Fast Chemical Reactions in Chaotic Flows: Predicting the Product - PowerPoint PPT Presentation

Fast Chemical Reactions in Chaotic Flows: Predicting the Product Growth Rate Yue-Kin Tsang Scripps Institution of Oceanography University of California, San Diego

Chemical Reactions in Fluids (WEDECO) Advanced Oxidation Processes water treatment process to remove (Climate Change Science Program) pharmaceutical contaminants using ozone and other reagents Air Pollution Modeling Ozone Hole Modeling Fluid flow (mixing) affects progress of reactions

Irreversible Bimolecular Reactions A + B → P Example: acid-base reaction (neutralization) HCl( aq ) + NaOH( aq ) → NaCl( aq ) + H 2 O( ℓ )

Irreversible Bimolecular Reactions A + B → P Example: acid-base reaction (neutralization) HCl( aq ) + NaOH( aq ) → NaCl( aq ) + H 2 O( ℓ ) (Paret and Tabeling, 1998) P. E. Arratia and J. P. Gollub, Phys. Rev. Lett., 96 , 024501 (2006) O. Paireau and P. Tabeling, Phys. Rev. E, 56 , 2287 (1997)

Advection-Diffusion-Reaction Equation Concentration fields: a ( x , t ), b ( x , t ) and p ( x , t ) ∂ a ∂ t + u · ∇ a = κ ∇ 2 a − γ ab ∂ b ∂ t + u · ∇ b = κ ∇ 2 b − γ ab ∂ p ∂ t + u · ∇ p = κ ∇ 2 p + γ ab � a ( x , 0) � = � b ( x , 0) � = 1

Advection-Diffusion-Reaction Equation Concentration fields: a ( x , t ), b ( x , t ) and p ( x , t ) ∂ a ∂ t + u · ∇ a = κ ∇ 2 a − γ ab ∂ b ∂ t + u · ∇ b = κ ∇ 2 b − γ ab ∂ p ∂ t + u · ∇ p = κ ∇ 2 p + γ ab � a ( x , 0) � = � b ( x , 0) � = 1 Fast reactions: reaction time ≪ advection time ≪ di ff usion time Goal: time dependence of mean product concentration � p � = 1 − � a �

Flow Model and Simulations u ( x , t ) √  2 U cos[ k f y + θ 1 ( n )] ˆ n τ < t � ( n + 1 i , 2 ) τ   =  √ 2 U cos[ k f x + θ 2 ( n )] ˆ  j , ( n + 1 2 ) τ < t � ( n + 1) τ   Domain size: 2 π L × 2 π L (scale separation ∼ k f L )

Progress of Reaction U = 0 . 22 , k f L = 1 0 10 -1 10 〈 a 〉 = 1 _ 〈 p 〉 1 / D 1/ Da = 0.22 a = 0 U = 0.17 . 5 0 1/ Da = 0.13 -2 10 -3 10 0 50 100 150 t/ τ

Progress of Reaction U = 0 . 22 , k f L = 1 0 10 exp( − λ t ) -1 10 〈 a 〉 = 1 _ 〈 p 〉 1 / D 1/ Da = 0.22 a = 0 U = 0.17 . 5 0 1/ Da = 0.13 -2 10 -3 10 0 50 100 150 t/ τ

Progress of Reaction U = 0 . 22 , k f L = 1 0 10 exp( − λ t ) -1 10 〈 a 〉 = 1 _ 〈 p 〉 1 / D 1/ Da = 0.22 a = 0 U = 0.17 . 5 0 1/ Da = 0.13 -2 10 ∂ t a = − γ ab ∂ t b = − γ ab -3 10 0 50 100 150 t/ τ

Relation to Decaying Passive Scalar ∂ a ∂ t + u · ∇ a = κ ∇ 2 a − γ ab ∂ b ∂ t + u · ∇ b = κ ∇ 2 b − γ ab φ ≡ a − b ∂φ ∂ t + u · ∇ φ = κ ∇ 2 φ ⇒ For infinitely fast reactions: a ( x , t ) and b ( x , t ) never overlap | φ | = | a − b | = a + b ( a � 0 , b � 0) ∴ � a � = � b � = �| φ |� 2

Verifying the Passive Scalar Approximation U = 0 . 22 , k f L = 1 0 10 e − λ t 〈 a 〉 = 1 _ 〈 p 〉 1/ Da = 0.50 1/ Da = 0.22 -1 � a � = 1 − � p � 10 U = 0.17 1/ Da = 0.13 �| φ |� 2 -2 10 0 10 20 30 40 50 60 70 t/ τ

Literature on Decaying Passive Scalar “Strange eigenmode” B. J. Bayly, in Nonlinear Phenomena in Atmospheric and Oceanic Sciences (1992) R. T. Pierrehumbert, Chaos, Solitons & Fractals 4 , 1091 (1994) Variance decay rate from finite-time Lyapunov exponent (local stretching) T. M. Antonsen, Jr., Z. Fan and E. Ott, Phys. Rev. Lett. 75 , 1751 (1995) Validity of local stretching theory, decay rate based on e ff ective di ff usivity J. Sukhatme and R. T. Pierrehumbert, Phys. Rev. E 66 , 056302 (2002) D. R. Fereday, P. H. Haynes, A. Wonhas, and J. C. Vassilicos, Phys. Rev. E 65 , 035301 (2002) J.-L. Thiffeault and S. Childress, Chaos 13 , 502 (2003) D. R. Fereday and P. H. Haynes, Phys. Fluids bf 16, 4359 (2004) Y.-K. Tsang, T. M. Antonsen, Jr. and E. Ott, Phys. Rev. E 71 , 066301 (2005) Experimental studies G. A. Voth, T. C. Saint, G. Dobler, and J. P. Gollub, Phys. Fluids 15 , 2560 (2003) Other: KAM surface, Kraichnan model, forced scalar, boundary e ff ects,...etc A. Pikovsky and O. Popovych, Europhys. Lett. 61 , 625 (2003) P. H. Haynes and J. Vanneste, Phys. Fluids 17 , 097103 (2005) T. A. Shaw, J.-L. Thiffeault, and C. R. Doering, Physica D 231 , 143 (2007) E. Gouillart, O. Dauchot, B. Dubrulle, S. Roux, and J.-L. Thiffeault, Phys. Rev. E 78 , 026211 (2008)

Finite-time Lyapunov Exponent Finite-time Lyapunov exponent, h δ x ( t ) t log | δ x ( t ) | h ( x , t ) = 1 | δ x (0) | ¯ δ x (0) t →∞ h ( x , t ) h = lim x probability density function of h , ρ ( h , t ) at large time : ρ ( h , t ) ∼ exp[ − tG ( h )] t → ∞ as -3 ) U = 0.22 , k f = 1 , τ = 10 U = 0.22 , k f = 1 , τ = 10 ( x 10 100 8 -2 ) ( x 10 t =150 τ 6 mean h 80 t =120 τ 5 6 4 t =90 τ 3 60 ρ ( h , t ) t =60 τ 0 50 100 150 G(h) t/ τ 4 t =30 τ 40 2 20 0 0 0.00 0.02 0.04 0.06 0.01 0.02 0.03 0.04 0.05 h h

Theory of Decaying Passive Scalar Strange eigenmodes : φ ( x , t ) = ˆ φ ( x , t ) e − ( s / 2) t where ˆ φ ( x , t ) is statistically stationary, hence �| φ | n � ∼ e − n ( s / 2) t Decay of scalar variance , � φ 2 � ∼ e − st as κ → 0: For k f L ≈ 1, s = min h [ h + G ( h )] For k f L ≫ 1, s = κ eff L 2 where κ eff ≫ κ is the effective (eddy) diffusivity of the flow

Predicting λ Recall for infinitely fast reactions, � a � = �| φ |� 2 and from the theory of passive scalar decay, �| φ | n � ∼ e − n ( s / 2) t Hence, 1 − � p � = � a � ∼ e − λ t gives λ = s / 2. For k f L ≈ 1, λ ≈ 1 2 min h [ h + G ( h )] For k f L ≫ 1, λ ≈ κ eff 2 L 2

Theory vs. Simulations: k f L = 1 τ = 10 U = 0.22 0.04 0.02 simulation theory 0.03 λ 0.02 λ 0.01 0.01 simulation theory 0 0.00 0.1 0.2 0.3 0.4 0.5 10 20 30 40 50 τ U assumptions: (1) infinitely fast reaction assumptions: (2) κ → 0 (more restrictive) 2 π/ k f an optimal velocity correlation time, τ ≈ U

Theory vs. Simulations: k f L = 5 U = 0.25 a ( x , t = 30 τ ) 0 10 〈 a 〉 = 1 _ 〈 p 〉 λ =0.0033 λ =0.0114 -1 10 k f L =5 k f L =1 0 20 40 60 80 t / τ For our flow model, κ eff = U 2 τ 8 . So the theoretical prediction is λ = κ eff 2 L 2 = 0 . 0031 .

Summary investigate the progress of fast bimolecular reactions in chaotic flows majority of product is formed during the exponential phase make prediction on the reactant decay (product creation) rate using decaying passive scalar theory τ = 10 U = 0 . 22 , k f L = 1 0 10 0.04 simulation exp( − λ t ) theory 0.03 -1 10 〈 a 〉 = 1 _ 〈 p 〉 1/ Da = 0.50 1/ Da = 0.22 λ 0.02 U = 0.17 1/ Da = 0.13 -2 10 ∂ t a = − γ ab 0.01 ∂ t b = − γ ab -3 0 10 0 50 100 150 0.1 0.2 0.3 0.4 0.5 t/ τ U Phys. Rev. E 80 , 026305 (2009) ( http://www-pord.ucsd.edu/ ∼ yktsang)