Flocculation in turbulent fiber suspension flows

Mats Nigam

Noss AB

Crowding number

223 v

v

Fiber suspensions are usually characterized in terms of the Crowding number,

lN c (average number of fibers in a sphere with diameter l)d

where c is the volume concentration, l the f

⎛ ⎞= ⎜ ⎟⎝ ⎠

iber length and d the fiber diameter.The following characterization is based on settling experiments by Martinez et al,

N 16 dilute, fibers settle independently16 N 60 flocculatedN 60

≤≤ ≤≥ networked

This presentation focuses on the intermediate range.

The effect of fibers in dilute suspensions

A fiber moves with a large whirl essentially as a solid body with negligible loss of turbulence kineticenergy to friction and deformation work.

Bending stiffness prevents the fiber from aligning with the small whirls resulting in loss of turbulence kinetic energy to bothfriction and deformation work.

The presence of fibers enhances the decay-rates of turbulent eddieswith length-scales smaller than the fiber length

Modified energy spectrum

2( ) /( )E luκ

lκ

IncreasingRe

fiber ”cut-off ” length

Kolmogorov length

Duffy & Lee (1978) measured the friction factor in pipe flow and found it to be constant for a range of Reynolds numbers.

Stationary homogeneous turbulence

x

y

z

( )tx

0x

A fluid particle performs a random walk with(Taylor, 1921)

0( )t t⟨ − ⟩x x :

where denotes a spatial average.

The length of the path traced out by the particle is

...⟨ ⟩

t:

In a suspension each fiber traces out a pathwhereas the distance from its original position onlygrows as

t:

t

high probability of entanglement (flocculation)in stationary homogeneous turbulence

Turbulent flows with mean shear and/or mean strain

Yield stress

Developing pipe flow

r

ay

x

( )u yy

laminar region

uniform coreturbulent region

inlet length

uniformflow

uniformcore

fully developedflow

boundary layer

Flocculated region of the flow field

1 ,wy y a ra

τ τ⎛ ⎞= − = −⎜ ⎟⎝ ⎠

y

w

ra

ττ

≤Flocculated region:

Assume that the shear stress is linear within theboundary layer,

( , ) 1 ( )( ) wyx y xx

τ τδ

⎛ ⎞= −⎜ ⎟⎝ ⎠

( ) ( )1 1 1( )y

w

x r xa a a x

τδ δτ

⎛ ⎞− ≤ ≤ − −⎜ ⎟

⎝ ⎠Flocculated region:

Downstream development of flocculated region

( , )x yτ

yτ

flocculated region

uniform coreturbulent

yτ yτ

( )xδ

xy

( )w xτ

Flocculated area / cross-section of pipe =

22

2

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

( ) ( )( ) 2 ,

y yc

w w

c

x x xA x for x xa a x a x

x xA x for x xa a

τ τδ δ δτ τ

δ δ

⎧ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎪ = − + ≤⎜ ⎟⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎨⎪ ⎛ ⎞= − ≥⎪ ⎜ ⎟

⎝ ⎠⎩

: ( )c w c ywhere x xτ τ=

Empirical method

( )

1/*

2 22 1

1

( )( )( ) ( )( )

2 ,

8.8

nw

**

w

xu x yu(x,y)Let, C n , with u xu x

and use von Karmans integral equation,

ddUU Udx dx

together with an empirical correlation for the inlet length,

l uaa

τν ρ

τδδ δρ

ν

⎛ ⎞= =⎜ ⎟⎝ ⎠

+ + =

⎛ ⎞= ⎜ ⎟⎝ ⎠

/ 6

1/ 7 6 /7( ) ( )( )

w

w

x l x x and (n=11)l x a l

τ δτ

⎛ ⎞ ⎛ ⎞⇒ = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Ratio of flocculated flux to total flux

6/7 2 2 7

6/7 12/ 7

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

f c

c

Q X X X T X T for X X T (T 1)A X O A X

Q n X X for X X

−⎧ − + ≤ = ≥⎛ ⎞= + ≈ = ⎨⎜ ⎟ − ≤ ≤⎝ ⎠ ⎩

, / ( )y wwhere X=x/l and T= lτ τ

X

T=2 T=1

T=3/4

T=1/2

T=1/4

( ) /fQ X Q( )y w and l

from empirical correlations

τ τ

Conclusions

• The presence of fibers enhances the deacy-rates of turbulent eddies with length-scales smaller than the fiber length

• Homogeneous turbulence induces flocculation• Deflocculation occurs in turbulent flow fields with mean shear and mean strain• The concept of a floc yield stress was introduced and an empirical model for flocculation in developing

pipe flow was formulated

In desperate need of

A useful model for the interaction between flocculation and turbulence