Fractal Models of the Permeability and Diffusivity of Fibrous Materials

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1 Fractal Models o the Permeability and Diusivity o Fibrous Materials Jintu Fan Vincent V. C. Woo Proessor in Fiber Science and Apparel Design Morgan Sesquicentennial Fellow and Department Chair Department o Fiber Science & Apparel Design College o Human Ecology

2 Department o Fiber Science & Apparel Design Applications o Fibrous Materials: Environment Filtration Geo-textiles Aircrats and spacecrats Buildings Health & living Tissue engineering scaolds Wound dressing Apparel Energy Insulation Fuel cell electrolyte membranes Fuel cell gas diusion layers

3 Permeation in Fibrous Media

4 Schematic o ibrous media composed o tortuous channels l t A cs q( ) l 0

5 Total number o pores/channels D N ( L R ) ( R R ). t scale max scale Number o channels with a radius o R i 1 N N ( R R ) ( R R ) B D i1 i max i1 max i ( D) R R B R R ; D D1 max i i1 i ( D) ln Ri1 ln Ri e R R ( D) ln R ln R R 4 R q( R) p. 8 i 1 ln(1 i1 i ) Ri1 Ri, i1 i i Flow rate through the channel with a radius o R (Hagen Poiseulle equation ) D R

6 Total low rate through all channels: Q ( R ) q( R )( N N ) t i i i1 i i0 C i0 i0 n n i0 q( R )( N N ) i i1 i n pr R ( D) R B R R 8 D 4 D1 max i i i1 i n pr DR B R R 8 D D3 max i i1 i D B pr R R 8 4 DC D 4D 4D max n 0 D D B 4 R max p Rmax D4 8 4 D C R scale D B 4 p Rmax ; 8 4 DC (3 D) ln Ri1 ln Ri e 1 R R (3 D) ln R ln R R i ln(1 i1 i Ri1 Ri ), i1 i i R

7 Mean velocity o luid permeation U Qt A t 1 2 D E pr 8 4 D1 C 1 2 D pr 8 4 D 1 2 max 2 max, Empirical Relations: 0.5 R p 1 r. 2 2ln (Sampson, 2003) ( ). (Tomadakis and Robertson, 2005) Darcy Law K U p, Fractal Relations: Rmax D R p R ( R) dr Rmin, Rmin D 1 D ln 2. ln R R max min

8 Non-Dimensional Permeability K/r Model Clarenburg and Piekaar (1968) Gostick et al. (2006) Wheat (1963) Kostornov and Shevchuk (1973) Ingmanson et al. (1959) Johnson (1998) K 2 D D 1 ( 0.11) 2 1. r D D 1 2ln Shou, D. Fan, J. & Ding, F. Physics Letters A 374 (2010)

9 More General Form o Permeability Model ater introducing the ractal dimension o tortuosity D T K d 4 d 4(2 d ) ( ) [ ] [ ] D D d d d 1/2 (1 D T )/ (3 T ) (1 ) (1 ) Xiao, B. et al, Electrochimica Acta 134 (2014)

10 Diusion through Fibrous Media Gas diusion through one channel: ( ) VC j R, p Dequ Rp L R Diusivity through a speciic channel o radius R: D equ 1 u Db. 3 2Rp 1 Kn Total diusion lux: Rmax Qd q( Rp) dn. Rmin p Empirical Relations: M KnKn 2R 3 1Kn 2R ( ). (Tomadakis and Robertson, 2005) Eective diusivity D e J A t R R max min 2 R Dequ A M p ( R) dn, Fractal relations R D max R R ( R) dr R Rmin D 1 min.

11 Diusion through Fibrous Media D e 2D Rmax 2R dr Rmin 2R 3 Db D 0.11 r 2 D 1 D ( ) D 1 Shou, D et al, Microluidics and Nanoluidics, (2014) 16:

12 Experimental Inverted Cup Method

13 Comparison o Fractal Model and Experimental Results Shou, D et al, Microluidics and Nanoluidics, (2014) 16:

14 Relative Permeabilities in Multiphase Flows in Fuel Cell GDL Relative water permeability: K ( S) d 4(2 d ) S [ ] 3 D d d (1 S), w, w (1 D )/2 w T, w, w krw( S) S K d 4(2 d ) (1 D )/2 Relative gas permeability: [ ] 3 D d d (1 ) T T T (1 D )/2 T d, g 4(2 d, g )[(1 S) ] K ( ) 3,, [1 (1 ) ] g S DT d g d g S krg ( S) (1 S) K d 4(2 d ) (1 DT )/2 [ ] 3 D d d (1 ) T

15 Relative Permeabilities k rw Present analytical model,eq.(5-34) Hao and Cheng, 2010 Dana and Skoczylas, 2002 Koido et al., 2008 Acosta et al., 2006 Gostick et al., 2007 Li, 2011 Levec et al., 1986 Specchia et al., 1977 Kumbur et al., S

16 Relative Permeabilities Present analytical model,eq.(5-41) Hao and Cheng, 2010 Dana and Skoczylas, 2002 Li, 2011 Nguyen et al., 2006 Owejan et al., 2006 k rg S

17 Relative Permeabilities k rw, k rg k rg, krg, k rw, krw, Xiao, B. et al, Electrochimica Acta 134 (2014) S

18 Optimization o Thermal Insulation (Conductivity over Diusivity) 1.0x10 4 Y 5 =(k tot,e /k g )/(D e /D b ) 8.0x x x x10 3 m = Xiao, B, et al, Fractals, Vol. 25, No. 3 (2017)

19 Optimization or Protective Outer Fabric 2.5x10 3 (Permeability over Diusivity) 2.0x10 3 m =10-3 Y 5 =(K e /K g )/(D e /D b ) 1.5x x x D T Xiao, B, et al, Fractals, Vol. 25, No. 3 (2017)

20 Directional Water Flow Human Skin Dierential Surace Properties 20

21 Directional Water Flow 21

22 Dierential spontaneous capillary low Capillary pressure: Permeability: Total resistance: Volume low rate: K 128 R eq F cos 1 P c min D 1 D 1 1D cos 2 D T F T L0 D 2 D T min 1 max A 2 D D P T max z Total liquid mass absorbed by the material k1 l j K APc Q R eq k1 j1 j k z l j j1 K D N z l j L j 1 k j 1 2 z j 1 z l j l 1 z 0 z l j z z Z k1 k1 j j j k j1 j1 M z A l z l k N k k k 3 k 2 k 1 time t t t t 2 t t 1 t 0 Kausik Bal, Jintu Fan, M.K. Sarkar, Lin Ye, Dierential spontaneous capillary low through heterogeneous porous media, International Journal o Heat and Mass Transer 54 (2011)

23 Dierential spontaneous capillary low Mass o water absorbed (g Flo w into layers o Fab ric A irst then to layers o Fab ric B (Exp) Flo w into layers o Fab ric B irst then to layers o Fab ric A (Exp) Flo w into layers o Fab ric A irst then to layers o Fab ric B (Th) Flo w into layers o Fab ric B irst then to layers o Fab ric A (Th) Time (s) A B φ λ max 450μ 350μ λ min 1μ 1μ L 8.8mm 8mm

24 Acknowledgement Shou, D: Ph.D Student, now Postdoc at Cornell University Xiao B: Ph.D Student, now Proessor at Sanming University, China Bal, K: Post Doc, now Proessor at University o Calcutta, India Sarkar, M. Ph.D Student, now R&D Manager at Hong Kong Ye, L, Collaborator, Proessor at Sydney University Ding F, Collaborator, Proessor at Hong Kong Polytechnic University, just moved to Ulsan National Institute o Science and Technology, Korea.

25 Thank you or your attention!