SUPPLEMENTARY MATERIAL
Multiscale modeling of oil uptake
Jean-Michaël Vauvre, Anna Patsioura, Régis Kesteloot, Olivier Vitrac
Jean-Michaël Vauvre
INRA, UMR 1145 Ingénierie Procédés Alimentaires, F-91300 Massy, France
AgroParisTech, UMR 1145 Ingénierie Procédés Alimentaires, F-91300 Massy, France
McCain Alimentaire S.A.S., Parc d’entreprises de la Motte du Bois, 62440 Harnes, France
Anna Patsioura
INRA, UMR 1145 Ingénierie Procédés Alimentaires, F-91300 Massy, France
AgroParisTech, UMR 1145 Ingénierie Procédés Alimentaires, F-91300 Massy, France
Régis Kesteloot
Régis Kesteloot conseil,
60 Avenue du Colonel Driant, 59130 Lambersart, France
Olivier Vitrac
INRA, UMR 1145 Ingénierie Procédés Alimentaires, F-91300 Massy, France
AgroParisTech, UMR 1145 Ingénierie Procédés Alimentaires, F-91300 Massy, France
Phone: +33169935063
Email: olivier.vitrac@agroparistech.fr
Topical area
Transport Phenomena and Fluid Mechanics
Keywords
Porous media, Spontaneous imbibition, Percolation model, Kinetic Monte-Carlo, Deep-frying
1
1
Common assumptions
The optical configuration of the experiments on an inverted microscope is reproduced in
Figure 1a-c. The aim of the 3D optical reconstruction was to rebuild the penetration depth of
oil within each cell. The main idea was to work with averaged properties along a significant
section, so-called Σ and of specific surface area AΣ, parallel to (x,y) so that composition
effects could be related to an increase of the length of the optical path T (see Figure 1c),
where fluorescent absorption and emission occurred. As a result, the attenuation of the
 z ,t 
excitation beam, E X , was assumed to be controlled by two effective absorption coefficients:
one associated with the scattering of light due to diffusion and refraction through crossed
semi-transparent phases, denoted  s T ,t  ; and one specifically associated to the specific
absorption by fluorophores, denoted  f
T ,t 
. Due to the deep-UV excitation at 295 nm,
absorbing fluorophores included phenolic residues in cell walls and 1,6-diphenyl-1,3,5hexatriene, they were denoted with subscripts 1 and 2, respectively.
By noting that collected intensity was also limited by the scattering of emitted fluorescence,
the following set of effective absorbing coefficients, denoted  X and  M , was used for
excitation and emission respectively:
 
 X








 M
where
 i i wall ,oil ,air
T ,t 
 s
T ,t 
 f
T ,t 
T

1 1   x, y,z 
 x , y , z ,t 
    òwall  s , wall  òoil
 s ,oil dxdy  dz
T 0 A  

T

1 1   x, y, z 
 x , y , z ,t 
    òwall
 f 1c1,wall  òoil
 f 2 c2,oil dxdy  dz
T 0 A  




T ,t 
 s

(1)
T ,t 
are the volume fractions of each phase verifying òsolid  òoil  òair  1 ;
{ci }i 1..2 are the volume concentrations in fluorophores and { f ,i }i 1,2 the corresponding
specific fluorescent absorption coefficients; { ns ,i }i  wall ,oil are the non-fluorescent absorption
coefficients and were assumed independent of the emission wavelength without loss of
generality.
Similarly, en effective quantum fluorescence yield, denoted  , was assumed to vary with
the local composition in the horizontal plane z at time t. It related the local excitation intensity
z ,t
absorbed to generate fluorescence, E Xf  , to the local intensity of the fluorescence emission,
 z ,t 
EMf
as:

T ,t 
T ,t 

dEMf
T ,t 
dEXf

T
1 
0

 x, y , z 
wall
 A   ò


 x , y , z ,t 
1 Xf 1c1, wall  òoil
 2 Xf 2c2,oil dxdy  dz

s
 T ,t 
(2)
Since the volume fraction of cell walls remained unchanged during experiment, the invasion
of oil of the considered plane caused an increase of 
the measured fluorescence.
2
T ,t 
with time and consequently, to
Figure 1. Simplified description of excitation and emission pathways. a) Experimental setup,
where the incident excitation EX0 is scattered by the sample (intensity EXs) and used to
generate fluorescence (intensity EXf). The residual excitation EX is lost at the top of the
sample. The corresponding emission of fluorescence (EMf) is scattered by the sample
(intensity EMs). Only the fraction enclosed within the solid angle  (with intensity I) is
measured by the detector. b) Scheme of the local conversion of excitation into fluorescent
emission. c) Corresponding light paths T are as used in Eq. (7). d) Honeycomb arrangement of
cells; e) and corresponding step-wise integration of absorption and emission intensities when
oil fills successively cells of layer 1 (step 1) and of layer 2 (step 2). It is important to notice
that air is not represented as it is associated to  s  Ki  0 .
2
Photon counting balance
As the speed of light is much faster than the percolation of oil, both excitation of light and
emission fluorescence fluxes were assumed at steady state. By choosing the top surface as
reference position z  0 , inverse modeling aims at relating the thickness of the optical path, T,
3
I 
z
0
z ,t 
T
where fluorescence occurs, with the intensity of the fluorescence I T ,t   
dz collected
t
by the detector. The variation of the fluorescent intensity generated when the optical path is
T,t
I  
increased of dT,
, is related to the variation of the intensity absorbed and emitted by
T t
T ,t 
the fluorophores,
E Xf

EMf
T ,t 
and
T
T
t
, respectively, via Eq. (2) and the following photon
t
balances:
I T ,t 
T

t , measured

I T ,t 
T

t , emited light
in direction of the detector
T ,t 
T 
 EMf
4 T
 s
T ,t 
I T ,t 
T
I T ,t  
t
t , scattered light
in direction of the detector
T 


4
 T ,t 
(3)
T ,t 
E Xf
T
 s
T ,t 
I T ,t 
t
where  T  is the solid angle, which the lens makes with Σ at position z. Similarly, the photon
balance applied to the excitation beam leads to a variation of the light flux, lost by the top of
the sample,
E XT ,t 
, as:
T t
T ,t 
E Xf
E XT ,t 

T t
T
 s
T ,t 
E X
T ,t 
(4)
t
 z 
 T ,t 

4
and 1, yields to a generalized Beer-Lambert law written as the photon balance between the
absorbed and emitted light fluxes:
The linear combination of Eqs. (3) and (4), respectively to the weights K 

d K T ,t  EXT ,t   I T ,t 


dT
t
dK T ,t  T ,t 
EX    s
dT t
 T ,t 
K
T ,t 
T ,t 
EXT ,t   I T ,t 


(5)
It is worth noticing that Eq. (5) neglects the effect of wavelength on attenuation constant
T ,t 
. It can be integrated along T as soon as variables are separable, that is when K T ,t 
can be assumed constant. According to Eq. (11) in 1, the variation of the solid angle associated
to a source much smaller than the detector (lens) and close to the detector is indeed


 1
T C
 , where C is the thickness of the quartz coverslip (0.17 mm)
 1
2
2
4 2 
T  C   Rlens 

s
and Rlens is the diameter of the lens (12 mm). By considering thin cell layers (total thickness of
300 µm), the variation of  between z  0 and z  T is low. The variation of   t  is of 6%.
When the optical absorption and composition properties vary significantly along T, the
strategy will consist in decomposing the integral over N successive optical paths such as
N
T   Ti . When K is constant on each domain and equal to K i , one gets:
i 1
4
Ti
N


 Ki EXT ,t   I T ,t 
i 1 Ti1
 T ,t 
Ki E X
I
 T ,t 

N
Ti
 
i 1 Ti1
Ti ,t 
s
(6)
dT
with T0  0 .
It is worth noticing that Eq. (6) is commutative for all optical paths with similar thicknesses
and optical properties. As a result, a step-wise integration procedure can be devised by
collecting all phases with similar optical properties and filling homogeneously the capillary:
oil or cell walls. This strategy is illustrated in Figure 1d-e as an equivalent dissipative network
for excitation and emission. Pure air does not have significant effect on light scattering
 T ,t 
 Ki  0 . If N is now the number of dense phases assumed
because it is associated to  s
to be homogeneous, Eq. (6) is integrated as:
N
 Ki E XTi ,t   I Ti ,t  
ln


 s ,i Ti




 K E Ti1 ,t   I Ti1 ,t  
i 1
i 1
 i X

N
(7)
with Ti the length of the optical path for each solid phase i, intensity at top surface I 0,t   0 ,
 0,t 
incident excitation at bottom surface EX
 EX 0 and apparent scattering coefficient  s ,i for
the phase i.
3
Application of Eq. (7) to the identification of the oil front z
Eq. (7) is used with the following convention: i  1 for bottom cell wall, and i  2 for oil. The
position of the oil front, z, is defined in this study as the equivalent thickness of the optical
path filled with oil, T2  z , regardless oil is on the top or at the bottom of the considered
cell. According to Eq. (7), two sensors should be required to retrieve unambiguously z: one
T ,t
sensor to measure the collected fluorescence, I  N  , and another to measure the transmitted
 T ,t 
light across the product, EX N .
In this study, we inferred a reconstitution strategy based on the sequential observations on the
cell filling process using a single detector and reported in Patsioura, et al. 2. For each stack of
two cells (see Figure 1c), the inverse problem –finding the position of the oil front– was
broken into two sequential problems: finding z when only a cell of the first layer was partly
filled with oil and finding z  z1 when the cells of the top layer are already filled with oil and
where z1 is the position on the cell wall between the top layer (layer 1) and the bottom layer
(layer 2).
Step 1: For 0  t  t z1 , only optical paths crossing the layer 1 are considered: N  2 in Eq. (7)
with i  1 for the bottom cell wall between layers 1 and 2 (see Figure 1d) and i  2 for the oil
filling the top cell. For t  t z , the top cell is saturated (filled) with oil.
1
Step 2: For t  t z  tlag , where tlag is a significant lag time, we get N  3 in Eq. (7) with
1
i  1 for cell wall, i  2 for oil height in the top cell with T2  z1 and i  3 for oil height in
the bottom cell with T3  z .
The application of Eq. (7) for the top cell leads to:
 z    z  
K 2 EX ,2  I 2
t
t

K1EX 0 exp   s ,1
0  t  t z1
t
 exp   s ,2 z   with 
t 
T1
0  z  z1



5
(8)
Under low scattering and low fluorescent conditions (i.e.  f ,i z   s ,i z  0 for i = 1, 2),



the right-hand side in Eq. (8) becomes exp   s ,i z t   1   s ,i z t     s ,i
2

z t  2 . As a
 z   ,
result, the transmitted light and collected fluorescence across a single cell, EX ,2  and I 2
t
z 
t
respectively, are approximated by a linear function of z  t  .
4
Literature Cited
1.
Shelyuto V. Exact analytic results for the solid angle in systems with axial symmetry.
Z. angew. Math. Phys. 1989;40(4):608-612.
2.
Patsioura A, Vauvre JM, Kesteloot R, Jamme F, Hume P, Vitrac O. Microscopic
imaging of biphasic air-oil flow in French-fries using Synchrotron radiation. submitted to
AIChE J. 2014.
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