7. Well-Stirred Reactors IV

Transcription

1 7. Well-Stirred Reactors IV The Belousov-Zhabotinsky Reaction: Models and Experiments Oregonator [based on the FKN mechanism; Field, R. J. & Noyes, R. M.: Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60, (1974), urlhttp://dx.doi.org/ / ; Tyson, J. J. & Fife, P. C.: Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73, (1980), ] Q + W U + P (R1) U + W 2P (R2) Q + U 2U + 2V (R3) 2U Q + P (R4) B + V νw (R5) Adv PChem 7.1

2 Q = BrO 3 B = BrMA (bromomalonic acid) P = HOBr U = HBrO 2 V is the oxidized form of the catalyst, typically Ce 4+, Fe 3+, or Ru(bpy) 3 3+ W = Br ν is an adjustable stoichiometric parameter B, Q, and P are pool chemicals dimensionless rate equations three-variable Oregonator model: ɛ du = qw uw + u u2 (1a) dv = u v ɛ dw = qw uw + hv (1c) (1b) h = 2ν ɛ ɛ 1, q 1, h 1 = w evolves on the fastest time scale; eliminate by Adv PChem 7.2

3 quasi-steady state approximation dw = 0 w = hv u + q two-variable Oregonator model: du = 1 ( u u 2 hv u q ) ɛ u + q dv = u v (2a) (2b) non-trivial steady state: u = 1 h q + v = u 1 2h + h 2 + 2q + 6hq + q 2 2 Adv PChem 7.3

4 Jacobian [ ] 1 1 2u 2hqu J = ɛ (u + q) 2 h u q ɛ u + q 1 1 the two-variable Oregonator is a pure activatorinhibitor system; U(HBrO 2 ) = activator, V (the oxidized form of the catalyst) = inhibitor Hopf condition T = trj = 0 ɛ H = 1 2u 2hqu (u + q) 2 evaluate the Hopf condition for a specific value of q, q = Adv PChem 7.4

5 ɛ unstable h simulations q = , ɛ = : h H;l = , h H;u = h = 0.58, (u(0), v(0)) = (1.2u,0.8v) u t relaxation oscillations the low-h Hopf bifurcation is subcritical Adv PChem 7.5

6 oscillations for h = 2/3 u t the high-h Hopf bifurcation is supercritical, h = 2.106: (u(0), v(0)) = (1.2u, 0.8v) steady state (u, v) = ( , ) h = 2.105: (u(0), v(0)) = (1.2u, 0.8v) smallamplitude limit cycle u t the two-variable Oregonator cannot display chaos; the three-variable Oregonator does not display chaos Adv PChem 7.6

7 Showalter Noyes Bar-Eli Model [Showalter, K.; Noyes, R. M. & Bar-Eli, K.: A modified Oregonator model exhibiting complicated limit cycle behavior in a flow system, J. Chem. Phys., 69, (1978), http: //dx.doi.org/ / ; Barkley, D.; Ringland, J. & Turner, J. S.: Observations of a torus in a model of the Belousov Zhabotinskii reaction, J. Chem. Phys., 87, (1987), aip.org/link/?jcp/87/3812/1] BrO 3 + Br + 2H + k 1 k 1 HBrO 2 + HOBr (R6) HBrO 2 + Br + H + k 2 k 2 2HOBr (R7) BrO 3 + HBrO 2 + H + k 3 k 3 2BrO 2 (R8) Ce 3+ + BrO 2 + H + k 4 k 4 HBrO 2 + Ce 4+ (R9) 2HBrO 2 k 5 k 5 BrO 3 + HOBr + H + (R10) Adv PChem 7.7

8 Ce 4+ k 6 g Br + Ce 3+ (R11) A + Y X + Y A + X κ 1 κ 1 X + P (R12) κ 2 κ 2 2P (R13) κ 3 κ 3 2W (R14) C + W κ 4 κ 4 X + Z (R15) 2X Z κ 5 κ 5 A + P (R16) κ 6 g Y + C (R17) with A = BrO 3 P = HOBr W = BrO 2 X = HBrO 2 Adv PChem 7.8

9 Y = Br Z = Ce 4+ C = Ce 3+ κ i = h i ([H + ])k i h 1 = [H + ] 2 h 2 = h 3 = h 4 = h 5 = [H + ] h 1 = h 2 = h 3 = h 4 = h 5 = h 6 = 1 v 1 = κ 1 [A][Y] κ 1 [X][P] v 2 = κ 2 [X][Y] κ 2 [P] 2 v 3 = κ 3 [A][X] κ 3 [W] 2 v 4 = κ 4 [C][W] κ 4 [X][Z] v 5 = κ 5 [X] 2 κ 5 [A][P] v 6 = κ 6 [Z] rate equations for CSTR d[a] = v 1 v 3 + v 5 + α ( [A] 0 [A] ) Adv PChem 7.9

10 d[p] = v 1 + 2v 2 + v 5 α[p] d[w] = 2v 3 v 4 α[w] d[x] = v 1 v 2 v 3 + v 4 2v 5 α[x] d[y] = v 1 v 2 + g v 6 + α ( [Y] 0 [Y] ) d[z] = v 4 v 6 α[z] d[c] = v 4 + v 6 + α ( [C] 0 [C] ) the model describes mixed-mode oscillations and chaotic behavior Barkley et al, Figure 1, varying [Y] 0 : steady state Hopf bifurcation limit cycle secondary Hopf bifurcation torus (quasiperiodic behavior); bistability between torus and stable steady state created by a saddle-node bifurcation; torus is destroyed by colliding with an unstable torus; unstable limit cycle undergoes a period-doubling bifurcation, before terminat- Adv PChem 7.10

11 ing in a saddle-loop bifurcation (homoclinic orbit) of the unstable steady state of the middle branch of the steady state curve review of experimental and numerical evidence for chaos in the BZ reaction: Argoul, F.; Arneodo, A.; Richetti, P.; Roux, J. C. & Swinney, H. L.: Chemical chaos: from hints to confirmation, Acc. Chem. Res., 20, (1987), doi/abs/ /ar00144a002 periodic chaotic sequences are common in experiments on the BZ reaction a simple model of chaotic behavior in the BZ reaction: Gyorgyi, L. & Field, R. J.: A three-variable model of deterministic chaos in the Belousov Zhabotinsky reaction, Nature, 355, (1992); doi.org/ /355808a0 read Chapter 8 in: Epstein, I. R. & Pojman, J. A., An Introduction to Nonlinear Chemical Dynamics (Oxford University Press, 1998) Adv PChem 7.11

12 The Chlorite Iodide Malonic Acid Reaction: and Experiments Models chlorite iodide malonic acid (CIMA) reaction belongs to the group of systematically designed chemical oscillating reactions pioneered by the Brandeis group among the very few reactions that can display transient oscillations in batch reactors major variant of the CIMA reaction is the simpler chlorine dioxide iodine malonic acid (CDIMA) reaction: after an initial induction period, where the chlorite, ClO 2, and iodide, I, are rapidly consumed to produce chlorine dioxide, ClO 2, and iodine, I 2, the dynamics of the CIMA reaction is given by that of the CDIMA reaction empirical rate law description of the CDIMA reaction: Lengyel-Epstein-Rabai (LRE) model [Lengyel, I.; Rabai, G. & Epstein, I. R.: Batch oscillation in the reaction of chlorine dioxide with iodine and malonic acid, J. Am. Chem. Soc., 112, (1990), Lengyel, I.; Rabai, G. & Epstein, I. R.: Experimental and modeling study of oscillations in the chlorine Adv PChem 7.12

13 dioxide-iodine-malonic acid reaction, J. Am. Chem. Soc., 112, (1990), /ja00181a011] MA + I 2 IMA + I + H +, (R18) ClO 2 + I ClO I 2, (R19) ClO 2 + 4I + 4H + Cl + 2I 2 + 2H 2 O, (R20) S + I 2 + I SI 3, (R21) reaction velocities: v 18 = k 1a[MA][I 2 ], (3) k 1b + [I 2 ] v 19 = k 2 [ClO 2 ][I ], (4) v 20 = k 3a [ClO 2 ][I ][H + ] + k 3b[ClO 2 ][I 2 ][I ] α + [I ] 2, (5) v 21 = k 4 [S][I 2 ][I ] k 4 [SI 3 ]. (6) MA= malonic acid, CH 2 (COOH) 2 IMA = iodomalonic acid, CHI(COOH) 2 S = starch or another substrate that binds triiodide ion I 3 [starch forms a deep blue complex with I3 ; Adv PChem 7.13

14 this color corresponds to the reduced state of the system; the oxidized state has a yellow color] α empirical parameter independent variables: [MA], [I 2 ], [ClO 2 ], [I ], [ClO 2 ], [S], and [SI 3 ] concentration of H + considered to be constant, Cl and IMA are inert products rate equations of the LER model for the CDIMA reaction in a well-stirred reactor without external feeds: d[ma] d[i 2 ] d[clo 2 ] d[i ] d[clo 2 ] d[s] = v 18 = v v v 20 v 21 = v 19 = v 18 v 19 4v 20 v 21 = v 19 v 20 = v 21 Adv PChem 7.14

15 d[si 3 ] = v 21 experiments = [I ] and [ClO 2 ] undergo much larger changes than the concentrations of the input species MA, ClO 2, and I 2 = simplified model [Lengyel, I. & Epstein, I. R.: Modeling of Turing Structures in the Chlorite- Iodide-Malonic Acid-Starch Reaction System, Science, 251, (1991), /science ; Lengyel, I. & Epstein, I. R.: A chemical approach to designing Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci. USA, 89, (1992), pnas.org/cgi/content/abstract/89/9/3977] Lengyel-Epstein (LE) model A U (R22) U V (R23) 4U + V D (R24) S + U W (R25) Adv PChem 7.15

16 where U = I, V = ClO 2, and W = SI 3 rate equations du = k 1 k 2u 4k u v 3 α + u 2 k 4u + k 4 w dv = k 2u k u v 3 α + u 2 dw = k 4u k 4 w k 1 = k 1a [MA] k 2 = k 2 [ClO 2 ] k 3 = k 3b [I 2 ] k 4 = k 4 [S][I 2 ] substrate is present in large excess, [S] can be considered constant add the first and last rate equation d(u + w ) = k 1 k 2u 4k u v 3 α + u 2 Adv PChem 7.16

17 fast equilibrium assumption for W w = k 4 k 4 u = ( ) d u + k 4 u k 4 = k 1 k 2u 4k u v 3 α + u 2 σ du = k 1 k 2u 4k u v 3 α + u 2 σ = 1 + k 4 k 4 = 1 + k 4 k 4 [S][I 2 ] = 1 + K [S][I 2 ] experimental range: agent: σ = 1 1 σ < 1000; no complexing nondimensionalize the rate equations u = αu v = k 2α k v 3 Adv PChem 7.17

18 t = t k 2 σ du = f u(u, v) = a u 4 uv 1 + u 2 ( dv = f v(u, v) = b u uv ) 1 + u 2 a = k 1 k 2 α b = k 3 k 2 α a [CH 2 COOH 2 ]/[ClO 2 ] b [I 2 ]/[ClO 2 ] steady states a u 4 u v 1 + u 2 = 0 u u v 1 + u 2 = 0 u v 1 + u 2 = u Adv PChem 7.18

19 a u 4u = 0 u = a 5 ( u 1 v ) 1 + u 2 = 0 v = 1 + u 2 = 1 + a2 25 unique steady state! stability ( ) ( ) J11 J J = 12 A11 /σ A = 12 /σ J 21 J 22 A 21 A 22 A 11 = f u u A 12 = f u v A 21 = f v u = 3a2 125 (u,v) 25 + a 2 = 20a (u,v) 25 + a 2 = b 2a2 (u,v) 25 + a 2 Adv PChem 7.19

20 A 22 = f v v = b 5a (u,v) 25 + a 2 detj = = 25ba σ(25 + a 2 ) > 0 = (u, v) cannot undergo a stationary bifurcation Hopf condition T = trj = 1 σ 3a bσa = 0 b H = 3a aσ for a > a 125/3 3a a 2 b 5a 25 + a 2 = 0 the steady is stable, T < 0, for b > b H, and unstable, T > 0, for b < b H ; the LE model has a limit cycle for b < b H Adv PChem 7.20

21 The LE system is a pure activator-inhibitor system; iodide ion, U, is the activator, A 11 > 0, and chlorite ion, V, is the inhibitor, A 22 < 0, and A 12 < 0 and A 21 > 0 consider a = 50.0 and σ = 1.0; then b H = 29.5 (u, v) = (10.0,101.0) the Hopf bifurcation is subcritical b = b H = 29.5, (u(0), v(0)) = (1.1u,0.9v) u t oscillations with finite amplitude Adv PChem 7.21