c + = Diffusion Diffusion 2 6.82 10 -6 v c D c 10 - - PowerPoint PPT Presentation

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c + = Diffusion Diffusion 2 6.82 10 -6 v c D c 10 - - PowerPoint PPT Presentation

Feb 25, 2024 417 likes •521 views

! Diffusivity ! Diffusivity ! Diffusion Equation ! Diffusion Equation ! Diffusion to a Wall ! Diffusion to a Wall ! Deposition Velocity, Diffusion ! Deposition Velocity, Diffusion Boundary Layer, Diffusion Force Boundary Layer, Diffusion

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SLIDE 1

1

  • G. Ahmadi

ME 437/537

  • G. Ahmadi

ME 437/537

! ! Diffusivity Diffusivity ! ! Diffusion Equation Diffusion Equation ! ! Diffusion to a Wall Diffusion to a Wall ! ! Deposition Velocity, Diffusion Deposition Velocity, Diffusion Boundary Layer, Diffusion Force Boundary Layer, Diffusion Force ! ! Diffusion to a Flat Plate Diffusion to a Flat Plate ! ! Diffusion in Tube Diffusion in Tube ! ! Taylor Diffusion Taylor Diffusion

  • G. Ahmadi

ME 437/537

Fick’s Fick’s Law Law dx dc D J − =

c D c t c

2

∇ = ∇ ⋅ + ∂ ∂ v

d 3 kTC D

c

πµ =

Diffusion Diffusion Equation Equation Diffusivity Diffusivity

  • G. Ahmadi

ME 437/537

) m ( d µ

) s / cm ( D

2

10-2 5.24 × 10-4 10-1 6.82 ×10-6 1 2.74 × 10-7 10 2.38 ×10-8 Table of Particle Mass Diffusivity Table of Particle Mass Diffusivity

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SLIDE 2

2

  • G. Ahmadi

ME 437/537

Dt 2 s 2 =

t d kT 2

3 2

πµ = θ

kT 2 3 u m 2 1

2 =

m / kT 3 u 2 =

} kT ) x x ( mg exp{ C C − − =

Mean Square Mean Square Displacement Displacement Brownian Motion Brownian Motion

  • f Rotation
  • f Rotation

Particle Fluctuation Particle Fluctuation Energy Energy Concentration in Concentration in gravitational field gravitational field

  • G. Ahmadi

ME 437/537

Effect of Effect of Particle Mass Particle Mass Particle Mean Particle Mean Free Path Free Path

] t / ) e 1 ( 1 [ Dt 2 s

/ t 2 τ −

− τ − =

m / kT 8 π τ ≈ λ α

  • G. Ahmadi

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2 2

y c D t c ∂ ∂ = ∂ ∂ y

C ) , y ( C =

) t , ( C =

Dt 4 y = η

Dt 4 1 c y c y c η ∂ ∂ = ∂ η ∂ η ∂ ∂ = ∂ ∂

Dt 4 1 c y c

2 2 2 2

η ∂ ∂ = ∂ ∂

Similarity Variable Similarity Variable

  • G. Ahmadi

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t 2 c Dt 4 t 2 y c t c t c η η ∂ ∂ − = − η ∂ ∂ = ∂ η ∂ η ∂ ∂ = ∂ ∂

d dc 2 d c d

2 2

= η η + η A ln ) d dc ln(

2 +

η − = η

2

Ae d dc

η −

= η B d e A c

1

2 1

+ η = ∫

η η −

Similarity Equation Similarity Equation

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SLIDE 3

3

  • G. Ahmadi

ME 437/537

η

C ) ( C = ∞ → η ) ( C = = η

) Dt 4 / y ( erf C ) t , y ( C =

∫

ξ ξ −

ξ π = ξ d e 2 ) ( erf

2

) ( erf = 1 ) ( erf = ∞

  • G. Ahmadi

ME 437/537

0.0 0.2 0.4 0.6 0.8 1.0

C/Co

0.5 1 1.5 2 2.5 3

x

tD=0.0025 tD=0.062 tD=0.25 tD=1 tD=4

Variation of concentration profile with time. Variation of concentration profile with time.

y

  • G. Ahmadi

ME 437/537

c D

D t D C J u δ = π = =

Dt

c

π = δ

c D d

C / du 3 F πµ =

Diffusion Diffusion Velocity Velocity

Diffusion Diffusion Boundary Boundary Layer Layer

Diffusion Diffusion Force Force

  • G. Ahmadi

ME 437/537

dt t D C Jdt dN π = =

π = Dt 4 C N

Number Deposited Number Deposited in Time in Time dt dt Total Deposited in Total Deposited in Time Interval (0,t) Time Interval (0,t)

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SLIDE 4

4

  • G. Ahmadi

ME 437/537

u DL 4 C N

in

π =

π = Dt 4 C N

u / L t =

L R L R 2 N C C

2 in

  • ut

π π − = −

  • ut

C

in

C U R U

  • G. Ahmadi

ME 437/537 2 in

  • ut

uR DL 4 1 C C π − =

3 / 4 3 / 2 in

  • ut

177 . 2 . 1 56 . 2 1 C C φ + φ + φ − =

2

uR DL = φ

Concentration Ratio Concentration Ratio Detailed Analysis Detailed Analysis

  • G. Ahmadi

ME 437/537

Uo Uo Co Co y x δc δ

c v u = = =

y = ∞ → y c c , U u = =

  • G. Ahmadi

ME 437/537 2 2

dy u y u v x u u ∂ ν = ∂ ∂ + ∂ ∂

y v x u = ∂ ∂ + ∂ ∂

2 2

y c D y c v x c u ∂ ∂ = ∂ ∂ + ∂ ∂

Momentum Mass Concentration

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SLIDE 5

5

  • G. Ahmadi

ME 437/537

x U y ν = η

) ( ' f U u η =

) ( c c η =

' ' ' f 2 ' ' ff = +

' fc S 2 1 " c

c

= +

D Sc ν =

Momentum/Mass Concentration

Blasius Equation

) ( f x Uo η ν = ψ

  • G. Ahmadi

ME 437/537

) ( ' f ) ( f = = 1 ) ( ' f = ∞

) ( c = c ) ( c = ∞

U x 5 ν = δ

332 . ) ( ' ' f = γ =

... 2 ~ f

2 +

η γ

Boundary Boundary Conditions Conditions Blasius Blasius Solution Solution Near the Near the Plate Plate

  • G. Ahmadi

ME 437/537

∫ ∫

∞ η

γ − γ − =

3 c 1 3 c 1

dz ) z s exp( dz ) z s exp( C C

12

1

γ = γ

∫

η

γ − γ =

3 c 1 3 c 1

dz )] z s [exp( 89 . s c c

  • G. Ahmadi

ME 437/537

0.0 0.2 0.4 0.6 0.8 1.0

C/Co

0.5 1 1.5 2 2.5 3

z

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SLIDE 6

6

  • G. Ahmadi

ME 437/537

vx U s Dc 34 . vx U 89 . ) s ( Dc y c D J

3 c 3 c 1 y

= γ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ =

= 3 c 3 c c

s 6 . U vx s 3 J Dc δ ≈ ≈ = δ

eL 3 c L

R s Dc 68 . Jdx I = = ∫

ν = L U R

  • eL

Diffusion Diffusion Boundary Layer Boundary Layer Total Total Diffusion Diffusion

  • G. Ahmadi

ME 437/537

u c R y r z δc

) R r 1 ( u u

2 2

− = r R y − =

... R y 2 u ~ u +

Laminar Laminar Flow Flow

  • G. Ahmadi

ME 437/537

2 2

y c D x c y R u 2 ∂ ∂ = ∂ ∂

c =

y =

∞ → y

c c =

3 3

x y DR u = η

' c 3 2 ' ' c

2 =

η +

Diffusion Diffusion Equation Equation

Boundary Condition Boundary Condition

Similarity Similarity Variable Variable

  • G. Ahmadi

ME 437/537

∫ ∫

∞ η

η η − η η − =

1 3 1 1 3 1

d } 9 2 exp{ d } 9 2 exp{ c c ∫

∞ =

η η − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ =

1 3 1 3 3 y

d ) 9 2 exp( DR u x Dc y c D J

3

DRx u D c 67 . J =

Concentration Concentration Profile Profile

Wall Flux Wall Flux

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SLIDE 7

7

  • G. Ahmadi

ME 437/537 3 eR c 2 3 / 1 eR 3 / 1 c 3 2 c

R s x R 67 . 1 R s 67 . x R J Dc = = = δ ν = R u R

eR 3 2 L

DR L u DR c 01 . 2 Jdx R 2 I π = π =

∫

Total Flux Total Flux

Diffusion Diffusion Boundary Boundary layer layer

  • G. Ahmadi

ME 437/537

) x c r c r 1 r c ( D x c ) r ( u t c

2 2 2 2

∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂

) R r 1 ( U 2 ) r ( u

2 2

− =

) r c r 1 r c ( D x c ) R r 2 1 ( U t c

2 2 2 2

∂ ∂ + ∂ ∂ = ∂ ∂ − + ∂ ∂

Neglecting Axial Neglecting Axial Diffusion Diffusion

  • G. Ahmadi

ME 437/537

r c

R

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ t c ≈ ∂ ∂

x c const x c ∂ ∂ = = ∂ ∂

) r c r 1 r c ( D x c ) R r 2 1 ( U

2 2 2 2

∂ ∂ + ∂ ∂ = ∂ ∂ −

) R r 2 1 R r ( x c D 4 UR c c

4 4 2 2 2

  • −

∂ ∂ + =

For For

Moving Moving Frame Frame Diffusion Diffusion In Moving In Moving Frame Frame

Solution Solution

  • G. Ahmadi

ME 437/537

[ ]

r

  • c

c

=

=

∫ ∫ ∫

= π π = =

R 2 R 2 A

crdr R 2 rcdr 2 R 1 cdA A 1 c

) R r 2 1 R r 3 1 ( x c D 4 U R c c

4 4 2 2 2

− + − ∂ ∂ + =

x c D 48 U R ) R ( rdr ) U u ( c 2 Q

R 2 2 2 c

∂ ∂ π − = − π =

∫

Concentration Concentration Total Total Flux Flux

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SLIDE 8

8

  • G. Ahmadi

ME 437/537

x c ) D 48 U R ( R Q J

2 2 2

∂ ∂ − = π =

D 48 U R D

2 2 eff = 2 2 eff

x c D J x t c ∂ ∂ = ∂ ∂ − = ∂ ∂

. const x c ≠ ∂ ∂

14 P R L

e >>

>>

c e e

S R D UR 2 P = =

Effective (Taylor) Effective (Taylor) Diffusivity Diffusivity Flux Per Flux Per Unit Area Unit Area

Range of Validity Range of Validity

  • G. Ahmadi

ME 437/537

) x ( R N c

2 δ

π = } t D 4 ) Ut x ( exp{ t D 1 R N 2 1 c

eff 2 eff 2

− − π π =

t =

  • G. Ahmadi

ME 437/537

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

C/Co

1 2 3 4

x Variations of concentration along the tube at different times. Variations of concentration along the tube at different times.

  • G. Ahmadi

ME 437/537

! ! Mass diffusion decreases with size Mass diffusion decreases with size ! ! Diffusion Boundary Layer is Diffusion Boundary Layer is generally smaller that momentum generally smaller that momentum boundary layer boundary layer ! ! Convective diffusion in a tube Convective diffusion in a tube ! ! Taylor diffusion in a tube Taylor diffusion in a tube