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

! Diffusion to a Cylinder in ! Diffusion to a Cylinder in Cross Flow Cross Flow ! Deposition Velocity ! Deposition Velocity ! Interception ! Interception ! Filtration ! Filtration ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi a a Diffusion Diffusion Equation ∂ ∂ ∂ ∂ Equation 2 v c c c 1 c + = + θ v D ( ) ∂ θ ∂ ∂ ∂ r 2 r r r r r r r a Stream Stream ψ = θ − + AUa sin [ ( 2 ln 1 ) ] a a r Function Function Boundary Boundary d = + = r a , c 0 Conditions Conditions 2 1 = Re << A 1 = ∞ = − r , c c 2 ( 2 ln ) R ∞ e ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 1

∂ ψ ∂ ψ x = = − y Stream Function Stream Function Diffusion Diffusion u , v a ∂ ∂ y x Equation Equation ∂ ∂ ∂ c c ∂ ∂ ∂ 2 = c c c ψ D [ u ] Using x and ψ + = Using x and ∂ ∂ ψ ∂ ψ u v D x ∂ ∂ ∂ 2 x y y ψ ≈ 2 y x = = 2 AaUy sin x y 0 , c 0 = = Boundary y , x . Boundary 1 1 1 1 a a = ∞ = Conditions Conditions y , c c ∞ ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ψ ψ χ ∫ Let Let = ψ = Similarity Equation ξ = 1 / 2 Similarity Equation 1 sin x dx , 1 1 1 χ 2 AaU 2 / 3 Diffusion Equation Diffusion Equation AP dc d dc − ξ = ξ 1 / 2 e ( ) ∂ ∂ ∂ c D c ξ ξ ξ 3 d d d = ψ 1 / 2 ( ) ∂ χ ∂ ψ 1 ∂ ψ aAU 1 1 ξ 1 / 3 ψ = = c ( AP ) 2 2 Ua 0 , c 0 ∫ = ∞ − = = ⋅ e 3 P R S c exp{ AP z } dz 1 e e c e D 1 . 45 9 ψ = ∞ = , c c 0 ∞ 1 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 2

h ( 2 a ) 100 Sherwood Number Sherwood Number = = 1 / 3 sh 1 . 17 ( AP e ) Sh/A 1/3 D 10 1 Collection Efficiency Collection Efficiency 0 π η /A 1/3 h ( 2 a ) c ∞ = 0 η = − 1 / 3 2 / 3 3 . 68 A P R e ( 2 a ) Uc 0 ∞ 1E-1 1E0 1E1 1E2 1E3 1E4 Pe − η 2 / 3 ~ d Variations of Sherwood number and collection efficiency with Peclet Variations of Sherwood number and collection efficiency with Peclet number . number . R ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi dz → ∞ P No Diffusion No Diffusion 2a e C 1 C C-dC C 2 U a π / 2 υ dz ∫ π d v | dx = − η a c R = dc [ ( 2 a ) c ] ( ) = η = y d / 2 1 ln 1 4 2 4 3 R π 2 η = − = 2 a 0 2 AR 2 a R 1 2 3 2 UL c R Ua Collection by one fiber 2 No. of fibers ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 3

Empirical Equation Empirical Equation 1E+06 Eq (25) 1E+04 Eq (21) η = + 1 / 3 1 / 3 3 ( RP ) 1 . 3 RP 0 . 7 ( RP ) η 1E+02 r RP r e e e e Eq (20) 1E+00 1E-02 η ∝ 2 → ∞ R P e R 1E-04 0.1 1 10 100 RPe^(1/3) η ∝ P − P e → 2 / 3 0 Variation of filter collection efficiency. Variation of filter collection efficiency. R e ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ! 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 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 4