homogenisation techniques dedicated to paper industry ... · j.-f. bloch 1 homogenisation...

J.-F. Bloch

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Homogenisation techniques

dedicated to paper industry:

theory and applications.

Bloch, J.-F.

Paris, 30 septembre 2010

J.-F. Bloch

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Part I –

Homogeneization theory applied

to paper

Part II –

REV and X-Ray microtomography

Outline

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Part I - CONTENT

• I - Introduction

• II - Method presentation

• III - Main Results

• IV - Example

• V - Conclusion

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z

y

x

Papier

Feutre

I - INTRODUCTION

Paper

AIM :

Temperature field in paper during hot

pressing

Felt

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I - INTRODUCTION (2)

Paper and Felt: deformable porous media

Non saturated: (non /) wetting phases

Complex structure at the pore scale

Macroscopic modelling:

Homogeneization

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I - INTRODUCTION (3)

Homogeneization:

microscale to macroscale

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II – METHOD

PRESENTATION

The medium is assumed as periodic.

Random media yield similar

macroscopic description modelling.

AURIAULT J.-L. (1991) ”Heterogeneous medium. Is an equivalent macroscopic

description possible?”, Int. J. Engng. Sci., 29, 7, pages 785-795.

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II – METHOD

PRESENTATION:

Multiscale expansion method

- without macroscopic prerequisites,

- the macroscopic law,

- the effective parameters,

- the validity domain of the model.

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II – METHOD

PRESENTATION:

Multiscale expansion method

pore dimension / sample size

model quality.

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II – METHOD

PRESENTATION:

Main steps

1 – Physical phenomena at the microscopic scale.

2 – Two different scales are defined ( ).

3 – Physical Equations at microscopic scale.

4 – The dimensionless parameters are expressed in

function of .

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II – METHOD

PRESENTATION:

Main steps (2)5 – Asymptotic expansions in power of are

introduced.

6 – Problems at different orders are solved to

determine the successive approximations of the

variables.

7 – Physical quantities are evaluated from the

dimensionless ones.

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II – METHOD

PRESENTATION

• The scale ratio = l / L

• For a paper web, the characteristic lengths

l =10 m and L = 1 mm

= 10-2

• Dimensionless parameters are evaluated in

term of .

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II – METHOD

PRESENTATION

lk TT

kkkpkk

kpkTλ.TCρ.v

t

TCρ

k

k

Energy balance for a domain (k = w, a, s):

Flux conservation of heat on the interface:

NX

TN

X

Ti

j

lli

j

kk ijij

Temperatures are continuous on each interface

(no resistance) :

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II – METHOD

PRESENTATION

- in k :

Dimensionless variables: X = X*. XR

Dimensionless parameters:

i

kkkke

j

kk

i

kkk

y

TCVP

y

T

yt

TCP

iij

Ny

TN

y

TNi

j

lli

j

kkl k ij

ij

- in kl:

Rw

Ra

λλ

λN

aw

Rw

Rs

swN

R

jij

i

R

p

X

T

X

t

TC

P (R)

(*)

R

jij

i

R

iip

X

T

X

X

TVC

Pe

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II – METHOD

PRESENTATION- Reference Values

(J . m-1 . s-1 . K-1) (J . kg-1 . K-1) (kg.m-3)

s Cps s

0.33 1.33.10 3 1.5.10

w Cpw w

0.602 4.18.10 3 10 3

a Cpa a

0.026 10 3 1.23

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II – METHOD

PRESENTATION

1O0.548NεO0.043Nswaw λλ

a1-

w

2w

w PεOτλ

lCP

ws

2s

s POC

Pl

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II – METHOD

PRESENTATIONEach physical quantity is then looked as:

....φε.φεε.φφφ 332210

Homogenisation process

approximated macroscopic models.

Model accuracies = O( )

The larger the scale separation is,

the better is the result (approximation).

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III - MAIN RESULTS

A - Small Péclet number:

diffusion - convection

εOx

Tλ

xt

TC

j

effij

isw,

dy

I1

swj

kIkjij

effij

sw Ω sΩ wsw, dΩCdΩCΩ

1C

A-1: Pe O ( 2) P = O ( 2)

With :

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III – MAIN RESULTS (2)

Ox

T

x0

j

effij

i

A-2 : Pe O ( 2) P O ( 3)

A-3 : Pe = O ( ) P = O ( 2)

Ox

TVC

x

T

xt

TC

ii

j

effij

iw,s

J.-F. BLOCH, J.-L. AURIAULT, (1998) ‘Heat Transfer in Nonsaturated Porous Media:

Modelling by Homogenisation’, TiPM, 30: 301–321.

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III - MAIN RESULTS (3)

Ox

TVC

x

T

x0

ii

j

effij

i

A-4 : Pe = O ( ) P O ( 3)

A-5 : Pea,w = O ( ) Pes = O ( 2) P = O ( 2)

Ox

TVC

x

T

xt

TC

i+i

j

effij

iw,s wa

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III - MAIN RESULTS (4)

B - 2 : Pe = O (1) P = O ( 2)

B - 3 : Pe = O (1) P O ( 3)

2

iΩi

j

*eff**

isw, εO

x

TCV

x

Tλ

xε

t

TCε

ij

dVCy

I1

sw,

jIII(0)i

0

l

jIII

ljijeff***

ij

2

iΩi

j

*eff**

i

εOx

TCV

x

Tλ

xε0

ij

B - Higher Péclet number : dispersion

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III - MAIN RESULTS (5)

B - 1 : Pe = O (1) P = O ( )

B - Higher Péclet number : dispersion

2

iΩi

j

*eff*

i

i

2

ΩΩiΩΩΩ

εOx

TCV

x

Tλ

xε

xt

TχCε

t

TCεC

ij

swIIasw

dVCy

I1

w,s

j

(0)i

0

l

jljij

*eff*ij II

II

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III - MAIN RESULTS (6)

Pe O ( -1)

This case is NOT homogeneizable

no equivalent medium exists !

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z

y

x

Papier

Feutre

IV - EXAMPLE:

Hot pressing of a paper web

O10O5.0

10.10.10.10#

.V.C.Pe 2

6233p l

ii

j

effij

i x

TVC

x

T

x0

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IV – EXAMPLE:

Assumptions

ii

j

effij

i x

TVC

x

T

x0

ii

jijw

i x

TVC

x

TI

x0

ws

ii

jijw

i x

TVC

x

TI

x0

ii

jijsww

i x

TVC

x

TInn

x0

sw O

wpwwapaa VCVC

ns + nw # 1.

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IV - EXAMPLE:

Temperature field in paper and felt

Troll = 50 °C

Mach. Speed:10 m.s-1,

Paper thick.: 320 m,

Felt thick.: 2.5 mm,

Nip Length: 2.5 cm.

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IV - EXAMPLE

Detail of temperature field in paper during hot

pressing.

PAPER

FELT

Troll = 50 °C

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V – CONCLUSIONS

/ Part I• Same microscopic physics, different law

structures, with different effective coefficients.

• Evaluating dimensionless numbers ( ) :

macroscopic model catalogue, their effective

parameters and their validity domains are

obtained.

• Physical characteristic values dedicated to the

studied process.

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V - CONCLUSION / Part I (2)

• The equivalent description corresponds to a

medium that reacts globally to the considered

physical excitation like the microscopic medium

studied.

• If the medium cannot be homogenised,

macroscopic properties are not intrinsic to the

media.

• Applicable to any porous medium that satisfies

the microscopic hypotheses in use here.

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Part II

3D structure

of Papers

/ VER

(X-Ray Microtomography)

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Estimation of paper physical properties

based on synchrotron

X-ray microtomography.

Jean-Francis Bloch

Sabine Rolland du Roscoat

Maxime Decain, Christian Geindreau

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Content

(Partie Non présentée)

I. Context

II. Materials and Methods

III. Estimation of paper structure / VER

Deterministic approach

Statistical approach

I. Estimation of paper properties

II. Conclusions and perspectives

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I. Context

• Synchrotron X-ray microtomography gives

access to the inner structure of samples at a

micrometric scale

– Quantification of the structure

– Estimation of physical properties

• Comparison to experimental data

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I. Position of the problem

• May Physical properties be simulated from

X-Ray microtomography?

• Analysed volume is much smaller than the

characteristic lengths / physical properties?

Problem of the representativity for:

• Porosity and specific surface area

• Effective thermal conductivity and

permeability

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Tomographic data

REV ? No …

Yes

Estimation of physical properties

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How Long is Long enough

/ structural properties ?

Property

(porosity)

REV

Volume

size

Small samples (mm) !?

II.4. Representative Elementary

VolumeREV

• Definition

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II.4. Applied methods

Two approachs

Deterministic (classical)1

Statistical2

1 W.J. Drugan, J.R. Willis, "A micromecanics-based nonlocal constitutive equation and estimates of

representative volume element size for elastic composites.", J. Mech, Phys Solids, Vol. 44, No 4, 1996,

pp 497-524.

2 T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, "Determination of the size of the representative

volume element for random composites: statistical and numerical approach", International Journal of

Solids and Structures 40, 2003, pp 3647-3679.

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III.5. Conclusions

/ Statistical REV

• Estimation of properties on volumes smaller

than the deterministic REV

saving computing time in the case of the

estimation of physical properties

Rolland du Roscoat, S., Decain, M., Thibault, X., Geindreau, C., Bloch J.-F. Estimation of

microstructural properties from synchrotron X-Ray microtomography and determination of

the REV size in paper materials. Acta Materiala, 55(8), 2007, 2841-2850

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Content

I. Context

II. Materials and Methods

III. Estimation of paper structure / VER

Deterministic approach

Statistical approach

IV. Estimation of paper properties

V. Conclusions & perspectives

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IV.1. Estimation of permeability

• Estimation of tensor of permeability

– No penetration of fluid into fibres

– Pressure gradient is imposed at the interface

– Estimation of the local fluid velocity

– Deducing the permeability

from Darcy Law

Koivu, V., Geindreau, C., Decain, M., Mattila, K., Bloch, J.-F., Kataja, M., Transport

properties of heterogeneous materials combining computerized x-ray micro-tomography and

direct numerical simulations, International Journal of Computational Fluid Dynamics,

23(10), 2010, 713-721