2006_CompA_perm_Advani.pdf

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Permeability characterization of dual scale fibrous porous media

Nina Kuentzer a, Pavel Simacek a, Suresh G. Advani a,*, Shawn Walsh b

a Department of Mechanical Engineering, Center for Composite Materials, University of Delaware, Newark, DE 19716, USAb Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5066, USA

Received 17 June 2005; received in revised form 7 December 2005; accepted 7 December 2005

Abstract

The proper characterization of fabrics used in liquid composite molding (LCM) is integral to accurately model the flow through theseporous preforms. The dual-scale nature of many fabrics has brought about a need for a methodology, which characterizes not only thebulk permeability of the preform, but the micro-scale permeability of the fiber tows. These two permeability values can then be used inLCM simulations that can separately track the bulk flow front progression and the saturation of the fiber tows in preforms that exhibitdual-scale porosity. A three dimensional simulation called liquid injection molding simulation (LIMS) has been developed at the Uni-versity of Delaware that can predict the impregnation of the fiber preform with resin in closed molding processes such as resin transfermolding (RTM) and vacuum assisted resin transfer molding (VARTM). To address the dual-scale porosity, standard 2D or 3D meshelements are combined with 1D elements, which are attached at each node and represent the fiber tows. This implementation allowsfor the interactions between the bulk and micro flow, and can predict the saturation of the fiber tows, along with the movement ofthe bulk resin flow front. However, it does require two permeability inputs: one for the elements representing the bulk preform andanother for the 1D elements representing the fiber tows. A methodology is proposed to determine the bulk permeability and a parameterthat is closely associated with the micro permeability of the tows for dual-scale fabrics. This is accomplished by comparing the inlet pres-

sure profiles of one-dimensional constant flow rate injection RTM experiments with a simulation of flow in a dual-scale fabric. The meth-odology is validated and characterizations for four different fabrics are performed to demonstrate the versatility and limitation of themethodology. 2006 Elsevier Ltd. All rights reserved.

Keywords: A. Fabrics/textiles; E. Cutting

1. Introduction

It has been known for some time that woven andstitched fabrics inherently induce a dual-scale flow behav-

ior [1,2]. These porous preforms are comprised of distinctbulk- and micro-flow regions, which exhibit distinctly dif-ferent impregnation rates and, therefore, each correspondto a different permeability value. The pores of the bulkregion will fill more rapidly, while the micro space betweenfibers within the tows, which are bundles of individualfibers, will saturate more slowly, due to the smaller size

of the pores. The size of the pores between the tows, thediameter of the fibers within a tow, and the architectureof the preform all play a role in the determination of thepreform permeability.

The distinction between these separate bulk- and micro-flow regions is important, due to the delay that occurs inthe complete saturation of the preform. The bulk or themacro flow front first fills the more permeable poresbetween the tows; then, the fluid from the bulk region fillsthe less permeable pores within the tows, at which point themore slowly advancing micro flow front dictates the fillingprogression. For this reason, the location of the bulk flowfront seldom coincides with the boundary of the preformthat is fully saturated with fluid. For such preforms, a par-tially-saturated zone exists and can be defined as the region

1359-835X/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compositesa.2005.12.005

* Corresponding author. Tel.: +1 302 831 8975; fax: +1 302 831 8149.E-mail addresses: [email protected], [email protected] (S.G. Ad-

vani).

www.elsevier.com/locate/compositesa

Composites: Part A 37 (2006) 20572068
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in which the pores of the bulk region may be filled, but thefiber tows are only partially filled. In one-dimensional flow,one can define a partially-saturated length, Ls, which is

the distance between the bulk flow front and the fully-saturated front, as depicted in Fig. 1. The tows ahead ofthe bulk flow front are completely unsaturated and thetows behind the fully-saturated front are completelysaturated.

The partially-saturated length decreases as the micropermeability of the fiber tows gets closer to the bulk perme-ability of the preform. Single-scale fabrics, such as mostrandom mats, only have one length scale and, therefore,only one permeability value. They are the most extremecase where the micro permeability is inherently equal tothat of the bulk permeability, since there are no distinct

fiber tows; accordingly, these fabrics have no partially-sat-urated length.

In order to characterize the permeability of a dual-scalefabric preform, the pressure history at the injection gate ofa one-dimensional, linear constant flow rate experiment isused to determine the bulk permeability and a parameterthat is associated with the tow permeability. Darcys law,which is used to describe flow through porous media, mustbe correctly applied to the bulk and the tow regions due tothe presence of a delayed impregnation in the tows for suchfabrics. For single-scale fabrics, the inlet pressure for one-dimensional constant flow rate experiment exhibits a linearbehavior with time. When experimentally evaluating dual-scale fabrics, this linear trend only holds for two limitingcases.

On one extreme the permeability of the tows may be ofthe same order as the bulk permeability; therefore, bothregions fill simultaneously and the pressure as a functionof time grows linearly. At the other extreme, the permeabil-ity of the tows may be many orders of magnitude smallerthan the bulk permeability, in which case the tows remaincompletely empty (act as solid rods) in the time it takes tofill the bulk regions. For this case too, the pressure as afunction of time grows linearly, but more quickly thanthe previous case. The permeability that is measured from

the linear pressure profile for the former case represents the

saturated bulk permeability as the tows are completelyfilled while in the later case it is the unsaturated bulk per-meability as the tows are completely dry.

For the linear cases outlined above, the conventionalDarcys law is valid and can be expressed as

hmdi K

l rp. 1

This relation expresses the average Darcy flow velocity vec-tor, hmdi, as a function of the permeability tensor of thepreform, K, the viscosity of the test fluid, l, and the pres-sure gradient, $p. When applying (1) to a planar one-dimensional flow experiment, Darcys law can be simplifiedto 1D and re-expressed as

hmdi K

l

dp

dx; 2

where the bulk permeability in the flow direction, K, is ascalar value. Since the test fluid is injected into the preformthrough a mold cross-section, A, at a constant flow rate, Q,the definition of velocity, hmdi

Q

A, can additionally be re-

lated to Darcys law. The 1D Darcy relation can then bere-cast in terms of the changes in the injection pressure,Pin, and the position of the bulk flow front, x, with respectto time:

hmdi K

l

m

dxdt

; 3

where dPindt

m is the slope of the inlet pressure profile withrespect to time for a constant flow rate injection experi-ment. To enable one to find the bulk permeability, the flow

front velocity, mff, is expressed in terms of the Darcy aver-age velocity by dividing it by the available porous volume,/, as dictated by mass conservation:

dx

dt mff

hmdi

/. 4

Eq. (4) is substituted into Eq. (3) and the definition ofvelocity, Q

A, is implemented. The rearrangement of terms

results in an explicit expression for the bulk permeabilityof the preform in the direction of the flow:

K l

m/

Q

A

2

. 5

Clearly only one permeability value can be ascertainedfrom Eq. (5). The micro-scale impregnation is in effect ne-glected, which is a valid assumption for a single-scale pre-form or a dual-scale preform in which the tows exhibit thebehavior of the two limiting cases mentioned in which thetows are either completely unsaturated or completely satu-rated. For these cases, one expects that for a constant flowrate injection, the change in pressure at the inlet with re-spect to time will be linear, resulting in a constant valuefor the slope, m.

It is evident from Eq. (5) that a steeper slope will result

in a lower permeability value. When the fiber tows of a

Flowfront

Fully-saturated

front

Fiber tows

Ls UnsaturatedSaturated

Fig. 1. A schematic that depicts the saturated, partially saturated (Ls),and unsaturated flow regions within dual-scale preforms, and highlightsthe delayed impregnation of the fiber tows.

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preform remain completely unsaturated as fluid flowsthrough a dual-scale fabric, the pressure grows linearlyand quickly. The dry tows are providing additional resis-tance, resulting in a steeper slope and hence a lower perme-ability value. If the tows are completely saturated, it issuspected that the fluid inside the tows is stagnant and only

the fluid outside the tows is flowing. There is less resistanceto the flow in this case since the tows are filled with fluidand provide lubrication for the moving liquid. The pressurestill grows linearly, but not as quickly as in the former case,resulting in a less steep slope and hence a larger permeabil-ity value as compared to the unsaturated case. Such find-ings were also reported by Breard et al. [3].

The region between the two extreme linear cases repre-sents the true nature of the flow. The tows at the inceptionmay be completely unsaturated; therefore, the pressureprofile with respect to time at the inlet follows the lineartrajectory of a completely unsaturated preform. As thefiber tows start to draw the resin, this profile becomes

non-linear and once the tows are completely saturated willapproach the linear behavior exhibited by a completely sat-urated preform. The impregnation of fluid into the fibertows during this non-linear phase is addressed by applyingmass conservation with a sink term to the bulk flow:

r hmdi qp;s. 6

The fiber tows act as sinks, q, which drain away fluid fromthe advancing bulk flow front and these sinks are a func-tion of the pressure, p, and the degree of saturation, s.The saturation, or percent of filled fiber tows, is drivenby the pressure of the resin surrounding the tow, in addi-

tion to capillary wicking effects; the wicking effects are as-sumed to be negligible over the filling period. If the fibertows fill at the same rate as the regions around them thesink term is zero; therefore, the mass conservation equationis homogeneous and will simplify to the single scale porousmedia physics of

r hmdi 0. 7

Eq. (7) also describes the mass balance inside individual fi-ber tows, because during the filling of the individual towsno fluid sink is present at the micro scale.

This important sink term is incorporated into the finiteelement/control volume based numerical simulation calledLIMS that predicts the flow of resin in fibrous preforms [4].This is accomplished in the LIMS environment by attach-ing 1D elements to each node of the 2D or 3D elements.The 1D elements represent the micro-flow regions and2D or 3D elements represent the bulk regions as shownin Fig. 2.

The two types of elements are assigned separate perme-ability values corresponding to the bulk permeability of thepreform and the micro permeability of the fiber tow,respectively. Since the 1D elements share the nodes andconsequently pressure values with the 2D or 3D mesh,the micro flow is coupled with the bulk flow. The satura-

tion of the tows can be tracked, as well as the impact this

impregnation has on the injection pressure, which can bemonitored at the inlet injection node. This numerical meth-odology is completely outlined in the work of Simacek andAdvani [4].

This model is chosen, as it allows one to calculate thepressure profile at the inlet by specifying the bulk perme-

ability and a fitting parameter related to the tow permeabil-ity. In this manuscript, a methodology is introduced thatallows one to estimate the bulk permeability and themicro-permeability parameter of dual-scale preforms thatcan be used in LIMS, by matching the inlet pressure profilefrom LIMS, to the data obtained from experiments.

2. Background

2.1. Modeling

Modeling the dual-scale nature of fibrous preforms byimplementing not only the bulk flow, but also the microflow, is not a new trend. Parnas and Phelan [1] recognized,early on, the importance of evaluating the global flowfront along with the local impregnation of the fiber tows.The model incorporated sinks, which take away fluidfrom the advancing flow front, and was a key developmentsince it coupled the flow at different length scales by com-bining 1D radial flow into the tow with 1D linear flowalong the length of the preform. Sadiq et al. [2] experimen-tally investigated the model, and observed key trends,which reinforced the importance of the need for suchmodels.

Many further models have been developed, which eval-

uate the unsaturated flow through porous preforms and

Fig. 2. Bulk- and micro-flow interactions are modeled in LIMS byattaching 1D elements (which represent the fiber tows) to the 2D or 3Delements (which represent the bulk preform).

N. Kuentzer et al. / Composites: Part A 37 (2006) 20572068 2059


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explore how the bulk flow behavior is influenced by theimpregnation of the fiber tows [513]. Such models havethe ability to be transformed into flow simulations, in orderto numerically incorporate the sink that causes the non-lin-earity in the pressure profile; however, for accurate predic-tions one would have to supply two values of the

permeability, one for the bulk flow and one for the impreg-nation of the tows.

2.2. Experimental investigations

Experiments to determine the in-plane bulk permeabilityhave been developed and various issues in measurementtechniques have been addressed [5,9,1422]. For dual-scalepreforms, Parnas et al. [14] conducted 1D linear and 2Dradial flow experiments at a constant injection pressure.The unsaturated and saturated permeability values for ran-dom mat and woven fabric were compared; a differencebetween the two values was observed. Luce et al. [15]

extended this work by conducting similar experiments,but utilizing preforms composed of two types of fabric.The impact of compaction of the preform was additionallyaddressed. Binetruy et al. [5] also conducted constant injec-tion pressure experiments and highlighted the differencebetween the saturated and unsaturated regions of the pre-form. They concluded that the micro flow did not impactthe saturated region, but that the unsaturated region isimpacted by the impregnation of the tows and accordinglyaffects the overall permeability of the preform.

Slade et al. [9] conducted 1D linear experiments at aconstant flow rate, in order to evaluate the resulting pres-

sure profile over time. The pressure results were used todetermine both the unsaturated and saturated permeabilityvalues of random mat and stitched biaxial fabric. Babu andPillai [16] also examined the pressure plots resulting from1D constant flow rate experiments; this work studied whateffect compression has on the pressure profile of biaxialstitched, triaxial stitched, biaxial woven, and unidirectionalfabrics.

However, no method has been established in which boththe bulk and micro permeability values can be determinedfrom one experiment. This paper proposes a methodologyto recover both the bulk permeability and the micro-per-meability parameter of different preforms by conductingone RTM constant flow rate injection experiment. The

experimental data is used to analytically determine the bulkpermeability and is compared to the numerical results gen-erated by LIMS to determine the micro-permeabilityparameter. Both values can be used in LIMS environmentto accurately predict the flow in closed mold processes.

3. Experimental procedure: bulk permeability

determination

In order to measure the permeability of different pre-forms, one-dimensional constant flow rate experimentsare conducted. A closed mold is used, which consists ofone 1/2 in.-thick aluminum lower mold half and one

2 in.-thick clear acrylic top mold half that acts as a viewingwindow. A 1/8 in.-thick aluminum spacer separates the twohalves and provides the cavity in which the layers of fabricare placed. O-rings are fitted to grooves machined in bothmold halves in order to seal the mold and prevent any fluidfrom leaking. The mold is secured with 26 bolts, which areequally spaced and torqued. The mold is additionally rein-forced to minimize any deflections by clamping stiffeningbars, which stand 2 in. on end, to the mold exterior. Aschematic of the mold is shown in Fig. 3.

In order to achieve an experiment free of race-trackingalong the edges of the preform, care is taken when cutting

each fabric layer. Race-tracking occurs when a channel oflower resistance is created along the edges of a preformwithin a mold cavity, as a result of frayed fabric edges orincorrect preform size [17]. A frame that is identical indimensions to the mold cavity is used as a cutting templateto ensure the width of the layers is as exact as possible. Thefabric is cut to ensure the inlet port is left uncovered so thatthe test fluid first fills the empty space within the cavity and

Aluminumbottom

plate

Aluminum

spacer plate

SIDE VIEW TOP VIEW

Acrylictopplate

Steelreinforcing

bars

Inletport

Crossbolts

Fabric preform

Fig. 3. Schematic of mold set-up for 1D flow experiments.



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initiates a planar one-dimensional flow through the pre-form. The vent port is also not covered by the fabric, soair can escape without any resistance or restriction. Thecut preforms are 8 1/8 in. wide and 12 in. long in the flowdirection.

Each experiment is videotaped to highlight three impor-

tant criteria: (a) the progression of the flow front isstraight, even, and steady; (b) the flow is free of edge effectssuch as race-tracking; and (c) no layers are washed away bythe flow. The transparent top plate facilitates this proce-dure. The bulk permeability measurements can be con-ducted in the presence of race-tracking [17], but thenumerical LIMS analysis to which the experimental datais compared, assumes no edge effects; therefore, the exper-iments are conducted to match this assumption and if thiscriterion is not met, the data is discarded. The data is alsodiscarded if one of the other two criteria is not met.

Vegetable oil, either pure or colored by the addition of anoil-based dye, is used as the test fluid in this study. The vis-

cosity of the oil is measured before each experiment using aBrookfield viscometer. A Radius brand injection machinemoves a piston via a stepper motor, in order to provide aconstant flow rate. Once the residual gas is expelled frompiston cylinder, the oil is injected into the closed moldthrough the inlet port. The inlet pressure is recorded usinga pressure transducer connected to the Radius injection sys-tem. The pressure data is collected over the duration of theexperiment, which is complete once the bulk flow reachesthe end of the fabric and the vent. The injection is continuedand the resin is allowed to bleed through the vent until thepressure reaches a steady state value, signifying the com-

plete saturation of the fabric. A schematic of the 1D con-trolled flow system is shown in Fig. 4.

For each experiment, the inlet pressure data is plottedagainst time. The plot begins at time zero, which corre-sponds to when the oil enters the preform; the plot endswhen the bulk flow front reaches the end of the preformat time, tf. The plot, an example of which is shown inFig. 5, can be subdivided into three regions.

Linear region (i) is representative of the time frame dur-ing which fluid fills the bulk region, yet no micro poreshave had time to be impregnated. Non-linear region (ii)is representative of the pressure response as the fluid ofthe bulk flow front begins to impregnate the fiber tows.

Linear region (iii) is representative of the period duringwhich a constant partially-saturated length ensues.

As previously discussed, there are two flow regions indual-scale fabrics: the flow through the larger pores ofthe bulk region and the flow through the smaller poreswithin the tows. In essence, the flow through both of theseregions initially develops, but after a certain time framebecomes fully developed. The micro pores are more diffi-cult to impregnate, so the time it takes the flow to fullydevelop will be longer than the time it takes the flow inthe bulk region to fully develop. In order for the constantpartially-saturated length to exist in region (iii), the mold

must be long enough to ensure that the micro flow frontfully develops, ensuring the presence of the fully-saturatedfront. Once both fronts fully develop, the fully-saturatedfront and the bulk flow front are then traveling at the samespeed. The complete mathematical proof of this phenome-non is available in the Appendix of [23]. The partially-sat-urated length, which is the distance between the two fronts,is thus constant and the associated pressure drop is linear.

Fig. 4. Schematic of 1D controlled flow system used to determine permeability values of fabric preforms.

Time (s)

Pressure(psi)

(i) (ii) (iii)

t=0 tf

Fig. 5. Dual-scale fabrics exhibit three flow regions; (i) an initial linearregion, (ii) a non-linear region, and (iii) a final linear region, whichrepresents the flow that has a constant partially-saturated length.



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Using an excel template, the final experimental datapoint is fixed and preceding data points are added as tomaximize the fit (R2 value) of the linear best fit line passingthrough the data points. The distinction between region (ii)and (iii) can thus be made, since the addition of non-linearpoints from region (ii) would not improve, but impede the

fit. Linear region (iii) is used to determine the bulk perme-ability of the preform. The slope of the best fit line in region(iii), which is also determined by the excel template is useddirectly in Eq. (5) to determine the bulk permeability.

Accordingly, a minimum requirement of the experimentis that an appropriate combination of fiber volume frac-tion, viscosity, and flow rate are used in order to yield anexperiment in which the tows of the fabric begin to filland the constant partially-saturated length region isreached. The pressure plot therefore begins linearly,becomes non-linear, and finally reaches a linear slope dur-ing the final portion of the experiment. If this requirementis not met, the bulk permeability value of interest cannot be

determined, since region (iii) will not exist.

4. Numerical procedure: micro-permeability parameter

determination

The 1D constant flow rate RTM experiment is modeledto simulate the experimental conditions and parameters.Pressure data is generated and collected over the courseof the injection, just as if a physical experiment were beingconducted. A rectangular mesh that is composed of square,2D elements is generated to replicate the preform containedwithin the mold used in the experiment. The mesh size is 83

elements in length by 60 elements in width. These 2D ele-ments are assigned a fiber volume fraction, Vbulkf , and a per-meability value, Kbulk, within LIMS, both of whichcorrespond to the bulk-impregnation regions of thepreform.

One-dimensional bar elements are then attached toevery 2D element node. These 1D elements are assigned afiber volume fraction, Vtowf , and a permeability value, Ktow,representing the micro-impregnation regions within thefiber tows. By the addition of these 1D elements to the

existing 2D elements, the interactions between the bulk-and micro-flow regions are enabled.

A length, l, is set for each 1D element, each of whichalso has an associated initial default cross-sectional area,Atow. The area is adjusted to Atow-adj. by the LIMS model,based on the set l and Vtowf and the input V

bulkf :

Atowadj: VbulkVbulkfVtowf

2l

; 8

where Vbulk is the volume of a 2D element. Next, Vbulkf isadjusted to Vbulkadj:f in order to accommodate for the addi-tional volume of the tows, which are attached to the exist-ing mesh; this preserves the overall porous volume of themesh:

Vbulkadj:f V

bulkf 1 2

1 VtowfVtowf

. 9

The set length of the 1D elements is chosen based on theassumption that the tow permeability is four to five ordersof magnitude smaller than the bulk permeability. The bulkpermeability values are of the order 1E9 and 1E10 m2

and accordingly, when l is set to the order 1E3 m, theresulting tow permeability is of the order 1E14 m2. Thespecific value of l does not matter, since the value of thetow permeability would just need to be accordingly alteredto reach the same Ktow/h

2 parameter. But, the representa-tive length of the 1D elements is chosen such that thetow permeability is in line with values which make physicalsense. The parameters used to run each simulation, as wellas the values of the parameters that are modified, are com-piled in Table 1. The complete details governing the satura-

tion program approach are explicitly outlined in [4].For each experiment, a LIMS file of the mesh is created

to match the experimental parameters. The viscosity, l, ofthe test fluid is first input into LIMS. Then, Vbulkf is set andKbulk, which is determined from the experimental pressureprofile over time, as discussed in the previous section, isassigned to the 2D elements. The 1D elements are nextattached to each node of the 2D element mesh. The con-stant flow rate injection simulation is finally conducted inLIMS, where the constant flow rate boundary condition

Table 1

2D and 1D element parameters corresponding to LIMS flow simulation

Fabric types 2D elements 1D elements

Vbulkf (%) Vbulkadj:f (%) V

towf (%): 80

l (m): 0.002Atow (m2): 0.001

Atow-adj. (1E03 m2)

Woven E-glassWGa 40 60.0 0.0191WGb 40 60.0 0.0191Woven carbonWCa 35 52.5 0.0167WCb 35 52.5 0.0167Stitched E-glass 1SG1a 54 81.0 0.0387SG1b 54 81.0 0.0387Stitched E-glass 2SG2a 31 46.5 0.0148SG2b 39 59.1 0.0188



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is imposed at the edge nodes of the 2D mesh, which arealong the width of the preform.

In order to determine the micro-permeability parameter,the pressure and corresponding time data that is outputfrom the simulation is matched to the experimental databy changing only a fitting parameter in the simulation asso-

ciated with the tow permeability, Ktow/h2

, [4]. This para-meter lumps together the fiber tow permeability, thecross-section characteristic dimension of the fiber tow, h,and possibly a shape factor of the fiber tow cross-section.The fiber tow is assumed to have a rectangular cross-sec-tion; therefore, the rounded corners that are not taken intoaccount for an elliptical cross-section are lumped into theparameter. The designation h is equivalent to l/2. This isdone since the 1D elements fill in only one direction inthe simulation, whereas filling would occur in two direc-tions in a physical rectangular tow.

The actual permeability, Ktow, of the micro-flow regionsinside the tows is not evaluated. The parameters set in the

simulation for the 1D elements, such as land thus Atow-adj.,only estimate the actual physical parameters of the fabricsused in the experiments. Since a non-dimensional fittingparameter is achieved, either h or Ktow can be altered toachieve the best fit. But, as long as the same input param-eters are used in the simulation environment with theexperimentally obtained value of Ktow/h

2, the bulk- andmicro-flow interactions can be properly modeled.

Initially, the Ktow/h2 parameter is estimated and

assigned to the 1D bar elements; many simulations are con-ducted from which a general starting point is established inorder to reduce the number of necessary iterations. The

corresponding inlet pressure profile that is generated iscompared with the experimental data. The next value isselected, by either doubling or halving the guess, dependingon how the data from the initial guess compares with theexperimental data. Fig. 6 illustrates the two possible sce-narios; case (a) shows the simulation plot lies above theexperimental plot, which corresponds to an initial guessthat was too small and therefore must be doubled, whilecase (b) shows the opposite case which corresponds to aguess for the tow-permeability parameter that was toolarge and must be halved. The direction of increasingtow-permeability parameter is shown to clarify the choice,

either doubling or halving, according to this bisectionmethod.

This iterative process is continued until two curves cor-responding to two different tow-permeability parametersflank the experimental data curve. This resulting intervalis then subdivided into multiple equally spaced segments.

Each region is defined by a tow-permeability parameterand the pressure and time data that corresponds to eachvalue is simulated in LIMS. Multiple segments are gener-ated, as opposed to purely bisecting the interval, so thatthe minimization location can be graphically verified.

The generated time, tj, and the corresponding pressuredata, pnum (tj) is input into an excel template, where jbeginsat the first data point and increments to the final number ofdata points, N. The pressure values are first interpolated ateach experimental data time step, so both sets of data canbe compared at the same value of time:

pnumtexp pnumtj

tj1 texp

tj1 tj pnumtj1

texp tj

tj1 tj .

10

The experimental time, texp, falls between the numericaltime steps, tj and tj+1: tj6 texp 6 tj+1.

A simple point wise linear regression scheme is thenimplemented. The template is used to locally determinethe set of numerically generated pressure data that best fitsthe experimental pressure data, pexp. The difference in thepressures is taken at each time step, squared, and summedover the length of the experiment from texp = 0 to tf:

Xtf

texp0

pnumtexp pexptexp

2. 11

The segment where this error is globally minimized is thenidentified. The new interval is further subdivided intoequally spaced segments. The segments are progressivelyrefined as the procedure is iterated in order to convergeon the value of the tow-permeability fitting parameter.

5. Validation

The model of the one-dimensional constant flow rateRTM experiment is created in LIMS and used as a tool

Time (s)

Pressure(psi)

Pressure(psi)

Case (a):

guess too

small

Time (s)

Case (b):

guess too

large

IncreasingKtow/h

2

(a) (b)

Fig. 6. During estimation of the value ofKtow/h2 two scenarios arise: case (a) the guess for Ktow/h

2 is too small and case (b) the guess for Ktow/h2 is too

large. Bisection method is used to converge on the best match.



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to validate the methodology for determining the bulkpermeability and the micro-permeability parameter. Bothof these values are chosen and assigned to the 2D and1D elements in the finite element mesh, respectively. Theone-dimensional constant flow rate simulation is then per-formed using LIMS. The pressure plot is constructed as a

function of time, by recording the pressure output fromthe simulation at an inlet node. This plot is then viewedas if it was obtained from a physical experiment.

This experiment is then processed, as if it were any typ-ical physical experiment. The bulk permeability is deter-mined by using the slope of the linear region (iii) shownin Fig. 5, as discussed in the experimental procedure. Themicro-permeability parameter is determined by followingthe sequence of steps outlined by the flowchart in Fig. 7.The values obtained for the bulk permeability and themicro-permeability parameter are then compared with theselected values to verify the methodology.

The methodology is tested for different ratios of perme-

abilities to ensure that it is sufficiently robust. The bulk per-meability value is selected to be of the order of 1E09 m2,to which permeability values of different orders, 1E12 m2,1E13 m2, 1E14 m2, and 1E15 m2 are assigned to the1D elements. This is done to ensure the methodology hasthe ability to recover the permeability values, irrespectiveof the permeability ratio. Both values can be exactlyextracted when imposing the above methodology. One lim-itation, which is exposed, is that the values cannot berecovered if no tows fill with fluid, as in the case whenthe permeability value of the 1D elements is of the order1E15 m2. This result corroborates the minimum require-

ment discussed in the numerical procedure: if the tows ofthe preform do not even begin to saturate, a saturated bulkpermeability cannot be accurately determined. Numericalissues also arise when one goes beyond six orders of mag-

nitude difference between permeability assigned to 2D ele-ments that represent the bulk perform and the 1D elementsthat represent the fiber tows; but, this is usually unlikely inmost performs, as this would preclude filling of the fibertows.

6. Experimental results

Four fabrics are evaluated in this study: 9 oz/yd2 biaxialwoven carbon (WC), 24 oz/yd2 biaxial woven E-glass(WG), and two types of biaxial stitched E-glass, 96 (SG1)and 18 (SG2) oz /yd2, respectively. Different fabrics areevaluated to show how they deviate from a baseline sin-gle-scale medium and exhibit dual-scale behavior. Fig. 8highlights the different architectures of the four differentfabrics.

Random mat fabric is included as a reference fabric,since it is known to exhibit a single-scale Darcian flow

behavior. The random mat preform has only one lengthscale and accordingly exhibits a linear pressure profile.The woven and stitched fabrics have slight non-linearitiesand bend away from the single-scale trend over theduration of the experiment, since these preforms havetwo length scales. The non-linearity is induced by thepresence of the fiber tows, which act as sinks into whichthe fluid of the advancing flow front ingresses. The non-linearity in these preforms is slight, but not insignificant.This transition of the pressure from a linear, to a non-linear, and back to a linear profile, as is described inFig. 5 and as can be seen in Fig. 9, enables the determina-

tion of the micro-permeability fitting parameter that canbe used in LIMS to properly model the flow inside thetows along with bulk flow through different fabrics. Thenormalized plots for both the reference random mat fabric

1. INPUT estimated value of

Ktow/h2

into LIMS mesh.

No

No

Yes

Yes

2. RUN 1D constant flow rate flow

simulation using LIMS.

3. COMPARE simulated pressureplot over time to experimental plot.

to

to

3. COMPARE forall segments.

2. RUNfor all

segments.

Halve

value.

Double

value.

Is Ktow/h2

o large?

Yes 5. DETERMINE region

where Ktow/h2

is

minimized.

Does new Ktow/h2

=

previous Ktow/h2?

4. IMPLEMENT inter-

polation and linear reg-

ression scheme for all.

Is Ktow/h2

o small?

Subdivide region

equally.

Fig. 7. Flowchart of procedure to determine the tow-permeability parameter.



9/12

and an example of one woven E-glass fabric are shown in

Fig. 9 to highlight the differences in the inlet pressureprofiles.Multiple experiments are performed for each type of

fabric to evaluate repeatability. Four types of fabrics thatall exhibit dual-scale porosity are characterized and theresults of two data sets are included for each fabric. Table2, which uses the same fabric acronyms as Table 1, outlinesthe bulk permeability values that are determined, as well asthe values of the micro-permeability parameter that areobtained from matching the experimental data to the dataoutput from LIMS. In addition, the value K* that repre-

sents the ratio of the bulk permeability to the micro-perme-ability parameter is tabulated:

K Kbulkh2

Ktow

. 12

The partially-saturated length, Ls, generated by LIMS anda non-dimensional constant Ls

2

K, which aides in addressing

what parameters are impacting the partially-saturatedlength, are also incorporated into the results table.

7. Discussion

The data sets corresponding to the first three types offabric (WG, WC, and SG1) are determined from experi-ments conducted at the same fiber volume fraction. Inthe case of the stitched glass preforms, the bulk permeabil-ity varies approximately 30%. This variation decreases toapproximately 15% in the case of the woven glass fiberand is only a few percent in the case of the carbon fiber pre-forms. Other researchers have also reported that perme-ability values exhibit a large scatter from one experimentto the next [18,20].

The way the fabric layers nest together plays an impor-tant role in these variances. Layers may end up placedeither bundle to bundle or bundle to gap, as shown inFig. 10. One configuration may preferentially allow fluidto more easily permeate through the bulk region as com-pared to the other, which causes the deviations. In the caseof the carbon, seven plies in the mold result in a fiber vol-

ume fraction of 35%, where as only five plies of glassoccupy 40% of the mold. Deviations in stacking are thusnot impacting the bulk region of the carbon to the extent

Fig. 8. The four fabrics that are characterized: (i) 9 oz/yd2 biaxial woven carbon, (ii) 24 oz/yd2 biaxial woven E-glass, (iii) 96 and (iv) 18 oz/yd2 biaxialstitched E-glass.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.4 0.6 0.8 1.00.2

T/Tfill

P/Pm

ax

RM-experimentalWGb-experimentalWGb-LIMS

Fig. 9. A woven glass fabric has a non-linear pressure profile as comparedto a linear pressure profile of random mat.

Table 2Bulk permeability, micro-permeability parameter, K*, Ls, and Ls2/K*

results for each experiment

Fabric Vf(%)

Kbulk1E10 (m2)

Ktow/h2

1E08K*

(m2)Ls(m)

Ls/L Ls2/K*

WGa 40 12.6 2.3 0.055 0.257 0.892 1.21WGb 40 10.7 3.7 0.029 0.188 0.653 1.22

WCa 35 8.6 1.3 0.066 0.272 0.944 1.12WCb 35 8.3 2.3 0.036 0.199 0.691 1.10

SG1a 54 20.6 8.6 0.024 0.210 0.729 1.84SG1b 54 14.2 15.5 0.009 0.129 0.448 1.82

SG2a 31 12.7 1.8 0.071 0.267 0.927 1.01

SG2b 39 7.2 3.3 0.022 0.163 0.566 1.22

A B

Fig. 10. When fabric layers are placed together they stack either (A)bundle to bundle or (B) bundle to gap, creating different preferential flow

paths.



10/12

the glass does. The thicker and less pliable E-glass pliescannot mesh as well as the carbon plies, and more prefer-ential bulk flow regions are therefore likely to exist. Addi-tionally, more fluid is going into the tows in the case of theglass, which impacts the overall bulk permeability; the fluidtravels more readily through the smaller proportion of bulk

porous volume. The glass is thus more permeable than car-bon, though there is less porous volume for the fluid toflow through.

For both stitched fabrics, the tows of one layer are per-pendicular to the tows of the preceding layer. The first typeof stitched glass preform (SG1) is made up of seven joinedlayers, while the second type of stitched glass preform(SG2) is made up of only two layers. Thus, for SG1, onlytwo plies are needed to achieve a fiber volume fraction of54% within the mold, whereas four and five layers of theSG2 fabric occupy only 31% and 39% of the mold. Theindividual SG1 plies are accordingly less flexible and havemuch more freedom in how they nest with the subse-

quently-stacked ply, causing the greater bulk permeabilitydeviations from one experiment to the next.

The micro-permeability parameter cannot be directlycompared from one fabric to the next, since the parameteris impacted by size of individual fibers and fiber tows, theshape of fiber-tow cross-section and the fiber volume frac-tion of the fiber tow. Accordingly, the Atow-adj. values aredifferent in the LIMS environment for each fabric. But,the parameter can be compared between the experimentsfor each type of fabric, since the experiments are conductedat the same fiber volume fraction and the simulations areconducted corresponding to the same Atow-adj. parameters.

A trend which is exposed is that less permeable bulkregions correspond to a more permeable parameter interms of the micro pores. A hypothesis for this result is thatas the oblong tows are also compressed and become widerand thinner in cross-section, the pores of the bulk regionreduce in size. Thus, it is harder to infiltrate the pores ofthe bulk region, but the tows are able to fill more easily,since the fluid does not need to travel as far to reach allportions of the tow. The second set of stitched glass fabricdata (SG2), which is determined from experiments con-ducted at two different fiber volume fractions, also demon-

strates the trend that the micro-permeability parameterincreases, as the bulk preform becomes less permeable.As the number of plies increases, the bulk permeability alsodecreases as anticipated.

When comparing the tow-permeability parameter rele-vant at the micro scale to an equivalent parameter perti-

nent at the bulk scale, namely Kbulk/L2

, where L is thelength of the preform, the ratio of the two values is of inter-est. Table 3 outlines the separate values, in which the finaldata column reveals the ratio of the parameter from themicro to the bulk region.

This ratio reveals information regarding the differenttimes scales in the two flow regions. The tows may needbetween 1.18 and 9.05 times as long to fill as the bulkregions, depending on the architecture of the fabric. Thisratio is important, as it alludes to necessary processingrequirements of the different fabrics. Though these timerequirements are not directly applicable for the character-ization of the materials, they must be considered when

resin is infused into the fabrics, so as to ensure all regionsare saturated with fluid before the part cures.

As the bulk flow front progresses along the length of thepreform, a region near the inlet gate exists in which at least99% of the micro pores are saturated. Additionally, aregion near the vent exists where less than 1% of the micropores are saturated. The region in between, is the partially-saturated region. As depicted in Fig. 11, each shade corre-lates to a varying degree of saturation, and all shadesrepresent the partially-saturated length, Ls. This lengthidentifies the region in which flow is continuing to ingressinto the fiber tows and lies between one and 99% satura-

tion: 1% < Ls < 99%.The degree of saturation output from LIMS has the

ability to be mapped to a contour plot; this is depicted inFig. 11. The partially-saturated length, Ls can be clearlymeasured from the 99% to the 1% demarcation on the leftside and right side, respectively. The values less than 1% arenot included in the value of Ls, because they are deemedcompletely unsaturated, just as the values above 99% arenot included, as they are considered completely saturated.This partially-saturated length is generated through LIMSsince it is not possible to know exactly which regions are

Table 3The ratio of the tow-permeability parameter to an equivalent parameter relevant at the bulk scale reveals information regarding the time scales in the twodifferent flow regions

Fabric Kbulk (m2) 1.E10 Kbulk

L2Ktow (m

2) 1.E14 Ktowh2

KtowL2

h2Kbulk

WGa 12.6 1.5E08 2.30 2.3E08 1.51WGb 10.7 1.3E08 3.70 3.7E08 2.87

WCa 8.6 1.0E08 1.30 1.3E08 1.25WCb 8.3 1.0E08 2.30 2.3E08 2.30

SG1a 20.6 2.5E08 8.60 8.6E08 3.46SG1b 14.2 1.7E08 15.50 1.6E08 9.05

SG2a 12.7 1.5E08 1.80 1.8E08 1.18

SG2b 7.2 8.7E09 3.30 3.3E08 3.80



11/12

saturated to which degree from the current experimentalset-up.

If K*, which is the ratio of the bulk permeability to thetow-permeability parameter, increases, one would expectLs to increase as well, as Ls will be impacted by changesin either the bulk permeability or the micro-permeabilityparameter. For example, K* increases ifKtow/h

2 is reduced.A lower value of the tow-permeability parameter increasesthe partially-saturated length Ls. This is an anticipatedresult, since the partially-saturated region should increaseas the micro-permeability parameter moves further away

from the value of the bulk permeability. On the other hand,Ls will decrease as the micro-permeability parametermoves closer and closer to the value of the bulk permeabil-ity. This result is again anticipated, since no partially-satu-rated length would be present in a single-scale medium.

The notion of the partially-saturated length providesvaluable information about the experiments that are con-ducted. Ls2/K* is an important parameter since it remainssteady for experiments conducted on the same type of fab-ric, at the same fiber volume fraction. As the fiber volumefraction in the mold increases, results for all four differenttypes of fabric that possess very different architecturesshow that the Ls2/K* parameter also increases. Both the

K* vs. Ls/L and Ls2/K* vs. Vbulkf trends arising from Table2 are plotted in Fig. 12.

The average summed least squares error for each type offabric was found to be 4.38 (WC), 1.45 (WG), 0.26 (SG1),and 0.58 (SG2). These results lead to the hypothesis thatthe experimentally obtained pressure profiles of the stitched

glass fabrics provide the best fit for the model. It can beobserved too, that the three fabrics with the lowest errorare all comprised of E-glass. The smaller diameter of theindividual carbon fibers within each tow, which correspondto smaller pores, may alter the sink strength in a mannernot captured by the model to the extent the sink strengthof the glass tows has the ability to be captured in the cur-rent LIMS model.

8. Conclusions

By reproducing the injection pressure history at the inletof a one-dimensional constant flow rate injection experi-

ment using LIMS numerical analysis in which 1D elementsrepresent the fiber tows and 2D elements represent the bulkpreform, a methodology has been developed, which deter-mines both the bulk permeability and a tow-permeabilityparameter for that fiber preform. This model confirms withexperiments that although Darcys law is linear, non-linearbehavior is exhibited as a fluid injected at a constant rateflows into a dual-scale fabric in which the tow and bulkpermeabilities are different. This resulting non-linearbehavior is due to the way the fluid at the bulk flow frontenters the fiber tows at a delayed pace; the tows are accord-ingly considered fluid sinks. The pores at the micro scale

are much smaller and cannot be infiltrated as easily asthe pores of the bulk region.The notion of the partially-saturated length has also

provided valuable information about the experiments thathave been conducted. By knowing the partially-saturatedlength, along with the bulk permeability of the preform,the tow-permeability parameter can be obtained by usingthe knowledge that the non-dimensional Ls2/K* term is aconstant for a specific fabric. In addition, one could gaugethe relative time scales to infiltrate bulk regions and towsin a fabric at different fiber volume fractions. Experimen-tal data, along with numerically generated LIMS data,provide a methodology to couple the bulk- and micro-flow

0.8

1.0

1.2

1.4

1.6

1.8

2.0

30% 35% 40% 45% 50% 55%

Vf

Ls

2/K*

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.4 0.6 0.8 1.0

Ls/L

K*(m2)

Fig. 12. Trends exhibited by parameters K* vs. Ls/L and non-dimensional parameters Ls2/K* vs. Vf.

Fig. 11. An example of a contour plot that enables the determination ofthe partially-saturated length (Ls), which is the region between 1% and99% saturation.



12/12

interactions, which are so crucial in dual-scale preformsused in so many LCM applications.

Acknowledgements

This work was partially funded by the Army Research

Laboratory, grant number DAAD 19-01-2-0005, and byAdvanced Materials Intelligent Processing Center estab-lished at University of Delaware by Office of Naval Re-search, grant N00014-04-1-0891.

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