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

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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

SLIDE 2

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

Department of Fiber Science & Apparel Design

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Permeation in Fibrous Media

SLIDE 4

Schematic of fibrous media composed of tortuous channels

cs

A

l

( ) q 

t

l

SLIDE 5

max

( ) ( ) .

D t scale scale

N L R R R  

Total number of pores/channels

 

 

 

1 1

1 max 1 max 1 max 1 ( ) ln ln 1 1

( ) ( ) ( ) ; 1 ln(1 ) , ( ) ln ln

i i i i i

D D i i i i D D i i i R D R R R R i i i i i

N N R R R R D R R B R R R R e B D R R R

 

         

           

Number of channels with a radius of

1  i

R

4

( ) . 8 R q R p    

Flow rate through the channel with a radius of R (Hagen–Poiseulle equation )

SLIDE 6

   

 

1 1 4 1 max 1 3 max 1 4 4 max

( ) ( )( ) ( )( ) ( ) 8 8 8 4

n t i i i i i n i i i i n D D i i i i i n D D i i i i D D D n

Q R q R N N q R N N pR R D R B R R pR DR B R R D B pR R R D C       

             

                  

   

 

 

1 1

4 max max 4 4 max (3 ) ln ln 1 1

8 4 ; 8 4 1 ln(1 ) , (3 ) ln ln

i i i i i

D D scale R D R R R R i i i i i

R D B p R D C R D B p R D C R R e C D R R R   

 

     

                          Total flow rate through all channels:

SLIDE 7

2 max 2 max

1 2 8 4 1 1 2 , 8 4 1

t t

Q U A D E pR D C D pR D                 

Mean velocity of fluid permeation

0.5

1 . 2 2ln

p f

R r              Empirical Relations: (Sampson, 2003)

max min

min

( ) , 1

R f p R f

D R Rf R dR R D    



 

max min

ln 2 . ln

f

D R R   

0.785

1 0.11 ( ) . 0.11     

(Tomadakis and Robertson, 2005) Fractal Relations: Darcy Law

, K U p   

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

• 6

10

• 4

10

• 2

10 10

2

10

4



K/r2 Model Clarenburg and Piekaar (1968) Gostick et al. (2006) Wheat (1963) Kostornov and Shevchuk (1973) Ingmanson et al. (1959) Johnson (1998)

Non-Dimensional Permeability

2

2 2 0.785

2 1 ( 0.11) 1 . 1.594 4 1 2ln

f

K D D D D r                         

Shou, D. Fan, J. & Ding, F. Physics Letters A 374 (2010) 1201–1204.

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More General Form of Permeability Model after introducing the fractal dimension of tortuosity DT

Xiao, B. et al, Electrochimica Acta 134 (2014) 222–231

(1 )/2 1/2 2 2

4 4(2 ) ( ) [ ] [ ] 128(3 ) (1 ) (1 )

T

f f f D f T f f f

d d d K D D d d d      



      

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Diffusion through Fibrous Media

Diffusivity through a specific channel of radius R: Total diffusion flux: Gas diffusion through one channel:

   

( ) ,

p equ p p

C j R D R L R  V

1

1 1 1 . 3 2 1

equ b p

u D D R Kn 



           

max min

( ) .

R d p R

Q q R dN  

Effective diffusivity

max min

2

( ) ,

R equ R M eff p t

R D R dN J D A A      

2 3 1 2

Kn M

Kn R Kn R            

Empirical Relations:

0.785

0.89 ( ) . 0.11    

(Tomadakis and Robertson, 2005) Fractal relations

max min

min.

( ) 1

R f R f

D R Rf R dR R D   



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Diffusion through Fibrous Media

 

   

 

max min

2 2 0.785

2 2 3 . 0.89 ( ) 1 100 1 0.11 1 2

f f

D R R eff b D f f f

R dR R D D D r D D      

 

              



Shou, D et al, Microfluidics and Nanofluidics, (2014) 16:381-389.

SLIDE 12

Experimental

Inverted Cup Method

SLIDE 13

Comparison of Fractal Model and Experimental Results

Shou, D et al, Microfluidics and Nanofluidics, (2014) 16:381-389.

SLIDE 14

Relative Permeabilities in Multiphase Flows in Fuel Cell GDL

, , (1 )/2 , , (1 )/2

4(2 ) [ ] 3 (1 ) ( ) ( ) 4(2 ) [ ] 3 (1 )

T T

f w f w D T f w f w w rw f f D T f f

d d S D d d S K S k S S d d K D d d      

 

         

Relative water permeability: Relative gas permeability:

(1 )/2 , , , , (1 )/2

4(2 )[(1 ) ] 3 [1 (1 ) ] ( ) ( ) (1 ) 4(2 ) [ ] 3 (1 )

T T

D f g f g T f g f g g rg f f D T f f

d d S D d d S K S k S S d d K D d d      

 

                      

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Relative Permeabilities

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 krw S

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., 2007

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Relative Permeabilities

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 krg S

Present analytical model,Eq.(5-41) Hao and Cheng, 2010 Dana and Skoczylas, 2002 Li, 2011 Nguyen et al., 2006 Owejan et al., 2006

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Relative Permeabilities

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

krg, krg,  krw, krw, 

krw , krg S

Xiao, B. et al, Electrochimica Acta 134 (2014) 222–231

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Optimization of Thermal Insulation (Conductivity over Diffusivity)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 2.0x10

3

4.0x10

3

6.0x10

3

8.0x10

3

1.0x10

4

m =10-3

Y5=(ktot,eff/kg)/(De/Db)



Xiao, B, et al, Fractals, Vol. 25, No. 3 (2017) 1750030.

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Optimization for Protective Outer Fabric

19

1.0 1.2 1.4 1.6 1.8 2.0 0.0 5.0x10

2

1.0x10

3

1.5x10

3

2.0x10

3

2.5x10

3

m =10-3



 Y5=(Keff/Kg)/(De/Db) DT

(Permeability over Diffusivity) Xiao, B, et al, Fractals, Vol. 25, No. 3 (2017) 1750030.

SLIDE 20

Directional Water Flow

20

Human Skin Differential Surface Properties

SLIDE 21

Directional Water Flow

21

SLIDE 22

Differential spontaneous capillary flow

Capillary pressure:

min

cos 1 1

f c f

D F P D        

          

2 2 min max max

1

1 1 128

cos 2

T f T

D D D f T

D T f

D K P

F L A D D

     

 

  



   

 

           

Permeability:

c

eq

AP Q R  

 

   

1 1 1 1

eq

k k j j j k

z

z l j l j R K K

   



       

 

 

Total resistance: Volume flow rate:

1 k  2 k  3 k 

k k  k N 

z 

 

1 z l 

 

2 1 j

z l j





 

1 k j

z l j





 

1 N j

z l j L



 



z z 

Z

time

t 

1

t t 

2

t t  t t 

1 k  2 k  3 k 

k k  k N 

z 

 

1 z l 

 

2 1 j

z l j





 

1 k j

z l j





 

1 N j

z l j L



 



z z 

Z

time

t 

1

t t 

2

t t  t t 

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.

 

 

1 1 1 1 k k j j j k j j

M z A l z l   

   

  

           

 

Total liquid mass absorbed by the material

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Differential spontaneous capillary flow

2 4 6 8 10 12 14 50 100 150 200

Time (s) Mass of water absorbed (g)

Flow into layers of Fabric A first then to layers of Fabric B (Exp) Flow into layers of Fabric B first then to layers of Fabric A (Exp) Flow into layers of Fabric A first then to layers of Fabric B (Th) Flow into layers of Fabric B first then to layers of Fabric A (Th)

A B φ 0.77 0.62 λmax 450μ 350μ λmin 1μ 1μ L 8.8mm 8mm

SLIDE 24

Acknowledgement

• Shou, D: Ph.D Student, now Postdoc at Cornell University
• Xiao B: Ph.D Student, now Professor at Sanming University,

China

• Bal, K: Post Doc, now Professor at University of Calcutta, India
• Sarkar, M. Ph.D Student, now R&D Manager at Hong Kong
• Ye, L, Collaborator, Professor at Sydney University
• Ding F, Collaborator, Professor at Hong Kong Polytechnic

University, just moved to Ulsan National Institute of Science and Technology, Korea.

SLIDE 25

Thank you for your attention !