Fractal Models of the Permeability and Diffusivity of Fibrous - PowerPoint PPT Presentation

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

Department of Fiber Science & Apparel Design Applications of Fibrous Materials:  Environment  Filtration  Geo-textiles  Aircrafts and spacecrafts  Buildings  Health & living  Tissue engineering scaffolds  Wound dressing  Apparel  Energy  Insulation  Fuel cell electrolyte membranes  Fuel cell gas diffusion layers

Permeation in Fibrous Media

Schematic of fibrous media composed of tortuous channels l t A cs q  ( ) l 0

Total number of pores/channels   D N L ( R ) ( R R ) . t scale max scale Number of channels with a radius of R  i 1    D D N N ( R R ) ( R R )   i 1 i max i 1 max i        D D 1 ( D R ) R B R R ;  max 1 i i i     R   ( D ) ln R ln R  i e 1 R R i 1 i     R R i 1 i  B ln(1 ) , i 1 i     ( D ) ln R ln R R  i 1 i i Flow rate through the channel with a radius of R  4 R   (Hagen – Poiseulle equation ) q R ( ) p .  8

Total flow rate through all channels: n     Q R ( ) q R ( )( N N )  t i i i 1 i  i 0 n     q R ( )( N N )  i i 1 i  i 0  n          D 4 D 1 pR R ( D R ) B R R   max i i i 1 i 8  i 0  n          D D 3 pR DR B R R   max i i 1 i 8  0 i    D B      D 4 D 4 D pR R R   max n 0 8 4 D C      D D B R     4   max  p R      max D 4   8 4 D C R   scale  D B   4 p R ;   max 8 4 D C       R (3 D ) ln R ln R  i R R e i 1 i 1     R R i 1 i  C ln(1 ) , i 1 i     (3 D ) ln R ln R R  i 1 i i

Empirical Relations: Mean velocity of fluid permeation        0.5 Q     R 1 r . t U  p f   2 2ln A t (Sampson, 2003)   1 2 D E   2 pR     max  8 4 D 1 C 1 0.11   0.785   ( ) . 1 2 D     2 pR , 0.11     max 8 4 D 1 (Tomadakis and Robertson, 2005) Fractal Relations: Darcy Law   D  R  max f R Rf R dR ( ) R ,  p min 1 R D min f  K ln     U p , 2 . D    f ln R R max min

Non-Dimensional Permeability 4 Model 10 Clarenburg and Piekaar (1968) Gostick et al. (2006) Wheat (1963) 2 10 Kostornov and Shevchuk (1973) Ingmanson et al. (1959) Johnson (1998) 0 10 K/r 2 -2 10    2     2     -4 0.785 K 2 D D 1 ( 0.11) 10       1 .     2     1.594 4 D D 1 2ln r f -6 10 0.3 0.4 0.5 0.6 0.7 0.8 0.9  Shou, D. Fan, J. & Ding, F. Physics Letters A 374 (2010) 1201 – 1204.

More General Form of Permeability Model after introducing the fractal dimension of tortuosity D T      4 4(2 ) d d d K   f f f 1/2 (1 D )/2 2 ( ) [ ] [ ] T        2 D 128(3 D d ) d d (1 ) (1 ) f T f f f Xiao, B. et al, Electrochimica Acta 134 (2014) 222 – 231

Diffusion through Fibrous Media Gas diffusion through one channel: Empirical Relations:         V C  Kn 2 R 3 j R ( ) D R ,      p equ p Kn L R p    M 1 Kn 2 R Diffusivity through a specific channel of radius R: 0.89  1     u 1 1 1 0.785    ( ) .   D D .     equ b   3 2 R 1 Kn   0.11 p Total diffusion flux: (Tomadakis and Robertson, 2005)   R max Q q R ( ) dN . d p R min Fractal relations Effective diffusivity D  R   max f R Rf R dR ( ) R  min.  D 1 R 2 R D ( ) R dN min   R f max equ  J R  min M D , eff A A p t 

Diffusion through Fibrous Media  2 D 2 R f  R max dR   2 R 3  R min D D . eff b 0.89        0.785 2 D   ( ) 1   f 100 D 1   0.11   f r           1 D 2 D   f f Shou, D et al, Microfluidics and Nanofluidics, (2014) 16:381-389.

Experimental Inverted Cup Method

Comparison of Fractal Model and Experimental Results Shou, D et al, Microfluidics and Nanofluidics, (2014) 16:381-389.

Relative Permeabilities in Multiphase Flows in Fuel Cell GDL Relative water permeability:   d 4(2 d ) S  f w , f w , (1 D )/2 [ ] T      3 D d d (1 S ) K ( ) S   T f w , f w , w k ( ) S S   rw d 4(2 d ) K  f f (1 D )/2 [ ] T      3 (1 ) D d d T f f Relative gas permeability:  (1 D )/2      T   4(2 )[(1 ) ] d d S f g , f g ,           3 D d d [1 (1 S ) ] K ( ) S      T f g , f g , g k ( ) S (1 S )   rg d 4(2 d ) K  f f (1 D )/2 [ ] T      3 D d d (1 ) T f f

Relative Permeabilities 1.0 Present analytical model,Eq.(5-34) Hao and Cheng, 2010 Dana and Skoczylas, 2002 0.8 Koido et al., 2008 Acosta et al., 2006 Gostick et al., 2007 0.6 Li, 2011 Levec et al., 1986 k rw Specchia et al., 1977 0.4 Kumbur et al., 2007 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 S

Relative Permeabilities 1.0 Present analytical model,Eq.(5-41) Hao and Cheng, 2010 Dana and Skoczylas, 2002 0.8 Li, 2011 Nguyen et al., 2006 Owejan et al., 2006 0.6 k rg 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 S

Relative Permeabilities 1.0 k rg ,   k rg ,   0.8 k rw ,   k rw ,   k rw , k rg 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 S Xiao, B. et al, Electrochimica Acta 134 (2014) 222 – 231

Optimization of Thermal Insulation (Conductivity over Diffusivity) 4 1.0x10 3 Y 5 =( k tot,eff / k g )/( D e / D b ) 8.0x10 m =10 -3 3 6.0x10 3 4.0x10 3 2.0x10 0.0 0.0 0.2 0.4 0.6 0.8 1.0  Xiao, B, et al, Fractals, Vol. 25, No. 3 (2017) 1750030.

Optimization for Protective Outer Fabric (Permeability over Diffusivity) 3 2.5x10   m =10 -3 3 2.0x10   Y 5 =( K eff / K g )/( D e / D b ) 3 1.5x10 3 1.0x10 2 5.0x10 0.0 1.0 1.2 1.4 1.6 1.8 2.0 D T Xiao, B, et al, Fractals, Vol. 25, No. 3 (2017) 1750030. 19

Directional Water Flow Human Skin Differential Surface Properties 20

Directional Water Flow 21

Differential spontaneous capillary flow         D F cos 1 Capillary pressure:   f   P  c      1 D  min f Z Z time time N N           z z l j l j L L   1     k k N N   j j 1 1 F cos        1 D 2 D D  L  T  T f D     0 f  2 D    min K T 1     k k           max A 2 D D   Permeability: z z l j l j 128 P T f   max   k k k k j j 1 1     z z z z t t t t  k 1     z l j   Total resistance:    k 1 l j        j 1  R z k  k    3 3 2 2 eq         K K      z z l j l j t t t t j 1 j k k  k  2 2 2 2   j j 1 1         z z l l 1 1 t t t t k  k  1 1 1 1 t  t  0 0 z  z  0 0 AP Volume flow rate:  c Q  R eq Total liquid mass       1 1 k   k     absorbed by the material           M z A l z l     j j j k   j 1 j 1 Kausik Bal, Jintu Fan, M.K. Sarkar, Lin Ye, Differential spontaneous capillary flow through heterogeneous porous media, International Journal of Heat and Mass Transfer 54 (2011) 3096 – 3099.