h and l are the small time and space increments and i is the spatial index (z ) il). The additive noise term ξx(t) in (2) plays the role of seeds for the growth of the waves. By suitably choosing ξx(t) we may obtain dynamical behavior similar to that observed in the differential-flow experiments. In this model, both time and space are discrete but the concentrations X and Y are continuous variables. Numerical solutions of eqs 2 will uniformly tend to those of the onedimensional reaction-flow-diffusion equations (eqs 1) as h, l f 0. In eqs 2 the noise process ξx(t) is taken to be Gaussian white noise with δ correlation and zero mean, i.e.
〈ξx(t)ξx(t′)〉 ) (ξ0x )2δtt′, 〈ξx(t)〉 ) 0 This one-dimensional model completely neglects the transverse effects; yet we are still able to reproduce the main dynamical features along the flow direction. One-dimensional simulations save computer time and allow us to collect an amount of data sufficient for studying the system’s statistical properties. In our previous one-dimensional simulations,2,3 the boundary condition was chosen for simplicity to be periodic. The use of periodic boundary conditions makes the instability absolute and causes spatially periodic traveling waves to evolve from the random initial conditions (cf. Figure 2 in ref 9) even in the presence of external noise. However, when traveling waves grow to their asymptotic value, boundary effects become important. The flow (Dirichlet) boundary conditions that correspond to the experiment were set as follows. At the inflow (i ) 0) we choose
X(0,t) ) X0 + ζ(t), Y(0,t) ) Y0
(3)
Here X0 and Y0 are steady-state concentrations of the local system (F(X0,Y0) ) G(X0,Y0) ) 0) and ζ(t) is a small-amplitude, uniformly distributed random number between 0 and 10-3 which, in addition to the spatially distributed noise term ξx acts as “seeds” that may develop into traveling waves in the course of their downstream propagation. The choice of the Dirichlet boundary conditions is suggested by the experimental setup (Figure 1): since the flow first passes slowly through the wide packed portion, it brings to the proper entrance of the tube the BZ intermediates in concentrations close to those of the steady state. At the outflow (i ) L), we choose a free boundary condition expressed as
X(L,t) ) VXR(L-1,t-1) + (1 - V)XR(L,t-1)
(4)
Here a superscript R indicates that X has been updated by the reaction step; i.e.,
XR(i,t) ) X(i,t) + ∫t
t+h
dt′F(X(i,t),Y(i,t))
This boundary condition allows the flow at i ) L to leave the tube with a constant velocity V without affecting the growth rate of trigger waves at any point within the tube. Figure 2b shows four snapshots of simulated waves from a central portion of the tube (reactor) using the parameters in ref 3 that represent the experiment (Figure 2a). Several wave fronts traveled down the reactor at a constant speed, and their spacing s exhibits the same kind of irregularity as the experiments. The planar wave fronts reflect the one-dimensional simulation. Figure 3b is a time series obtained at the halfway point (i ) L/2) of the reactor. The variability of the wave spacings and amplitudes qualitatively reproduces the experimental result (Figure 3a).
Statistical Properties of Waves In this model the steady state X0, Y0 is perturbed by small, random signals, (cf. eqs 2) which are meant to represent all the external perturabations to the system, such as impurity centers in the medium, gas bubbles generated in the reaction, inhomogeneities of the flow, etc. Small perturbations are amplified when the flow velocity exceeds a threshold value (which depends on Dx2,3) and traveling waves are formed. It can be shown that the diffusion term and the differential-flow term in eqs 1 have opposite effects: diffusion tends to smooth out small spatial inhomogeneities while differential flow tends to enhance them. If the latter dominates, perturbations grow. With initial conditions given by the reference state (X0, Y0) and with additive noises ξx and ζ, high-concentration peaks in both X and Y evolve in time and space. Figure 4b shows the calculated peak values of Xm(r) as a function of position along the tube: peaks from five snapshots (Figure 2) are superimposed in this panel. As the flow progresses from entrance to exit ports, one first observes an incubation period of slow growth, followed by a narrow zone of rapid exponential growth and by a broader zone of approximately linear growth, and finally by saturation. The peak amplitudes Xm show a wide scatter, particularly in the regions of growth. In the saturated, asymptotic regime, the scatter of the amplitudes is less pronounced than in the region of growth, but still appreciable. These results agree qualitatively with the experimental observations (Figures 2a, 3a, and 4a). This snapshot representation emphasizes the stochastic nature of birth and growth of the waves. On the other hand, each wave front, once it has reached a threshold, evolves smoothly and deterministically, as shown in Figure 4c. Here, the peak values Xm(t) of the individual waves are followed as a function of time. The exponential growth during the early phase agrees with the growth law
Xm(t) ) Xm(0) exp(λt), Xm(0) , 1
(5)
obtained by linearizing the governing eqs 1. It is valid for sufficiently small values of Xm(t). Since high-concentration peaks are formed competitively from random fluctuations near the reactor inlet, the separation between two adjacent peaks, s(t), is also distributed randomly (cf. Figure 2b). As the statistics of the seeding noise was changed, the asymptotic separations of the wave fronts (see Figure 3b) changed also. This observation is further evidence for the noiseinduced nature of the wave. This result is quite different from our previous2,3 and present simulations with periodic boundary conditions, which eventually impose a fixed wavelength on the pattern. Peak separations with different initial values tended to a common final value L/N where N is the number of highconcentration peaks, since the existing flow is reinjected into the tube. The statistics of the wave front separation s was studied as a function of flow rate. Figure 5 shows that the mean value 〈s〉 increases linearly with the flow from a finite intercept. The standard deviation σ(s), represented by the error bars, also increases with the flow. So does the relative deviation σ(s)/ 〈s〉, indicating that the waves become more irregular as the flow is increased above threshold. Finally, we studied the effect of recycling by replacing the flow boundary conditions (eqs 3 and 4) by periodic boundary conditions that correspond to the limiting case of 100% recycling. As Figure 6a shows, the resulting pattern is periodic in both time and amplitude, indicating that the system has now settled into a state of self-sustained, periodic operation that characterizes the absolute instability. The same periodic opera-
15814 J. Phys. Chem., Vol. 100, No. 39, 1996
Figure 5. Mean separation between high concentration peaks. Dotted line is a linear fit of the mean separations. Parameters are the sme as in Figure 2b. Error bars represent standard deviations.
Figure 6. Calculated snapshots of waves X(z) using (a) periodic boundary conditions and (b) free (flow) boundary conditions. Same parameters as in Figure 2b. V ) 0.015; L ) 1000l ∼18.7 cm.
tion persists when the appropriately modified flow model (eqs 2-4) is used with partial recycling.9,14 Without recycling, however, the system displays the irregular, noise-induced pattern (Figure 6b) that indicates convective instability. The selfsustained, periodic waves have been observed experimentally and numerically in a combustion reaction in a nonisothermal loop reactor in which part of the reaction heat was recycled.14 A relevant issue is the previously observed13 enhancement of the system’s productivity by differential flow and its dependence on the boundary condition. By comparing the space average 〈X〉 with X0, the concentration in the homogeneous reference state on obtains 〈X〉/X0 as a measure of the system’s change of productivity in the patterned state. For the periodic pattern, Figure 6a, we calculate an enhancement 〈X〉/X0 ) 66.8, while for the noise-induced pattern, Figure 6b, the enhancement is reduced to 〈X〉/X0 ) 50.0. Conclusions The fact that the nonisothermal packed bed reactor is convectively unstable was demonstrated earlier8 through the stability analysis of its nonuniform steady state: while all eigenvalues had negative real parts it nevertheless acts as a tuned amplifier of those perturbations that fall within a resonance band in frequency and wavenumber space.8 By physical analogy, one may infer that the isothermal packed bed reactor (Figure 1) is also convectively unstable. To confirm this without repeating the stability analysis, we take here an experimental
Wu et al. approach and focus on the stochastic characteristics of the system and on the growth dynamics of waves. The stochastic simulations are in essential agreement with experiment. This comparison ensures that the observed irregularities may be used as diagnostic pointers to the essential dynamics of the convective instability and that they are not experimental artifacts. For instance, were it not for the simulations in Figure 4b, the experimental results in Figure 4a would be little convincing. The irregularities of wave spacings (Figures 2a and 3a) and of amplitude are well reproduced by the calculations (Figures 2b and 3b). So is the wide scatter of the amplitudes of successive waves in their growth phase (Figure 4a,b). Both observations confirm that the waves are noise-induced structures. In contrast, the growth of individual waves proceeds smoothly (Figure 4c). In comparing Figure 2a and b one should keep in mind that the wave front irregularities transverse to the flow cannot be reproduced by the one-dimensional calculation. Preliminary computations using a two-dimensional analog of the coupled map (eq 2) gave nonplanar wave fronts that resemble qualitatively those in Figure 2a. They were rather time-consuming and were not pursued further. Finally, these results on convective instability are put in perspective computationally by introducing feedback in the form of periodic boundary conditions that are known11 to make the instability absolute. As a result, the waves shown in Figure 6a are periodic and have constant amplitudes. The present calculations show that the productivity enhancement in the patterned state depends not only on the system parameters but also on its boundary conditions. For the case studied in Figure 6, the decrease of the enhancement factor in going from periodic to flow boundary conditions probably reflects the increased wave spacing in the latter case, where the ignited, highly productive state contributes relatively less to the overall productivity. Comparison of Figure 6a and b concludes the demonstration of how the physical aspect of the instability is determined by the boundary conditions. References and Notes (1) A differential flow is a bulk flow in a mixture in which two or more components move with different velocities. (2) Rovinsky, A. B.; Menzinger, M. Phys. ReV. Lett. 1992, 69, 1193. (3) Rovinsky, A. B.; Menzinger, M. Phys. ReV. Lett. 1993, 70, 778. (4) Menzinger, M.; Rovinsky, A. B. The Differential Flow Instabilities In Chemical WaVes and Patterns; Kapral, R., Showalter, K., Eds.; Kluwer Academic Publishers: Dorderecht, The Netherlands, 1995. (5) Turing, A. Philos. Trans. R. Soc. London B 1952, 237, 37. (6) Rovinsky, A. B.; Zhabotinsky, A. M. J. Phys. Chem. 1984, 88, 6081. Aliev, R. R.; Rovinsky, A. B. J. Phys. Chem. 1992, 96, 732. (7) Yakhnin, V. Z.; Rovinsky, A.; Menzinger, M. J. Phys. Chem. 1994, 98, 2116; Chem. Eng. Sci. 1994, 49, 3257. (8) Yakhnin, V. Z.; Rovinsky, A.; Menzinger, M. Chem. Eng. Sci. 1995, 50, 2853. (9) Yakhnin, V. Z.; Rovinsky, A.; Menzinger, M. Chem. Eng. Sci. 1995, 50, 1591. (10) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon Press: Oxford, U.K., 1982. (11) (a) Briggs, R. J. Electron-Stream Interactions with Plasmas; MIT Press: Cambridge, MA, 1964. (b) Lifshits, E. M.; Pitaevskii, L. P. Physical Kinetics; Pergamon Press: Oxford, U.K., 1981. (12) Deissler, R. J. J. Stat. Phys. 1985, 40, 371. (13) Rovinsky, A. M.; Nakata, S.; Yakhnin, V. Z.; Menzinger, M. Phys. Lett. A, in press. (14) Lauschke, G.; Gilles, E. D. Chem. Eng. Sci. 1995, 49, 5359.
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