homogenisation techniques dedicated to paper industry ... · j.-f. bloch 1 homogenisation...
TRANSCRIPT
J.-F. Bloch
1
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