du Langevin Langevin + = u n ( t ) Equation Equation dt - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 437/537

• G. Ahmadi

ME 437/537

! ! Introduction to Aerosols Introduction to Aerosols ! ! Drag Forces Drag Forces ! ! Cunningham Corrections Cunningham Corrections ! ! Lift Forces Lift Forces ! ! Brownian Motion Brownian Motion ! ! Particle Deposition Mechanisms ! ! Gravitational Sedimentation ! ! Aerosol Coagulation

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A Particle under Random Molecular Impact A Particle under Random Molecular Impact

• G. Ahmadi

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) t ( n u dt du = β +

Langevin Langevin Equation Equation

τ = πµ = β / 1 m C / d 3

c

m kT 2 Snn π β =

N(t) = White Noise N(t) = White Noise Spectral Intensity Spectral Intensity

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• G. Ahmadi

ME 437/537

) t ( u t ( u ) ( R ) τ + = τ

∫

+∞ ∞ − ωτ

ω ω = τ d ) ( S e 2 1 ) ( R

uu i uu

∫

+∞ ∞ − ωτ −

τ τ π = ω d ) ( R e 1 ) ( S

uu i uu

Autocorrelation Autocorrelation Function Function Fourier Fourier Transform Transform

• G. Ahmadi

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) ( S | ) ( H | ) ( S

nn 2 uu

ω ω = ω

β + ω = ω i 1 ) ( H

2 2 uu

m / kT 2 ) ( S β + ω π β = ω

| | uu

e m kT ) ( R

τ β −

= τ

System System Function Function Response Power Response Power Spectrum Spectrum Autocorrelation Autocorrelation

• G. Ahmadi

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Mass Mass Diffusivity Diffusivity Position Position Variance Variance

) t ( x dt d 2 1 D

2

=

∫

=

t 1 1 dt

) t ( u ) t ( x

∫∫

τ τ τ − τ =

t t 2 1 2 1 uu 2

d d ) ( R ) t ( x

• G. Ahmadi

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Diffusivity Diffusivity Variance Variance

∫

τ τ τ − =

t uu 2

d ) ( R ) t ( 2 ) t ( x

∫

∞

τ τ =

uu

d ) ( R D

d 3 kTC m kT D

c

πµ = β =

3

• G. Ahmadi

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Probability Probability Density Density

Fokker Fokker-

• Planck Equation

Planck Equation

2 2

u f m kT ) uf ( u t f ∂ ∂ β = β ∂ ∂ − ∂ ∂

kT 2 mu2

e m / kT 2 1 f

−

π =

• G. Ahmadi

ME 437/537

Langevin Langevin Equation Equation

) t ( n m ) x ( F x x = − β + & & &

x ) x ( V ) x ( F ∂ ∂ − =

2 2

x f m kT ] f )) x ( F m 1 x [( x x ) f x ( t f & & & & ∂ ∂ β + − β ∂ ∂ + ∂ ∂ − = ∂ ∂

Fokker Fokker-

• Planck Equation

Planck Equation

• G. Ahmadi

ME 437/537

Probability Probability Density Density

Gravitational Gravitational Field Field

∫

− − =

x 1 2

]} m dx ) x ( F 2 x [ kT m exp{ C f &

)]} x ( V 2 x m [ kT 1 exp{ C f

2

+ − = &

kT ) x x ( mg kT 2 x m

2

e e C f

− − −

=

&

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! Choose a time step (∆t<<τ) ! Generate a sequence of uniform random numbers (0 <U<1) ! Transform to Gaussian random numbers.

) t ( n =

) t t ( S 2 ) t ( n ) t ( n

2 1 nn 2 1

− δ π =

White Noise

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• G. Ahmadi

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! ! ! Amplitude of the Brownian force is given by ! The generated sample of Brownian force need to be shifted by

2 1 1

U 2 cos U ln 2 G π − =

2 1 2

U 2 sin U ln 2 G π − =

t S G ) t ( n

nn i i

∆ π =

t U∆

• G. Ahmadi

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ni t ∆t U∆t

• G. Ahmadi

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ni t ∆t U∆t

• G. Ahmadi

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Brownian Force Brownian Force

Ounis, Ahmadi and McLaughlin (1991)

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Dt 2 ) t ( x 2 =

Simulations

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Mean + σ Mean - σ

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Mean + σ Mean - σ

• G. Ahmadi

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Mean + σ Mean - σ

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• G. Ahmadi

ME 437/537 Simulations

) Dt 4 z ( erfc N N

• t =
• G. Ahmadi

ME 437/537

) t ( ) ( 1 dt d n g u u u

p f p

+ + − τ =

Dt 2 ) t (

2 y

= σ

Equation of Motion Equation of Motion Equation of Motion

Variance Variance Variance

• G. Ahmadi

ME 437/537

• G. Ahmadi

ME 437/537