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 pharmaceutical contaminants using ozone and other reagents

(Climate Change Science Program)

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) + H2O(ℓ)

Irreversible Bimolecular Reactions

A + B → P Example: acid-base reaction (neutralization) HCl(aq) + NaOH(aq) → NaCl(aq) + H2O(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∇2a − γab ∂b ∂t + u · ∇b = κ∇2b − γab ∂p ∂t + u · ∇p = κ∇2p + γab

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∇2a − γab ∂b ∂t + u · ∇b = κ∇2b − γab ∂p ∂t + u · ∇p = κ∇2p + γab

a(x, 0) = b(x, 0) = 1

Fast reactions: reaction time ≪ advection time ≪ diffusion time Goal: time dependence of mean product concentration p = 1 − a

Flow Model and Simulations

u(x, t) =        √ 2 U cos[kf y + θ1(n)] ˆ i , nτ < t (n+ 1

2)τ

√ 2 U cos[kfx + θ2(n)] ˆ j , (n+ 1

2)τ < t (n+1)τ

Domain size: 2πL × 2πL (scale separation ∼ kfL)

Progress of Reaction

50 100 150 t/τ 10

• 3

10

• 2

10

• 1

10 〈a〉 = 1 _ 〈p〉

1/Da = 0.13 U = 0.17 1/Da = 0.22 1 / D a = . 5

U = 0.22 , kfL = 1

Progress of Reaction

50 100 150 t/τ 10

• 3

10

• 2

10

• 1

10 〈a〉 = 1 _ 〈p〉

1/Da = 0.13 U = 0.17 1/Da = 0.22 1 / D a = . 5

U = 0.22 , kfL = 1

exp(−λt)

Progress of Reaction

50 100 150 t/τ 10

• 3

10

• 2

10

• 1

10 〈a〉 = 1 _ 〈p〉

1/Da = 0.13 U = 0.17 1/Da = 0.22 1 / D a = . 5

U = 0.22 , kfL = 1

exp(−λt)

∂ta = −γab ∂tb = −γab

Relation to Decaying Passive Scalar

∂a ∂t + u · ∇a = κ∇2a − γab ∂b ∂t + u · ∇b = κ∇2b − γab φ ≡ a − b ⇒ ∂φ ∂t + u · ∇φ = κ∇2φ 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

10 20 30 40 50 60 70 t/τ 10

• 2

10

• 1

10 〈a〉 = 1 _ 〈p〉

1/Da = 0.13 U = 0.17 1/Da = 0.22 1/Da = 0.50

U = 0.22 , kfL = 1 a = 1 − p |φ| 2 e−λt

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 effective diffusivity

• 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 effects,...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 h(x, t) = 1 t log |δx(t)| |δx(0)| ¯ h = lim

t→∞ h(x, t)

x δx(0) δx(t)

probability density function of h, ρ(h, t) at large time: ρ(h, t) ∼ exp[−tG(h)] as t → ∞

0.00 0.02 0.04 0.06

h

20 40 60 80 100

ρ(h, t)

U = 0.22 , kf = 1 , τ = 10

0.01 0.02 0.03 0.04 0.05

h

2 4 6 8

G(h)

U = 0.22 , kf = 1 , τ = 10

50 100 150

t/τ

3 4 5 6

(x10

• 3)

(x10

• 2)

t=30τ t=60τ t=90τ t=120τ t=150τ

mean 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, φ2 ∼ e−st as κ → 0: For k fL ≈ 1, s = min

h [h + G(h)]

For k fL ≫ 1, s = κeff L2 where κeff ≫ 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−λt gives λ = s/2. For k fL ≈ 1, λ ≈ 1 2 min

h [h + G(h)]

For k fL ≫ 1, λ ≈ κeff 2L2

Theory vs. Simulations: kfL = 1

0.1 0.2 0.3 0.4 0.5 U 0.01 0.02 0.03 0.04 λ simulation theory τ = 10 10 20 30 40 50 τ 0.00 0.01 0.02 λ simulation theory U = 0.22

assumptions: (1) infinitely fast reaction assumptions: (2) κ → 0 (more restrictive) an optimal velocity correlation time, τ ≈ 2π/kf U

Theory vs. Simulations: kfL = 5

a(x, t = 30τ)

20 40 60 80 t/τ 10

• 1

10 〈a〉 = 1 _ 〈p〉 U = 0.25 kfL=1 kfL=5

λ=0.0033 λ=0.0114

For our flow model, κeff = U2τ 8 . So the theoretical prediction is λ = κeff 2L2 = 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

50 100 150 t/τ 10

• 3

10

• 2

10

• 1

10 〈a〉 = 1 _ 〈p〉

1/Da = 0.13 U = 0.17 1/Da = 0.22 1/Da = 0.50

0.1 0.2 0.3 0.4 0.5 U 0.01 0.02 0.03 0.04 λ simulation theory τ = 10

• Phys. Rev. E 80, 026305 (2009)

(http://www-pord.ucsd.edu/∼yktsang) U = 0.22 , kf L = 1 exp(−λt) ∂ta = −γab ∂tb = −γab