! Diffusion to a Cylinder in ! Diffusion to a Cylinder in Cross - - PowerPoint PPT Presentation

1

• G. Ahmadi

ME 437/537

• G. Ahmadi

ME 437/537

! ! Diffusion to a Cylinder in Diffusion to a Cylinder in Cross Flow Cross Flow ! ! Deposition Velocity Deposition Velocity ! ! Interception Interception ! ! Filtration Filtration

• G. Ahmadi

ME 437/537

Stream Stream Function Function

a

] r a ) 1 a r ln 2 ( a r [ sin AUa + − θ = ψ

) ln 2 ( 2 1

e

R A − =

1 Re <<

• G. Ahmadi

ME 437/537

Diffusion Diffusion Equation Equation

a ) r c r 1 r c ( D r c v c r v

2 2 r

∂ ∂ + ∂ ∂ = ∂ ∂ + θ ∂ ∂

θ

∞

= ∞ = = + = c c , r c , 2 d a r

Boundary Boundary Conditions Conditions

2

• G. Ahmadi

ME 437/537

Diffusion Diffusion Equation Equation

a

x y

2 2

y c D y c v x c u ∂ ∂ = ∂ ∂ + ∂ ∂

∞

= ∞ = = = c c , y c , y

Boundary Boundary Conditions Conditions

• G. Ahmadi

ME 437/537

Stream Function Stream Function

x v , y u ∂ ψ ∂ − = ∂ ψ ∂ =

Using x and Using x and ψ ψ

] c u [ D x c ψ ∂ ∂ ψ ∂ ∂ = ∂ ∂

1 2 1

x sin AaUy 2 ≈ ψ

. a x x , a y y

1 1

= =

• G. Ahmadi

ME 437/537

Let Let Diffusion Equation Diffusion Equation AaU 2 , dx x sin

1 1 1 2 / 1

ψ = ψ = χ ∫

) c ( aAU D c

1 2 / 1 1 1

ψ ∂ ∂ ψ ψ ∂ ∂ = χ ∂ ∂

∞

= ∞ = ψ = = ψ c c , c ,

1 1

• G. Ahmadi

ME 437/537

Similarity Equation Similarity Equation

3 / 2 1

χ ψ = ξ

) d dc ( d d d dc 3 AP

2 / 1 e

ξ ξ ξ = ξ ξ −

dz } z AP 9 2 exp{ 45 . 1 ) AP ( c c

3 e 3 / 1 e

∫

ξ ∞

− =

c e e

S R D Ua 2 P ⋅ = =

3

• G. Ahmadi

ME 437/537

Sherwood Number Sherwood Number

3 / 1 e)

AP ( 17 . 1 D ) a 2 ( h sh = =

3 / 2 e 3 / 1 R

P A 68 . 3 Uc ) a 2 ( c ) a 2 ( h

− ∞ ∞ =

π = η

Collection Efficiency Collection Efficiency 3 / 2 R

d ~

−

η

• G. Ahmadi

ME 437/537

1 10 100 1E-1 1E0 1E1 1E2 1E3 1E4

Pe

Sh/A1/3 η/A1/3

Variations of Sherwood number and collection efficiency with Variations of Sherwood number and collection efficiency with Peclet Peclet number . number .

• G. Ahmadi

ME 437/537

∞ →

e

P

No Diffusion No Diffusion

U

2 2 / 2 / d y R

AR 2 Ua dx | v = − = η

∫

π =

a 2 d R =

a

• G. Ahmadi

ME 437/537

C1 C2 C C-dC dz

3 2 1 4 3 4 2 1

fibers

• f

No. 2 fiber

• ne

by Collection R

) a dz ( ] c ) a 2 ( [ dc π υ η − =

2 1 R

c c ln UL 2 a π = η

2a

4

• G. Ahmadi

ME 437/537

3 3 / 1 e 3 / 1 e e r

) RP ( 7 . RP 3 . 1 ) RP ( + = η ∞ →

e

P

3 / 2 e R

P− ∝ η Pe →

2 R

R ∝ η

Empirical Equation Empirical Equation

• G. Ahmadi

ME 437/537 1E-04 1E-02 1E+00 1E+02 1E+04 1E+06 0.1 1 10 100

RPe^(1/3)

Eq (25) Eq (21) Eq (20)

Variation of filter collection efficiency. Variation of filter collection efficiency.

e rRP

η

• G. Ahmadi

ME 437/537

! ! Deposition by Diffusion to a Deposition by Diffusion to a Cylinder Cylinder ! ! Deposition by Interception to a Deposition by Interception to a Cylinder Cylinder ! ! Fiber Filter Efficiency Fiber Filter Efficiency

• G. Ahmadi

ME 437/537