the 6th Cornell conference on Fractals Huazhong University of - - PowerPoint PPT Presentation

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Boming Yu School of Physics Huazhong University of Sci. & Tech. yubm_2012@hust.edu.cn A review on flow resistance in microchannels with rough surfaces by fractal geometry theory and technique

http://blog.sciencenet.cn/?398451 Google Scholar: https://scholar.google.com/citations?user=_NmWuUQAAAAJ&hl=en

the 6th Cornell conference on Fractals

• n June 13–17, 2017, Cornell University

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Outlines

• 1. Introduction
• 2. Rough surface by fractal description
• 7. Concluding remarks
• 3. Models for simulating rough surfaces
• 4. Fractal geometry theory for rough surfaces
• 5. Flow resistance in micro channels
• 6. Other methodologies for flow resistance in

roughened channels

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• 1. Introduction

Rough surfaces widely exist in natures such as road surface, airplane surface, metal surface, tube surface, channel surface, earth surface, etc. Roughness of surfaces significantly influences the flow resistance when fluid flows through rough surfaces. Absolutely smooth surface does not exists!

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• 2. Rough surface by fractal description

2.1 Description of typical rough surfaces

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• A. Majumdar et al., Journal of Tribology, APRIL 1990,
• Vol. 112, p205
• A. Majumdar et al., ASME J. Tribol. 1991, 113: 1–11

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Suryaprakash Ganti, et al., Wear 180 (1995) 17-34 An NOP image at 4000 um scan length and an AFM image at 50 um scan length for a lapped steel surface.

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2.2 A self-affine fractal surface

Profile of a self-affine fractal surface

Weierstrass-Mandelbrot (W-M) function can be widely used to describe the profile of a rough surface ：

1 ; 2 1 ; 2 cos ) (

1

) 2 ( ) 1 (

   



   

   D x G x z

n n n D n D

• A. Majumdar et al., ASME J. Tribol. 1991, 113: 1–11

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• I. G. Main, et al., Geological Society, London,

Special Publications, 54: 81-96, 1990. Natural surfaces, real fractures in rock, such as dry hot rock. Rough surfaces of Fracture networks

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Heilbronner R., Keulen N. Grain size and grain shape analysis of fault rocks. Tectonophysics, 2006, 427(1):199-216.

Characters of fractures:

• -- Irregular
• -- Random
• -- Different apertures
• -- Different lengths
• -- extremely rough surfaces

Fractured networks

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Consider oil/gas/water flowing in such fractures/tubes, the effects of roughness of surfaces on flow in channels/fractures should be taken into accounted. (a) (b) (a) Cross-section of a micro-channel tube (b) A profile of a rough surface of a micro-tube (c) Fluid distributor (c)

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Artery or vena vessel

Antonets V.A.,et al. Fractal in the Fundamental and Applied

• Sciences. North-Holland: Elsevier, 1991. 59-71.

If fat is accumulated on the wall surface of artery, what will happen? High blood pressure happens!!!

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Afrin, N., et al. Int. J. Heat and Mass Transfer 54 (11): 2419-2426(2011).

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• 3. Models for simulating rough surfaces

Profile of a self-affine fractal surface Weierstrass-Mandelbrot (W-M) function can be used to describe the profile of a rough surface ：

1 ; 2 1 ; 2 cos ) (

1

) 2 ( ) 1 (

   



   

   D x G x z

n n n D n D

• A. Majumdar et al., ASME J. Tribol. 1991, 113: 1–11

3.1 Weierstrass-Mandelbrot (W-M) function where G is a characteristic length scale, D is the fractal dimension

• f the roughness profile, and

is the scaling parameter.



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3.2 Cantor model for rough surfaces

Rough surfaces can be characterized by fractal Cantor structures Cantor set

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Yongping Chen et al., Int. J. Heat and Fluid Flow 31, 622(2010) Thomas L. Warren et al., Wear 196, 1-15(1996)

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3.3 Random fractal spots for modeling rough surface

max

( ) ( / )D N L d d d  

Typical morphology

J.-H. Li, et al., Chin. Phys. Lett. 26 (11): 116101(2009)

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As D=1.25 and G=9.4610-13m, a rough surface by simulation

3.4 A rough surface simulated by Fractal- Monte Carlo method M.Q. Zou et al., Physica A 386, 176-186(2007).

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• 4. Fractal geometry theories for rough surfaces

Weierstrass-Mandelbrot (W-M) function can be used to describe the profile of a rough surface ：

1 ; 2 1 ; 2 cos ) (

1

) 2 ( ) 1 (

   



   

   D x G x z

n n n D n D

• A. Majumdar et al., ASME J. Tribol. 1991, 113: 1–11

4.1 Weierstrass-Mandelbrot (W-M) function where G is a characteristic length scale, D is the fractal dimension of the roughness profile, and is the scaling parameter.



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 

/2

~

f

D

N A a a





Mandelbrot in his book: The Fractal Geometry of Nature proposed that the cumulative size distribution of islands

• n earth follows the fractal scaling law:

where N is the total number of islands of area (A) greater than a, and Df is the fractal dimension of the surface.

• B. B. Mandelbrot, The Fractal Geometry of Nature,
• W. H. Freeman and Company, New York, 1983。

4.2 Model by extension of the fractal scaling law

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Marjumdar and Bhushan extended this power law to describe the contact spots on engineering surfaces, and the power-law relation is

 

/2 max

( ) /

f

D

N A a a a  

• A. Majumdar et al., Journal of Tribology, April 1990,
• Vol. 112, p205

2 max max

 g a 

2

 g a 

where and , and g is a geometry factor.

,



is a spot diameter.

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Yu et al. again extended the above equation to describe the pore size distribution in porous media by

f

D max )

( ) L ( N     

B.M. Yu, Analysis of flow in fractal porous media,

• Appl. Mech. Rev. 61, 050801(2008).

B.M. Yu and P. Cheng, Int. J. Heat Mass Transfer,

• V. 45, No. 14, 2983-2993(2002).

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• 5. Flow resistance in micro channels

Flow resistance is usually defined by

/ P L 

P 

where is the pressure difference, and L represents the straight length.

• r by Friction factor:

2

2 /( )

w m

f u   

where , and are respectively the wall shear, fluid density and mean velocity in a channel.

w





m

u

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5.1 Flow resistance for laminar flow in micro-channels with smooth surfaces For fully-developed, laminar, incompressible flow in a smooth rectangular microchannel with the height and width being respectively b and w, the equation of motion is

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2 2 2 2

1        u u dp z y dx

where u is the velocity in the x-direction,

 is the dynamic viscosity,

dp/dx is the pressure gradient along the flow direction,

x

Assume b<<W, then, Eq. (1) can be simplified as

2 2

1   d u dp dz dx

(1) (2)

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Due to the symmetry of the channel, the no-slip boundary condition on wall is

, 2 0, b z u du z dz            

(3) Solving Eq. (2) with the boundary condition Eq. (3) yields

2 2

1 ( z ) 2 4    dp b u dx

(4)

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The mean velocity over the cross section can be obtained as

2 2

1 1 / 2 12   



b m

dp b u udz b dx

(5) The volume flow (let w=1 and b<<w) rate is

3 2

• 2

12

b b

b dp Q udz dx   



(6)

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The wall shear in smooth channel:

2

2  



 

w b z

du b dp dz dx

(7a) Substituting Eq. (6) into Eq. (7a) yields

2

6

w

Q b   

(7b)

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From Eq. (6) we can obtain the pressure gradient across the length L as

3

12 = （ ）

S

P Q L b  

(8) Combining Eq. (5) and Eq. (7b) results in the fanning friction factor:

2

2 12

w m m

f u u b      

(9)

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The Reynolds number is

Re   

m h

u D

(10) Since b<<w, the hydraulic diameter Dh can be simplified as

b Dh 2 

24 24 / Re    

m h

f u D

(11) (12)

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The Poiseuille number Po for a fully developed laminar flow in an infinite plate channel is

Re 24

• P

f   

(13) Similarly, we can obtain the friction factor f for a fully developed laminar flow through a smooth circular tube

64 / Re f 

(14) (15) and the Poiseuille number Po:

is

Re 64 Po f   

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5.2 Flow resistance for laminar flow in micro-channels with rough surfaces by fractal geometry

max

( ) ( / )D N L d d d  

Typical morphology

J.-H. Li, et al., Chin. Phys. Lett. 26 (11): 116101(2009)

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The ratio of height to base diameter of conic peak is

h   

As shown in Fig. 1 (b), the base area for a conic peak/spot is The effective average height of conic roughness elements can be found to be 2 / 4 i i

S  

3 max 2

1 3 3 1

D s eff s

D h D    



    

(16) (17) (18)

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The non-slip boundary on the walls of roughened microchannels is

( / 2 ), 0,

eff R R

z b h u u z z             

Solving Eq. (2) with the boundary condition Eq. (19) yields (19)

2 2

1 [( ) ] 2 2

eff R

dp b u h z dx    

(20)

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The volume flow (when w=1 and b<<w) in roughened microchannel is

3 2 2

( 2 ) 12

eff eff

b h eff b R h

b h dp Q u dz dx 

  

  



(21) The pressure gradient in roughened microchannel is

3

12 = ( -2 ) （ ）

R eff

P Q L b h  

(22)

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Compared to the pressure gradient in smooth channels,

• Eq. (22) can be rewritten as

= ) （ ） （

R S R

P P F L L  

(23) where

3

1 (1 )

R r

F   

2 /

r eff

h b  

and where

r



is defined as the relative roughness in rectangular roughened microchannels.

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The relative increase of the pressure gradient is defined by

3

( ) ( ) 1 1 1 (1 ) ( )

R S R R r S

P P L L F P L            

(24) The friction factor in rough channels can be obtained as

24 Re

R R

f F 

(25) where FR>1, and friction factor is increased and similar results for flow in rough cylindrical tube.

24

R R

Po F =

and

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Reference

S.S Yang, et al., A fractal analysis of laminar flow resistance in roughened microchannels,

• Int. J. Heat Mass Transfer 77, 208-217(2014).

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• 6. Other methodologies for flow resistance in roughened

channels 6.1 Numerical simulations

Y.P. Chen, et al., Int. J. Heat and Fluid Flow 31 (2010) 622–629

The Gauss–Seidal iterative technique, with successive

• ver-relaxation to improve the convergence time.

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6.2 The Lattice Boltzmann method (LBM)

C.B. Zhang, et al., Int. J. Heat and Mass Transfer 70: 322 (2014) Schematic of gas flow heat transfer in a rough micriochannel.

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• 7. Concluding remarks

Analytical solution for flow resistance in roughened channels can be obtained based on fractal geometry, but it was impossible based on Euclid geometry. The flow resistance in roughened channels based on Weierstrass-Mandelbrot (W-M) function is open. The flow resistance in roughened channels based on Cantor set model is also open. The flow resistance in roughened natural fractures based on fractal geometry is also open.

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You are welcome to submit your original manuscript to Fractals journal for publication at: IF=1.22, 2014 IF=1.412, 2015 IF=1.540, 2016 IF is higher in the subject of Mathematics. Publishing original papers in: Fractals in Sciences; Fractals in Engineering; Fractals in Mathematics. http://www.worldscientific.com/worldscinet/fractals

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