Composites: Part A 31 (2000) 1331–1344
www.elsevier.com/locate/compositesa
Mathematical modelling of capillary micro-flow through woven fabrics
S. Amico, C. Lekakou*
The School of Mechanical and Materials Engineering, University of Surrey, Guildford, GU2 5XH, UK
Received 17 July 1999; accepted 2 February 2000
Abstract
The present work considers the capillary flow of Newtonian fluids along a single fibre yarn and through a plain-woven fabric. In the first
case, one-dimensional Darcy’s flow is considered through the micro-pores of the fibre yarns. In the second case, two-dimensional in-plane
network infiltration is considered through the micro-pores of the network of fibre yarns in the fabric. In both cases predictions include the
infiltration length as a function of time, the apparent permeability and the capillary pressure. The latter case also includes the number of
unsaturated transverse yarns and the degree of saturation. All predictions are compared to experimental measurements and the agreement is
very good. 䉷 2000 Elsevier Science Ltd. All rights reserved.
Keyword: Capillary micro-flow
1. Introduction
Resin transfer moulding (RTM) has been increasingly
used in the processing of composites. In this process, a
dry reinforcement or fibre preform is placed in the cavity
of a mould, the mould is closed and resin is forced to flow
through the fibrous reinforcement. Modelling the flow of
resin during the infiltration stage of the RTM process is
justified by the importance of this stage on the manufacturing of high quality parts. Furthermore, reliable modelling of
mould filling with an accurate flow description has the
potential of enabling a proper choice of mould and process
parameters, addressing critical issues in the process, such as
dry spots and voids [1].
Understanding of the flow requires the knowledge of the
permeability of the reinforcement, which is correlated to the
velocity of the flow front and the pressure gradient. Flow
models usually consider the flow as Darcy’s flow [2]
through a homogeneous porous medium. Although the
validity and applicability of this law through a structured
reinforcement have been questioned [3–5], Darcy’s law is
still by far the most used model to describe the flow of resin
in reinforcements during the processing of composites.
Permeability is a major property regarding the infiltration
of fibre reinforcement in RTM and, hence, many studies in
the literature have focused on experiments for the measure* Corresponding author. Tel.: ⫹ 44-1483-300800; fax: ⫹ 44-1483259508.
E-mail address: [email protected] (C. Lekakou).
ment of permeability and the analysis of experimental data
in these experiments. Amongst other factors, depending on
process conditions during the experiment of permeability
measurement, the influence of capillary pressure on the
overall flow can be of relative importance. In fact, many
of the discrepancies in the literature [6] about using Darcy’s
law to model the infiltration process in permeability experiments may be due to neglecting the effects of microscopic
flow phenomena in the interpretation of the macroscopic flow.
Regarding the low injection pressure cases, attempts have
been made to explain some of the discrepancies from
Darcy’s law by using generalised flow models including
non-Newtonian effects [7,8], capillary pressure effects [9]
and separation of the flow into macro- and micro-flow [9–
11]. In the latter studies of fibre reinforcements, micro-flow
is considered to be the flow inside the fibre yarns and macroflow is considered to be the flow in the macro-pores between
fibre yarns. Darcy’s law was employed for both macro- and
micro-flow [9–11], taking also into account capillary pressures and wetting effects [9].
Some published papers focus on microscopic studies of
void formation in composites manufacturing. Void formation has been related to resin impregnation and more specifically to viscous flow and wetting properties of the resin
related to the reinforcement. It has also been observed that
the measured global permeability may vary, depending on
the infiltrating fluid and its wetting properties [12,13]. In
flow visualisation studies [14–16] it has been observed
that at a low flow rate micro-flow was leading whereas at
a high flow rate macro-flow was leading.
1359-835X/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved.
PII: S1359-835 X( 00)00 033-6
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Nomenclature
Ay
Df
dH/dt
dP/dH
F
g
H
He
Hff
Hi
Ko
P
Pc
Pi
Qa,i
Qt,i
Qtot
u
t
Xff
Greeks
a a,i
a t,i
DXff,i
e
u
k
k mi,a
k mi,t
m
n
r
s
cross-sectional area of the yarn (m 2)
fibre filament diameter (m)
height change with time (m/s)
pressure gradient (Pa/m)
shape or form factor in the Young–Laplace equation
acceleration of gravity (m/s 2)
Height of fluid rise (m)
equilibrium height due to Pc (m)
height of the flow front (m)
height of a particular junction i (m)
Kozeny constant
pressure (Pa)
capillary pressure (Pa)
local pressure at a junction i (Pa)
axial flow rate at junction i (m 3/s)
transverse flow rate at junction i (m 3/s)
total flow rate (m 3/s)
superficial fluid velocity (m/s)
time (s)
horizontal position of the flow front of the transverse flow (m)
axial coefficient of flow at a network junction i
transverse coefficient of flow at a network junction i
horizontal distance between the transverse flow front and a network junction i (m)
porosity of the fibrous preform
contact angle between the fluid and the solid
permeability of the medium (m 2)
micro-axial permeability of the fibre yarn (m 2)
micro-transverse permeability of the fibre yarn (m 2)
fluid viscosity (Pa s)
interstitial velocity (m/s)
density of the fluid (kg/m 3)
surface tension of the wetting fluid (Pa m)
The prediction of permeability and wetting properties of a
fibre reinforcement would be a major advance for the design
stage of the processing of a composite product, especially
for RTM. This study focuses on woven fabrics. As several
layers of fabric are laid in an RTM mould, in-plane flow
during the infiltration stage is expected both in the macropores between fibre yarns and in the micro-pores within
fibre yarns. A general macro- and micro-infiltration
model, such as that suggested by Lekakou and Bader [9],
is expected to predict global permeability but such models
so far can provide only qualitative trends and need further
refinement and detailed validation to provide accurate
predictions of permeability. Because of the complex nature
of such models, which consist of a few sub-models, each
sub-model needs to be considered, refined and validated,
first, independently. The present study focuses on the
micro-flow through a woven fabric.
A mathematical model has been developed regarding
micro-flow through a two-dimensional (2D) network of
fibre yarns in woven fabrics. The term “network flow
models” as encountered in the literature so far, Refs. [17–
19] amongst others, describes the flow through a network of
tubes and corresponds to pore level simulations. The present
study considers Darcy’s network flow where the bonds of
the flow network consist of fibrous porous medium. It corresponds to micro-flow simulations at a level higher than the
micro-pores. The aim is to apply the developed Darcy’s
network micro-flow model for the prediction of the permeability and capillary pressure in flows through woven fabrics
and to validate the model with experimental flow data. In
order to be able to isolate the micro-flow through a network
of fibre yarns, capillary flow experiments through a single
layer of a plain-woven fabric were designed for model validation. This report follows an earlier study [20] which
describes the capillary experiments to follow the capillary
impregnation of an E-glass fibre yarn and a plain-woven
fabric. The experimental procedure and the obtained results
have been described in this earlier study. Apart from being
S. Amico, C. Lekakou / Composites: Part A 31 (2000) 1331–1344
1333
Fig. 1. Photograph of the infiltration of the fabric; (a) at the flow front and (b) close to the liquid surface of the container.
one of the sub-models for infiltration of an assembly of
woven fabrics in RTM, the model is potentially important
in RTM applications with high fibre volume fraction where
macro-pores are expected to be very small and the flow
through the network of fibre yarns in each fabric layer may
be a decisive factor in the infiltration of fabric reinforcement.
2. Modelling of Darcy’s flow through a fibre yarn
In this model, the fibre yarn is considered as a tow of
unidirectional fibres. The aim is to numerically evaluate
the vertical advancement of the fluid front along the axial
yarn direction. Darcy’s law is used to model the flow
through the yarn micro-pores, correlating flow-rate and
pressure drop according to Eq. (1).
dH
k dP
ˆ⫺
dt
m dH
…1†
k Pc ⫺ rgH
nˆ
me
H
…2†
uˆe
or
For resin systems which show highly non-Newtonian behaviour, a viscosity–shear rate relationship such as the power
law must be employed [21].
The theoretical value for the capillary pressure is estimated from the Young–Laplace equation as suggested by
Ahn and Seferis [22]:
Pc ˆ
F …1 ⫺ e†
s cos u
e
Df
…3†
where F is a dimensionless shape or form factor which
depends on the flow direction. For axial flow, it has been
derived [23] that F ˆ 4 on the basis of the principle of
hydraulic radius of a fibre bed. The micro-pores are assumed
to be uniform or with an averaged pore size distribution and
fibre twist in the yarn and any defects at the fibre surface
were neglected.
The analytical solution of Eq. (2) is rather complex [20],
requiring an iterative numerical procedure to evaluate the
height of fluid rise as a function of time. A procedure based
on the finite difference method is employed for the solution
of Eq. (2) in this study. The fluid is considered to advance in
length steps during the corresponding time steps. At every
new step, the time is increased by a constant value and the
length step is evaluated. Error estimation monitors whether
the length step is appropriate, i.e. error smaller than a given
tolerance. Otherwise, the length is re-evaluated until numerical convergence.
The model was implemented in a simple computer code
written in Fortran77. This program allows for different
Newtonian fluids by inputting the density and viscosity
values. The fibre radius, micro-porosity (i.e. porosity of
the yarn), micro-axial permeability and capillary pressure
are the other input parameters.
The Carman–Kozeny equation [23], given by relation
(4), is generally used to estimate the value of micro-axial
permeability on the basis of the value of micro-porosity.
kˆ
D f2
e3
16Ko …1 ⫺ e†2
…4†
The value used for the Kozeny constant, Ko, is discussed in
Section 4.1. It is important to remember at this point that
Gauvin et al. [24] stated that flow predictions are as good
as the accuracy of the permeability value entered in the
simulation.
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Fig. 2. Simplified representation of the network flow in the plain-woven fabric.
The value of capillary pressure was best estimated from
the Young–Laplace equation (3), for specific values of
micro-porosity, the surface tension of the liquid (measured
by using a DuNuoy ring apparatus [20]) and the contact
angle between the liquid and the E-glass fibre (measured
by the single fibre pull-out test using a dynamic contact
angle analyser, DCA-322 CAHN, which measures the
force exerted on a fibre slowly lowered in a beaker containing the fluid [20]). The capillary pressure is the driving force
responsible for the rise of the liquid in the yarn. As the
infiltration starts, the weight of the column of liquid in the
yarn acts against the capillary pressure, slowing down
the rate of infiltration (dH/dt).
The numerical time step used in the computer simulations
was usually of 1 s. However, if the final equilibrium position
of the flow is required, this time step may be increased so
that less computational effort is used.
3. Modelling the vertical capillary flow through a plainwoven fabric
It is necessary to understand the flow pattern in the plainwoven fabric in order to realistically model the flow. Fig. 1
shows an experiment of capillary infiltration of a vertical
piece of plain-woven fabric, Y0212 supplied by Fothergill
Engineered Fabrics, at the flow front (Fig. 1a) and close to
the liquid surface of the container (Fig. 1b). Three major
macroscopic features can be noticed:
1. The flow front is not homogeneous, with some yarns
showing further flow advancement. Since other factors
concerning the flow are supposed to be constant, this is
likely to happen due to characteristics of the individual
yarns, especially the pore distribution of individual yarns.
2. The flow progresses axially in the vertical yarns whereas
the horizontal yarns are filled in such a way that there are
at least three partially filled transverse yarns at any time.
The precise number cannot easily be estimated by visual
observation of the flow alone.
3. There are clear unfilled gaps between the yarns (macropores). These gaps can be seen even in the portion of the
fabric closest to the liquid surface of the container (Fig.
1b) and were confirmed by careful visual observation of
the infiltrated fabric. Therefore, the vertical capillary
infiltration process through a plain-woven fabric can be
modelled considering the macro-flow to be absent.
Since the model only considers the presence of microflow, the reinforcement is represented as a network of
porous fibre yarns linked at junctions. Fig. 2 shows a schematic representation of the fabric with its repeating basic
column.
Fig. 3 describes the pressures acting at a single junction i.
Flow in the transverse yarn is driven by the capillary pressure, acting at the flow front at a horizontal distance Xf f.
Flow in the axial yarn is driven by the capillary pressure
acting at the flow front (Hf f) but, in this case, the weight of
the column of liquid above the junction originates a counter
pressure Pi at this i junction (Hi), where Pi ˆ rg…Hff ⫺ Hi †:
The axial flow-rate, Qa,i, and transverse flow-rate, Qt,i, at a
given junction i will then, respectively, be:
Qa;i ˆ Ay
kmi;a …Pc ⫺ rg…Hff ⫺ Hi ††
m
…Hff ⫺ Hi †
…5†
Qt;i ˆ Ay
kmi;t …Pc ⫺ rg…Hff ⫺ Hi ††
m
DXff;i
…6†
S. Amico, C. Lekakou / Composites: Part A 31 (2000) 1331–1344
1335
Fig. 3. Flow advancement and pressures at a given junction i.
where DXff;i is the horizontal distance from the flow front to
the junction i and …Hff ⫺ Hi † is the height difference
between the flow front and the junction.
At every junction the flow splits between axial and transverse yarns according to the coefficients defined by Eqs. (7)
and (8), where aa;i ⫹ at;i ˆ 1:
aa;i ˆ
Qa;i
Qa;i ⫹ Qt;i
…7†
at;i ˆ
Qt;i
Qa;i ⫹ Qt;i
…8†
A similar analysis is carried out for each unsaturated junction (with a transverse yarn that has been reached by the
axial flow front but has not been completely filled yet)
throughout the duration of the experiment. For instance, if
for a given time there are three unsaturated junctions (Fig. 4)
according to mass balance, the total flow rate at that time
will be:
Qtot ˆ ‰Qt;i ⫹ Qt;i⫹1 ⫹ Qt;i⫹2 Š ⫹ Qa;i⫹2
…9†
Thus, it can be said that for a junction i the flow split will be
given by:
Qa;i ˆ Qa;i⫺1 aa;i
…10†
Qt;i ˆ Qa;i⫺1 at;i
…11†
The transverse flow will be responsible for filling the transverse yarns whilst the axial flow will proceed until the next
junction is reached and at this time a new flow split will
occur. A few models can be proposed, regarding the
unknowns a a and a t.
Model 1: a a is the same at all unsaturated junctions
throughout the infiltration process; this model ignores the
effect of the flow advancement in the axial and transverse
direction on the flow split. A value for a a is selected at the
beginning of a computer simulation and the flow-rate is split
at junctions according to Eqs. (10) and (11). This is the
simplest of the three models and a a is considered really as
an empirical constant, being estimated on the basis of the
values of the axial and transverse micro-permeabilities as a
first approximation. Accurate values of a a can only be
provided from the fitting of the final infiltration predictions
Fig. 4. Schematic representation of the flow distribution for three unsaturated junctions, junctions i, i ⫹ 1 and i ⫹ 2:
to the experimental data. The use of this model in this study
is to improve the understanding of the Darcy’s network flow
through the woven fabric.
In the following Models 2 and 3, the coefficients relevant
to the flow split at junctions are evaluated on the basis of
relations (12) and (13) which can be derived directly from
Eqs. (5)–(8):
aa;i ˆ
kmi;a DXff;i
kmi;a DXff;i ⫹ kmi;t …Hff ⫺ Hi †
at;i ˆ 1 ⫺ aa;i
…12†
…13†
More specifically:
Model 2: a a is the same at all junctions but varies with
infiltration time; this model considers an averaged flow
advancement length, expressed by average values of …Hff ⫺
Hi † and DXff;i over the region of unsaturated junctions near
the flow front. This model is expected to give good predictions of a a if the length of the unsaturated region near the
flow front is small in comparison to the total length of
infiltration in the axial direction.
Model 3: a a varies depending on junction and time
according to Eq. (12), thus, taking detailed account of all
effects on flow split. This model is the most comprehensive
model of all three models for a a and is expected to give very
good predictions.
Models 2 and 3 require a a, and consequently a t, to be
estimated regarding current local pressures and flow
distances as illustrated by Figs. 3 and 4. An iterative procedure is involved in Models 2 and 3, where for an estimated
flow advancement, the coefficient of flow split is evaluated
according to Eq. (12), the flow-rate is split according to Eqs.
(10) and (11) and new lengths of the flow advancement in
the axial and transverse directions are calculated on the
basis of the estimated flow-rate values. The procedure is
then repeated until numerical convergence has been
reached.
The model of Darcy’s network micro-flow was
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S. Amico, C. Lekakou / Composites: Part A 31 (2000) 1331–1344
permeability, the distance between two junctions and the
length of the transverse yarn are also used as input values.
The transverse micro-permeability of a fibre yarn is estimated as 19 times lower than the axial micro-permeability
[25].
In this program, the transient infiltration process is
regarded as a numerical sequence of steady state solutions
at each time step where the finite difference method is
employed. If there are junctions with unfilled transverse
fibre yarns (unsaturated junctions), the flow is split between
the axial and transverse yarns and a new height and transverse position of the flow front is calculated. If new junctions are reached, the number of unsaturated junctions is
increased. If a junction is fully filled in the transverse direction, that junction is considered saturated. This process is
repeated for each time step until equilibrium is reached.
At any time, the number of unsaturated junctions, the
time necessary for an individual junction to saturate, the
position of the flow front in the transverse unsaturated
yarns and the height of the axial flow front are known,
allowing the description of the flow.
Fig. 5. Flowchart of the computer program used to model the impregnation
of the woven fabric.
4. Results and discussion
4.1. Numerical studies for a single yarn
implemented in numerical form in a Fortran77 computer
program described in Fig. 5. The yarns are considered to be
uniform and equally spaced. Capillary pressure and permeability values are estimated according to experimental data
[20] and the Young–Laplace and Carman–Kozeny equations, respectively. The overall porosity is considered to
be constant, since the intricate architecture of the fabric is
expected to restrict variations in the cross-sectional area of
the yarns and the gap between them. Thus, the porosity of
the yarn has the same value for all runs, independently of the
infiltrating fluid. A few other parameters, namely the density
of the liquid, the radius of the yarn, the transverse micro-
Figs. 6 and 7 present a sensitivity analysis for the vertical
capillary flow of a model Newtonian fluid in an E-glass fibre
yarn. Fig. 6 illustrates the influence of the permeability
value on the final curve. As expected, an increase in k
increases the rate of the flow height rise. The experiment
takes less time to approach equilibrium. It is important to
mention that the three curves in this figure will eventually
reach the same equilibrium position, which is dependent
only on Pc. Likewise, if k is kept constant, an increase in
Pc also shifts the flow height curve upwards (Fig. 7).
Fig. 8 shows typical predicted flow curves for different
Fig. 6. Vertical axial capillary flow in a single yarn: curves for different permeability values; Pc ˆ 4013 Pa for all curves.
S. Amico, C. Lekakou / Composites: Part A 31 (2000) 1331–1344
1337
Fig. 7. Vertical axial capillary flow in a single yarn: curves for different Pc values; k ˆ 5:5 mm2 for all curves.
pairs of permeability and capillary pressure values as well as
experimental data from a capillary experiment [20]. In the
capillary flow experiments [20], a fibre yarn from an Eglass, plain woven fabric, Y0212 supplied by Fothergill
Engineered Fabrics, was suspended over the surface of a
Newtonian liquid with the lower end of the yarn, attached
to a small weight, was immersed in the liquid. The experimental data included the weight increase of the fibre yarn
being infiltrated and the progress of the flow front of the
liquid as it infiltrated the fibre yarn in vertical (upwards)
flow under the action of capillary pressure. From the relation
between the corresponding curves of weight and fluid height
rise with time [20], the area fraction of the pores of the fibre
yarn available for fluid flow was estimated and was considered to be equal to the micro-porosity of fibre yarn. The
experimental value of micro-porosity was then used in
simulations where the infiltration predictions were
compared to the corresponding experimental data. The
predicted curve in Fig. 8, which used data derived from
the experiments, Pc ˆ 352:3 Pa and k ˆ 78:9 mm2 ; gives
a fair representation of the experimental points. As
expected, whenever the values for Pc and k obtained from
a regression analysis of the experimental data are used as the
input values for simulation runs, the modelled curve reasonably represented the data for the duration of the experiment.
If purely theoretical values from the Young–Laplace
equation for capillary pressure …Pc ˆ 4443 Pa† and the
Carman–Kozeny equation for permeability …k ˆ
12:1 mm2 † are used where F ˆ 4 and Ko ˆ 0:30; the curve
overestimates the experimental data. This is attributed to the
fact that insufficient time has been allowed for the experiment, resulting in misleading estimations of Pc and k .
If one considers that the theoretical value of Pc from the
Young–Laplace equation was correct, i.e. e , F, s and Df
were correctly estimated or measured, a permeability value
of about 3.4 mm 2 could be found in order to match the data.
Fig. 8. Short-term vertical axial capillary flow of silicone oil in a single yarn: predictions for different pairs of capillary pressure and permeability values.
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S. Amico, C. Lekakou / Composites: Part A 31 (2000) 1331–1344
Fig. 9. Vertical axial capillary flow of a Newtonian epoxy resin in a single yarn—long-term experiment: predictions for different pairs of capillary pressure and
permeability values.
On the other hand if k from the Carman–Kozeny equation
was correct, i.e. e , Ko and Rf were correctly estimated or
measured, a Pc of about 1300 Pa could be predicted. Other
experimental data were analysed using the procedure of
changing the input values in an analogous way and similar
findings were reached.
Thus, it can be seen that several pairs of Pc and k can
satisfy the experimental data. Thus, for an accurate estimate
of Pc, for example, the permeability value k must be
previously known. The possible combinations of Pc and k
are much more restricted if a long-run experiment is
analysed (see Fig. 9). In this case, it can be seen that the
pair of P c ˆ 4019 Pa and k ˆ 5:5 × 10 ⫺12 m2 results in a
curve very close to the experimental data [20] and virtually
coincident to the pair Pc ˆ 4013 Pa and k ˆ 5:46 ×
10⫺12 m2 derived from the regression analysis of the experimental data.
This estimated Pc value agrees with the Pc value derived
from the Young–Laplace equation where F ˆ 4: A combination of short- and long-run experiments of capillary infiltration of the E-glass fibre yarn used in this study and the
comparison of experimental data with infiltration predictions yielded the conclusion that the best values of Pc can
be derived from the Young–Laplace equation, where F ˆ
4; and the best values for the micro-axial permeability can
be derived from the Carman–Kozeny equation, where Ko ˆ
1:58 for e in the range of 0.60–0.47 and Ko ˆ 5:77 for e in
the range of 0.20–0.30.
Considering the presented data it could be said that if
the experiment is not given sufficient time to approach
Fig. 10. Five short-term silicone oil experiments through a plain-woven fabric and predictions of Model 1 for different values of a a (0.01–0.999).
S. Amico, C. Lekakou / Composites: Part A 31 (2000) 1331–1344
1339
Fig. 11. Five short-term silicone oil experiments through a plain-woven fabric and predictions of Models 2 and 3.
equilibrium, the proposed model is likely to lead to an
underestimation of Pc, consequently increasing k . Although
a wide number of combinations of these parameters can fit
the experimental data for short infiltration time, such combinations are more restricted for longer infiltration times,
making the fitting procedure more reliable.
4.2. Numerical studies for the woven fabric
The network Darcy’s flow model for the infiltration of a
fabric layer was validated by the experimental data of
experiments [20] of the capillary flow of a Newtonian
fluid through an E-glass plain-woven fabric, Y0212
supplied by Fothergill Engineered Fabrics. In these experiments, a rectangular piece of fabric was suspended over the
surface of the Newtonian liquid, with one edge of the fabric
immersed in the liquid. The progress of the overall upward
flow of the liquid was monitored in the infiltration of the
fabric under the action of capillary pressure (also see Fig. 1).
The height versus time data for five experiments [20] carried
out for up to 5.5 h of infiltration are shown in Fig. 10. The
experimental points are reasonably close and minor deviations can be attributed to inherent characteristics of the
fabric or its handling prior to the experiment.
Fig. 10 also shows curves predicted by Model 1, when a a
was pre-set at 0.01, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70,
0.80, 0.90, 0.99, 0.995 and 0.999. Besides, it is shown what
the flow curve would be if the flow was modelled as through
a single yarn with the same porosity as that of the fabric
(upper curve in Fig. 10). The curves for a a in the range of
0.01–0.90 are practically coincident. This can be explained
by the fact that the faster the axial flow (for a higher a a) the
more junctions will be reached, and hence, the axial flow
becomes further reduced, specially as a t is low and there are
many unsaturated junctions. In the runs for a lower a a the
flow will not only reach junctions at a lower rate but also,
due to its higher a t, saturate those junctions faster.
Curves for a a in the range of 0.99–0.9999 shift from the
others due to the fact that the vast majority of the flow is
being used in the axial direction, tending to neglect the
presence of transverse flow, especially for aa ˆ 0:999:
Hence, by the end of 20 500 s all junctions reached by the
flow are unsaturated.
Models 2 and 3 are shown in Fig. 11. Model 2 (the same
a a for all junctions for a specific time) slightly overestimates the experimental data whilst Model 3 shows good
agreement with the experiment for most of the time.
When the time reaches around 20 500 s, Model 2 has
reached 55 junctions, with 4–5 of them still unsaturated at
the flow front and an a a within the range of 0.82–0.83.
The determination of a single a a for all unsaturated junctions (see Eq. (12)) requires the use of an average value for
the difference between the height of the axial flow front and
the height of a particular unsaturated junction …Hff ⫺ Hi †:
The same is required for the position of the transverse flow
front at each unsaturated junction (Xf f). The averaged values
will be responsible for keeping the a a varying in a particularly narrow range (0.82–0.83). The number of unsaturated
junctions, consequently, fluctuates between 4 and 5.
As could be expected, the use of a different a a for each
junction, according to Eq. (12), generates results closer to
the experimental ones for the curve of Model 3. Values of a a
were in the range of 0.67–0.81 depending on the position of
the junction. By the end of the experiment, there were
between 3 and 4 unsaturated junctions which agrees with
the photograph of the flow in Fig. 1a.
In order to verify the findings, a long-run experiment was
also analysed according to the three models. In the first
model, a a was pre-set to 0.01, 0.10, 0.20, 0.30, 0.40, 0.50,
0.60, 0.70, 0.80, 0.90, 0.99 and 0.99999 for each run and the
resulting curves can be seen in Fig. 12. Here again, the
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Fig. 12. Long-term silicone oil experiment through a plain-woven fabric and predictions of Model 1 for different values of a a (0.01–0.99999).
curves for a a in the range of 0.01–0.90 are practically coincident and, for higher a a values, the curve shifts upwards.
The histogram in Fig. 13 shows the number of unsaturated junctions by the time the run was interrupted. As
expected, the higher a a is, the more unsaturated junctions
are present. Runs for a low a a “use” most of the total flow to
close the unsaturated junctions, and at the moment of closing an unsaturated junction, the axial flow receives a boost
because there will be one less unsaturated junction. On the
long term, both factors tend to counter-balance, being
responsible for generally coincident flow behaviour for
runs with a dissimilar a a. For aa ˆ 0:99999 the number of
unsaturated junctions is equal to the number of open junc-
tions, which means that no junction was closed due to
saturation in the transverse direction.
If the findings from the photographs (Fig. 1) were to be
compared to this histogram, a value for a a of around 0.70
could be considered the best one in making the model agree
with the experimental observations (three unsaturated junctions at the flow front).
The results of Model 2 (Fig. 14) show a maximum height
error of approximately 10% in comparison with the experimental data over the period covered (approximately
19 days), 4–5 unsaturated junctions at the flow front and a
value for a a in the range of 0.82–0.83.
The curve predicted by Model 3 is presented in Fig. 15,
Fig. 13. Histogram showing the number of unsaturated junctions for different values of a a.
S. Amico, C. Lekakou / Composites: Part A 31 (2000) 1331–1344
1341
Fig. 14. Long-term silicone oil experiment through a plain-woven fabric and predictions of Model 2.
where 342 junctions were reached. The agreement was
excellent and the error was in the range of less than 1% in
comparison with the experimental data. As expected, the use
of an individual a a for each junction, which considers the
flow distance from that specific junction instead of an average value for all unsaturated junctions, generates results
closer to the experimental data. Values of a a were in the
range of 0.67–0.81 depending on the position of the junction and by the end of the experiment, there were between 3
and 4 unsaturated junctions. This is in accordance to what
was expected from the analysis of the photographs of the
actual flow (Fig. 1a) which was used to predict that at least
three junctions were partially filled at the flow front.
In order to illustrate how a a varies with time in Model 3,
data from the computer program for three consecutive junctions at three different time steps are shown in Table 1. The
number associated with the junction represents the time step
at which the junction was reached by the axial flow front.
The lower the number of the junction, the closer it is to the
liquid surface of the container and therefore, the earlier it
was reached.
Analysis of the table shows that for a particular time, the
further a junction is from the flow front, the higher a a is, e.g.
for the time step number 5764, aa;5161 ˆ 0:79 and aa;5631 ˆ
0:67: This characteristic cannot be so promptly derived from
Eqs. (7) and (8) since both Qa,i and Qt,i depend on the height
of the junction. Actually, both Qa,i and Qt,i increase with the
height of the junction but the percentile increase of Qt,i is
larger than Qa,i. The overall effect is to decrease a a for the
upper junctions.
Another feature is that a a increases with time for a junction, e.g. a a,5161 increases from 0.79 to 0.80. Since the total
Fig. 15. Long term silicone oil experiment through a plain-woven fabric and predictions of Model 3.
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Table 1
Values of a a for three different junctions at three different time steps
Junction
Table 2
Range of a a for three consecutive junctions and the time when the junctions
were open and closed
aa
Junction Opening
Time step 5764
Time step 5765
Time step 5916
0.791781
0.729168
0.671957
0.791896
0.729758
0.671958
0.805049
0.775094
0.683214
aa
5161
5393
5631
5161
5393
5631
flow rate decreases with time according to Eq. (1), both Qa,i
and Qt,i decrease with time and the percentile decrease of Qt,i
is larger than Qa,i, generating this effect.
Table 2 presents the same three junctions and their
respective values of a a at the time they were open and
closed. As above, a junction presents an increase in its a a
with time. Another expected result from this table is that the
further the junction is from the surface of the liquid of the
container, the longer it takes to be fully saturated (time
interval data). The values for time interval throughout the
duration of the modelling vary from 168 s for the first saturated junction—closest to the container—to 38180 s for the
338th junction (last saturated junction).
The predicted curve of capillary flow of the silicone oil
through the woven fabric up to equilibrium is shown in Fig.
16. In this graph, the final equilibrium position can be seen,
in this case, He ˆ 1:098 m—corresponding to Pc ˆ
9216 Pa; along with the experimental data. By the end of
the computational run 718 junctions had been reached by
the fluid, taking approximately 156 days to saturate the last
junction (the 714th one).
This curve allows one to predict the time necessary for
the equilibrium to be reached, when the final position (Hff) is
an arbitrary percentage of the expected equilibrium height
(Table 3). This table illustrates a well-known feature of
capillary experiments, the required long duration of experiments. Furthermore, the experimental data has to be
collected for a minimum period so that reliable Pc and
permeability values can be extracted [20,26]. In this case,
for 90% of the equilibrium height (He) to be reached, the
experiment would take approximately 156 days, and this
number would increase to around 403 days for Hff ˆ
0:99He :
Perhaps one of the most important parametric studies, due
to its practical value, is the influence of the viscosity on the
final curve. Fig. 17 shows three curves generated at different
input viscosity values (0.108, 0.120 and 0.132 mPa s). As it
is expected due to Darcy’s law, the higher the viscosity, the
slower the flow. It is important to mention that all three
Closing
Time (s) a a
0.7654342 20641
0.7654325 21569
0.7654303 22521
Time (s)
0.805049 23661
0.805044 24657
0.805079 25677
3020
3088
3156
curves will eventually reach the same final position—the
equilibrium height, which is dependent only on Pc. Taking
this into consideration, it is interesting to notice that, for the
studied time interval, a change of 10% in viscosity is
responsible for a shift of the curve of up to 22 mm (4.1%)
in height. Moreover, if a considerable change in viscosity of
the infiltrating fluid is likely to happen during experiments,
monitoring of viscosity has to be carried out to provide
appropriate data for the computer program for correct
modelling of the process.
5. Conclusions
The scope of this study has been to develop and validate a
model for a network Darcy’s micro-flow in the in-plane
infiltration of woven fabrics. After the model was implemented in an algorithm, the first task when running the
algorithm was to provide appropriate input values for
the micro-permeability and capillary pressure of the yarns
of the fabric for a known micro-porosity. The methodology
of these estimations was concluded from the simulations of
the infiltration of a single yarn.
The suggested numerical model for the axial capillary
flow through a single yarn was validated by demonstrating
that estimated values for Pc and k from the previously
reported capillary experiments do represent the experimental infiltration data for the duration of the experiment.
Although different combinations of Pc and k can fit the
data for the initial portion of the flow curve, the possibilities
are restricted for long run experiments, becoming more
accurate and proving the importance of the duration of the
experiment when mathematical curve fitting is used. The
conclusion from the single yarn infiltration simulations
was that the capillary pressure of a single fibre yarn can
be estimated from the Young–Laplace equation, with F ˆ
4 for axial flow in the yarn, and the axial micro-permeability
can be estimated from the Carman–Kozeny equation, with
Table 3
Range of a a for three consecutive junctions and the time when the junctions were open and closed
Time (s)
Time (days)
Time interval (s)
Hff ˆ 0:90He
Hff ˆ 0:99He
Hff ˆ 0:999He
Hff ˆ 0:9999He
1:35 × 107
⬇ 156
3:48 × 107
⬇ 403
5:69 × 107
⬇ 659
8:04 × 107
⬇ 931
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1343
Fig. 16. Modelled curve showing the final equilibrium position.
Ko ˆ 1:58 for e ˆ 0:60 ⫺ 0:47 and Ko ˆ 5:77 for e ˆ
0:20 ⫺ 0:30:
The Darcy’s network micro-flow model was tested by
using experiments where a sample of fabric was impregnated by a Newtonian fluid in upwards, vertical flow
under the action of capillary pressure. This type of experiment was selected because it produced no macro-flow, so
that the micro-flow model could be tested independently.
Three models of different degree of accuracy regarding
the evaluation of flow split coefficient, a a, at the network
junctions were tested. Model 1, which employed a constant
pre-set input value for a a, showed that the main effect of
accuracy of a a lies in the prediction of the number of unsaturated junctions behind the flow front: an a a value in the
range of 0.7–0.8 provided predictions close to the experimental data. Model 2, in which a a was evaluated inside the
algorithm while making the assumption that the unsaturated
region by the flow front is small compared to the whole
infiltrated region of fabric, yielded predictions which
differed by as much as 10% in comparison with the infiltration data of the experiment, for 4–5 unsaturated junctions at
the flow front. Model 3, in which a a was evaluated inside
the algorithm by taking all the terms of the model into
account, showed the best agreement with the experimental
results, since it allows a more appropriate distribution of the
flow, with a varying flow split factor, a a, in the range of
0.67–0.82, which was in accordance with experimental data
and photographs describing the number of unfilled transverse yarns at junctions by the flow front.
The successful validation of the network Darcy’s model
for woven fabrics means that one of the sub-models of the
infiltration of an assembly of woven fabrics in RTM is
Fig. 17. Sensitivity analysis regarding fluid viscosity (m values in Pa s).
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certified to yield independently accurate micro-infiltration
predictions. This is a step forward in the process of
assembling an overall macro- and micro-flow model for
the accurate quantitative prediction of permeability of
woven fabrics in RTM. The micro-infiltration sub-model
of this study is particularly important in cases of high
fibre volume fraction where micro-flow is significant.
Acknowledgements
[12]
[13]
[14]
[15]
This work is part of a PhD degree sponsored by CAPES,
Brazil.
[16]
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