3D Pore scale network model for the transport of liquid water, water vapor and
oxygen in polymeric films.
Jose E. Paza and Luis A. Seguraa
a Food Engineering Department, Universidad del Bío-Bío, Chillán, Chile ([email protected])
ABSTRACT
Polymeric materials are used in food packaging mainly due to their capability to control the exchange of low
molecular weight compounds between foodstuff and external environment. Indeed, the mass transport control
through the packaging film is extremely important to guarantee the proper shelf-life of the packaged
products. Most mathematical models developed to describe those phenomena are based on modifications of
Fick’s law in a continuous approach and the transport parameters such as diffusivity or permeability are the
result of the process history in these kinds of models. The objective of this work is to present a 3D pore scale
model for the transport of liquid water, water vapor and oxygen in polymeric films. The porous media is
represented by a three-dimensional cubic network with pore segments randomly assigned and the model is
solved using Monte Carlo method. As water vapor flows into the polymeric film, condensation of water
occurs at the pore walls of the network. Liquid in pore corners allows hydraulic connectivity throughout the
network at all time and capillary pressure is determined by augmented Young-Laplace equation. Here we
report pore-level distribution of liquid and vapor as transport phenomena advanced, effective water vapor,
liquid and oxygen diffusivity and absolute permeability are calculated. The vapor and oxygen diffusivity
diminish as vapor condensation occurs from 8.97x10-8 to 2.21x10-9 [cm2/s] and 7.83x10-8 to 1.89x10-9 [cm2/s]
respectively. The vapor and oxygen permeability diminish from 1.10x10-21
to 2.72x10-23
[m2] and 3.16x10
-21
to 7.67x10-23
[m2]. On the other hand, the hydraulic permeability increases from 4.49x10
-25 to 8.48x10
-21 [m
2].
The transport properties obtained by the model were compared with experimental results obtained by
specialized literature given a good agreement for the oxygen and water vapor.
Keywords: condensation; diffusivity; pore-level; oxygen; water vapor
INTRODUCTION
The deterioration of packaged foodstuffs largely depends on the transfers that may occur between the internal
environment of the packaged food and the external environment [1]. Polymers have been commonly used for
food packing due to their characteristics such as: low cost, easy manufacture, low weight, versatility in size
and shape, light weight among other properties. However, plastic materials due to their peculiar morphology
allow it mass transport of low molecular weight compounds such as permeant gases, water vapor, odors,
plastic residues, and additives within the environment/package/food system [2].
Many studies have been focused on the characterization and understanding of this process due to the
important effect of the mass transport in the quality of food.
The classical macroscopic mathematical models developed to describe the transport of water vapor and gases
through polymeric films have been described in terms of the solubilization-diffusion mechanism, governed
by both thermodynamics and kinetics factors [3-5]. Most mathematical models developed to describe those
phenomena are based on modifications of Fick’s law in a continuous approach and the transport parameters
such a diffusivity or permeability are the result of the process history. In this approach, as mass transfer
process advance it is impossible to obtain fluid distribution at pore level and a complete comprehension of
the involved transport mechanisms is still far from satisfactory.
There is another approach to solve the problem, the microscopic models. These kinds of models are based on
transport properties and physical characteristics at pore scale. Pore-level models have not yet used to model
the transport phenomena at nanopore-scale but they have been used to model the vapor water condensation
and transport phenomena at micropore-scale, mainly in oil reservoirs [6-7].
Pore network models are a suitable tool to understand the role of the pore structure on the transport
parameters. In-situ condensation is an important phenomenon that occurs at a pore-level when a gas phase is
present in the porous structure. During capillary condensation the pores are partially or completely blocked
by the condensed phase preventing the flow of non-condensed gases through the polymer. Therefore
condensable gases can be transported through porous media as a gas or liquid. Under certain conditions both
phases could be present, rendering the quantitative description of the transport a challenging problem. The
phenomena are not fully understood despite numerous studies published on gas transport by capillary
condensation, and there is a lack of agreement on how to predict the permeability if capillary condensation
occurs in small pores [8].
The objective of this study was to develop a 3D pore scale model for the transport of liquid water, water
vapor and oxygen at pore-level in polymeric films. A network model was selected to represent the pore space
and the model was solved using a Monte Carlo method based on a previous model by Bustos & Toledo [7].
In polymeric films, the water-vapor condensation and the gases transport occurs at the nanopore scale rather
than at the micropore scale, as in the Bustos & Toledo model [7]. This distinction is very important because
at the nanopore scale it is possible to find some transport mechanisms and condensation effects that are not
significant at the micropore scale, such as Knudsen diffusion and disjoining pressure.
MATERIALS & METHODS
A three-dimensional cubic network of pore segments represents porous media. Nodes at which the pore
segments are connected act only as volumeless junctions with infinite conductance. Pore segments are
rectilinear with polygonal cross sections circumscribing circles with distributed radii. Figure 1 displays a
typical portion of the pore network and corresponding parameters. Pore segment radii rt are randomly
assigned according to Log- Normal distribution function for chitosan film [9]. According to Assis & Da Silva
[10], the film throats radii range from 5 to 30 nm. The mean pore radius is 6.425 nm, with the standard
deviation of 1.825 nm.
Figure 1. Pore network (cross section) and parameters: rt represent the radii of the pore throats; L is the pore-body-center
to pore-body-center distance, Lb is the pore-body-side half-length and Lt is the pore-throat length, 1 and 2 are pore center
(nodes).
The water vapor and oxygen transport process is considered under isothermal condition. The gravitational
effects are negligible. As vapor flows into the polymeric film, condensation of water vapor occurs at pore
walls of the network. Liquid in pore corners allows hydraulic connectivity throughout the network at all time
and capillary pressure (Pc) is determined by augmented Young-Laplace equation (Eq.1).
( ) γHhPc 2+Π≡ [1]
Where Π(h) is the disjoining pressure, H is the mean curvature and γ is the interfacial tension. Using Monte
Carlo simulation, we find pore-level distribution of liquid (condensate) and vapor as transport phenomena
occurs and effective water vapor and oxygen diffusivity and water vapor permeability are determined. A
modified form of Poiseuille’s law defines the gas conductance, g, for each pore segment, i.e., pgQvo ∆⋅= ,
where voQ is the volumetric flow of gas (water vapor or oxygen) through the pore space and p∆ is the
pressure drop across the pore; g depends on the configuration adopted by the fluid phase [9, 11-12]. For any
given gas-condensate capillary pressure each phase develops its own flow network to which conductance can
be assigned in much the same way as for single-phase flow. Several approaches are available for computing
the pressure fields in either phase once the pore-level saturations are established. Here we used the direct
solution of performing a nodal-material balances for each phase (see, for instance, [7, 9, 13]). A nodal-
material balance for each phase leads to a system of linear equations of the form G·p=B, where G is a matrix
of conductances, p is a vector containing the unknown pressures and B is a vector dependent on the pore
pressures at the upper and lower boundaries of the network and the conductances of the throats connected to
these boundaries [7]. To find the distribution of nodal pressures in each flow network once an external
pressure gradient was imposed, we used an iterative solution to the system of equations. The system was
Liquid
βL
L
2rt 1
L
βL
2
Solid
Gas
(1-β)L/2
2rt (1-β)L/2
(1-β)L/2
optimally stored and solved with a conjugate-gradient method with successive overrelaxation. This method is
part of the ITPACK routine libraries, which are publicly available at the web site
http://rene.ma.utexas.edu/CNA/ITPACK. The relaxation parameter was chosen as 1.84. With the nodal pressure of a given flow network in hand, the flow rate everywhere was calculated and the
network conductance computed for the water-vapor and oxygen diffusivities from (Eq. 2) [14]:
vo N
vo
vo
Q LRTD
A M p
ρ =
∆
[2]
where Dvo is the water-vapor or oxygen effective diffusivity (m2/s), Qvo is the water-vapor or oxygen flux
(m3/s), (∆p/LN)vo is the pressure gradient across the network on the gas phase (Pa), A is the total network
cross-section (m2), R is the ideal gas constant (J/mol K), T is the system temperature (K), ρ is the gas density
(kg/m3) and M is the molecular weight of the gas (kg/mol). On the other hand, the liquid diffusivity (DL) is
obtained from the equation 3 [15]:
−=
l
cH
LdS
dPkD
µ
[3]
Where kH is the liquid permeability, µ is the liquid viscosity, and c ldP dS is the change of capillary pressure
with the network liquid saturation (Sl). The liquid permeability is computed from the Darcy’s law (Eq. 4):
pA
QLk N
H∆
=µ
[4]
where Q is the liquid flux (m3/s). For flow of gases, since the volumetric flow rate varies with pressure, it is
necessary to use either an integrated form of the equation or alternatively an average value of the flow rate. If
an average pressure is used, the volume at mean pressure has to be converted to a volume measured at one
atmosphere, so that Darcy's law may be expressed as [16]:
( )22
2
oi
ogvo
voPPA
PLQk
−=
µ
[5]
where kvo is the water-vapor or oxygen effective permeability (m2), Pi is the water-vapor or oxygen inlet
pressure of the network (Pa), Po is the water-vapor or oxygen outlet pressure of the network (Pa) and µg is the
gas viscosity (cP)
The mathematical model was developed using Fortran (Compaq Visual Fortran 6.6). All the
graphics results were obtained using SigmaPlot 9.0. The reported simulation results correspond to 95%-
confidence intervals around the mean of ten repetitions of each pore-size distribution. The 3D graphs were
made using the software Noesys 1.3 and T3D.
RESULTS & DISCUSSION
The mechanistic pore-level model of the transport of water vapor and oxygen is used here to find pore-level
distributions of the gas and liquid (condensate) and water vapor and oxygen permeability and diffusivity. Our
simulations results are compared with experimental results for water vapor and oxygen diffusivity given by
Del Nobile et al. [3-5], Buonocore et al. [17] and Van Krevelen [18].
Figure 2, shows images of liquid-gas patterns as the condensation occurs. At each condensation step, we
considered that a fixed volume of 100 (nm3) per each duct was condensed (as a condensation criterion), then
as the condensate volume increased some pores were sealed against water-vapor and oxygen transport; when
the percolation threshold was reached (the fraction of connected bonds was 0.248), the network was
disconnected from the water-vapor and oxygen transport.
p=1, GC=0, Sl (%)= 0 p=0.988, GC=1151, Sl (%)=2.77 p=0.941, GC=5842, Sl (%)= 3.00
p=0.858, GC= 14037,Sl (%) =3.27 p=0.648, GC= 34843, Sl (%) =3.67 p=0.248, GC= 74520, Sl (%) = 4.51
Figure 2 Condensation sequence for a three-dimensional pore network (40×20×40) with 32,000 pores and 99,200 of pore
throats ; p is the fraction of connected bonds; GC is the number of totally condensed pore throats and Sl is the network
liquid saturation (%).
Figure 3 shows liquid water, water-vapor and oxygen diffusivity curves from the simulation in the three-
dimensional network. As condensation advanced, some pores were sealed and both water-vapor and oxygen
diffusivities were diminished. On the other hand, as condensation occurs the liquid content in the pore
structure increases and the liquid permeability increases too. Here the effect of water re-evaporation and
oxygen transport in the liquid phase were not considered.
Network liquid saturation, %Sw
2 3 4 5 6 7
Effe
ctive
Diffu
siv
ity (
cm
2/s
)
0.0
2.0e-8
4.0e-8
6.0e-8
8.0e-8
1.0e-7
1.2e-7
1.4e-7
Oxygen
Water vapor
Water liquid
Figure 3. Effective diffusivity curves for three-dimensional pore network (40×20×40) obtained in this work (mean value
of ten repetitions).
Del Nobile et al. [3] present experimental values of water vapor diffusivity for three kinds of polyamides,
those values ranges between 0.5x10-9
and 5.5x10-9
(cm2/s). Specifically for quitosan films, Del Nobile et al.
[5] present two continuum mathematical models to estimate the water vapor diffusivity in function of the
water activity (aw). Their results ranges between 2x10-10 and 2.5x10-8 (cm2/s) for an ideal Fick’s model,
whereas using another modified Fick’s model their results are nearly to 2x10-10
(cm2/s). Buonocore at al. [17]
shows experimental water vapor diffusion coefficients for quitosan and their results ranges between 1x10-8
and 8x10-8
(cm2/s) for a water activity ranges between 0.3 and 0.7. On the other hand, Van Krevelen [18]
shows that the effective oxygen diffusivity for syntetic polymers ranges between 1x10-9
and 1x10-6
(cm2/s).
From Figure 3, we observe that from our model the effective water vapor diffusivity ranges between 6.4x10-9
and 4.1x10-8 (cm2/s) and the effective oxygen diffusivity obtained through the present model gives values
between 7.2x10-9
and 3.2x10-8
(cm2/s), which is agreeable with the experimental results presented by Del
Nobile et al. [5], Buonocore et al. [17] and Van Krevelen [18]. Then we can infer that the model proposed at
present work has a good predictive capability. It is important to note that the present model is based only in
morphological information of the porous medium, physical properties of the vapor water and oxygen and in
the general transport laws.
From figures 2 and 3 also it is possible to see the strong dependency of the mass transport mechanisms with
the liquid network saturation. As condensation process advances and pore throats are sealed for the transport
of water vapor and oxygen, the Knudsen transport mechanism becomes less important and other transport
mechanism like liquid transport becomes important.
Network liquid saturation, %Sw
2 3 4 5 6 7 8
Eff
ective
Pe
rme
ab
ility
(m
2)
0
2e-21
4e-21
6e-21
8e-21
1e-20
Oxygen
Water vapour
Water liquid
Figure 4. Effective permeability curves for three-dimensional pore network (40×20×40) obtained in this work (mean
value of ten repetitions).
In figure 4 we present simulation results of water-vapor, liquid water and oxygen effective permeabilities in
units of square meter (m2). A comparison between values of permeabilities obtained at present work with the
ones present in bibliography was not possible. The values reported in bibliography for the gas permeability of
films are measured gravimetrically according to the ASTM E96-80 or ASTM E96-95 standard and adapted to
edible films. These experimental results depend of the way of the experiments were carried out and also the
particular experimental conditions, presenting varied units of measurement, e.g. cm3/mm
2•s•atm. On the
other hand, the results of permeability obtained in this work (in m2) are based in a macroscopic law of
transportation (Darcy law). Moreover, this work is pioneer because is the first pore network model applied to
a packaging food material, for which, until now scientific comparable studies do not exist.
CONCLUSION
We have developed a simple pore-scale network model in order to study the water vapor, liquid water and
oxygen transport in polymeric films used in food packing. The model is based on microscopic properties.
The macroscopic transport properties obtained by the model were compared with experimental results
obtained by specialized literature given a good agreement between both. Then we can infer that the model
proposed at present work has a good predictive capability. This simple model can reproduce the transport
parameters values without any adjust parameters and only considering physical aspects of condensation
process and morphology and connectivity of the porous medium.
ACKNOWLEDGMENTS
The financial support from CONICYT-Chile through project FONDECYT N° 11060081 is gratefully
acknowledged.
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