Document Type : Research Article
Authors
K. N. Toosi University
Abstract
Keywords

Mechanics of Advanced Composite Structures 5 (2018) 83–90


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Capillary Effects on Surface Enhancement in a NonHomogeneous Fibrous Porous Medium
M.R. Shahnazari^{*}, M. Esfandiar
Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran 1999143344, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 2016‐08‐20 Revised 2017‐06‐04 Accepted 2017‐08‐22 
The evaluation of a free fluid surface in a porous medium has several mathematical applications that are important in industries using molds, particularly in the fluid injection process. The vacuumassisted resin transfer molding (VARTM) process is a promising technology in the primary composite industry. An accurate computational simulation of the VARTM process would be a costeffective tool in the manufacturing of composites. In this paper, capillary effects were incorporated into an existing resin transfer molding model to simulate VARTM processing. To increase the accuracy of the VARTM process simulation, the effect of capillary pressure on a surface without flow was studied using the boundary element method. The simulation results were close to the experimental data reported by other researchers. It can be concluded that better reliability and accuracy could be achieved from theoretical predictions by examining the effects of capillary pressure on flow injection into porous materials. 



Keywords: Vacuumassisted resin transfer molding Free surface Capillary pressure Porous media Finite element. 


© 2018 Published by Semnan University Press. All rights reserved. 
The problem of surface enhancement in porous media that can be infiltrated by fluids has become an important consideration in many industrial processes. This phenomenon has a significant effect on various applications, such as solidification in casting, crude oil, and extraction. Vafai and Srinivasan [1] analyzed the surface enhancement of a system containing two immiscible materials. Binetruy et al. [2] studied a twodimensional (2D) flow of a porous medium in a duct using a markerandcell (MAC) method. The phenomenological analysis of free surface transport through porous media was investigated by Chen and Vafai [3], and constant porosity is assumed in this analysis. Resin transfer molding (RTM) has become a widely used process to manufacture glassreinforced composites. A good description of the basic technical issues of RTM manufacturing can be found in a reference book by Cauchois [4]. RTM is a fabrication technique using closed molds that reduces the resin curing time. In this process, stacked, dry fibrous reinforcements are placed in the cavity of a rigid mold, and resin is injected at low pressure. The stiffness of the mold is often a concern during the manufacturing of large parts with a high fiber volume content. Although the filling time can be increased significantly, sometimes it is not possible to inject resin into the part. A higher injection pressure could reduce the injection time, but the cost of the required equipment may be too high for low volume productions.
Another limitation of injection pressure is point void avoidance in composite products. The liquid composite molding (LCM) process variant of vacuumassisted resin infusion (VARI) was first introduced in the Marco method [4]. It enables large parts to be successfully manufactured at a relatively low cost. In this process, stacked, dry fibrous reinforcements are placed between a stiff mold and a plastic bag. Resin is injected by gravity after a partial or total vacuum is achieved in the cavity containing the reinforcement. Most of the vacuumdriven techniques that are now commonly used in many industrial applications are based on trialanderror testing.
Williams et al. [5] have reviewed the main developments in infusion composite manufacturing process. From the VARI process developed by Group Lotus Cars Ltd. [6] to Seemann’s composite resin infusion molding process (SCRIMP) [7], many variants of resin infusion are now used by various companies. The main advantages of these techniques lie in their tooling cost, which is lower than that of RTM that uses closed molds. An original and optimized approach to resin infusion was developed by Kaizen Technologies under the name of the Kaizen Infusion System (KIS). The behavior of resin during the infusion process is not yet fully understood, and the processing strategy used in many applications is not always optimal [8, 9].
Capillary pressure is defined as the energy per unit volume of a porous medium needed for replacing a gas (or vacuum) with a liquid. Capillary effects have been shown to be a determinant in the mechanisms of void formation for the infiltration of fabrics [2, 10]. There is an optimal infiltration velocity that produces composites with minimal void content. Below that velocity, capillary forces dominate, and the infiltration process leads to void formation in the intertow region. But above the optimal value, viscous flow dominates, and voids are mainly created within the fiber tows [11].
Several studies have been conducted to determine the magnitude of capillary pressure developed in synthetic fabric infiltration. Batch et al. [12] investigated the capillary impregnation of aligned fibrous beds and found an optimal fluid speed at which micro and macro fluid speed with minimized void formation.
Patel and Lee [10] analyzed resinfiber wettability by conducting a wicking test and measuring capillary pressure using different test fluids. As a general trend, they found an increase in capillary pressure with a decrease in matrixfiber surface tension. Capillary pressures obtained for dioctyl phthalate (DOP) fluid and glass fiber (40–60 vol%) were in the order of 10 kPa. Following the same experimental approach, Amico and Lekakou [13, 14] conducted capillary rise experiments on glass fiber bundles with epoxy resin and silicon oil. They found a good correlation between the experimental and theoretical values of equilibrium capillary pressure for epoxy resin (9.6 KPa). Nevertheless, they did not explore the dynamic effects of capillary pressure. Verrey et al. [15] conducted infiltration experiments on noncrimp fabrics with different test fluids. They found negative values of capillary pressure for polyethylene glycol and lauryllactam 12 and positive values for an epoxy resin. However, in the last case, they found that for low numbers of capillaries (low fluid velocity), the resin changed its behavior from nonwetting to wetting, resulting in flows enhanced by capillary forces. A number of studies have been devoted to the experimental methods dedicated to measuring static and dynamic contact angles between single fibers and liquid, on both synthetic [1620] and natural fibers [2125].
Using a flow analysis network (FAN) method, Stoll et al. [26] have simulated the RTM process. The results were presented as mold filling time and free fluid surface state based on time. Acheson et al. [9] presented a 2D simulation for mold filling. The capillary effect has an important role in simulating processes with a low flow rate or low pressure injection, such as the VARTM process. The capillary effect during LCM with natural fibers has been investigated by Francucci et al. [27]. Setaguchi et al. [28] have focused on the wetting process in a glass fiber–resin system. They illustrated changes in wetting behavior by comparing the flow rate at the resin tip with that in the resin bulk.
Numerical simulation of resin injection can assist in positioning the inlet ports and vacuum intakes, especially for large and complex parts. Optimal injection strategies can be studied on a virtual model prior to prototype testing, consequently helping to reduce process setup costs.
The goal of this investigation is to verify the VARTM process with sufficiently accurate simulations of resin infusion, which can be performed using numerical tools that were originally developed for classical RTM, that is, to simulate the injection of resin in rigid molds. In this paper, the simulation of a VARTM process for a nonhomogeneous porous medium has been studied. The verified parameters included flow front patterns and infiltration times. To increase the accuracy of the presented simulation, the effect of capillary pressure on a surface without flow has been studied. Using the boundary element method (BEM), the fluid flow with or without capillary effects was analyzed, and the numerical results were compared with the results of other studies. By comparing the results obtained from two cases with other results, it was found that the results can be improved by paying attention to the capillary effects.
The flow of resin throughout the reinforcement process is well described by Darcy’s law, which predicts a linear relation between the local flux density and the applied pressure gradient. Darcy’s law is generalized to three dimensions, and the continuity equations are as follows:
(1) 

, 
(2) 
where is the local flux density (or superficial velocity), is the permeability tensor, is the resin viscosity, is the pressure of the resin, is the (local) resin density, and is the gravity vector [29]. The substitution of Darcy’s law (Eq. 1) in Eq. 2 results in a formulation of the continuity equation, where the only unknown parameter is pressure (a threedimensional [3D] scalar quantity) within the fluid in the mold:
(3) 
The permeability of a preform depends on several factors, the main one being its porosity. Several analytical models have been proposed to predict the permeability of a fibrous reinforcement [30]. Usually, experimental measurements are required to obtain the value of this key parameter. The most commonly used empirical model to describe the permeability of fiber porosity is as power low function. Therefore, by considering a nonhomogenous porous medium, the constant permeability can be shown in terms of porosity values, as follows:
, 
(4) 
where and can be estimated by experimental measurements [31]. Also, the porous media porosity, , can be determined by
, 
(5) 
where is the areal weight or superficial density, is the density, and is the thickness of the preform. The areal weight was determined by measuring the weight of the preform, which was divided by the area to yield units of grams per square meter. The density of the preform was assumed to be equal to the density of the fiber itself, which was provided by the manufacturer [31].
In a 50kPa vacuum, the fiber porosity was . Based on the experiments of Loos and Sayre [29], we assumed that the equivalent permeabilities were 5.26 E–11m^{2} and 8.73 E–11 m^{2} for 2.97 mm and 5.94 mm thicknesses, respectively.
The mathematical models for filling an LCM mold were solved with a set of partial differential equations. The finite element formulation was based on the procedure outlined by Reddy [32] and expressed for an element as
, 
(6) 
where and are as follows:
(7) 

(8) 
In Eqs. (6), (7) and (8), is the domain of an element, is the surface of an element, is the pressure at each node, is the pressure due to gravity at each node, is the specified flux through the face of an element, and is a linear interpolation function [29].
The process simulation model was used to investigate the resin infiltration of a 60.96cmby30.48cm preform during the VARTM process. The media was subdivided into a finite number of tetrahedrons, triangles (a 2D shell element with constant thickness), and runners (a onedimensional [1D] line element with a constant crosssection of an arbitrary shape). Figs. 1 and 2 show the finite element meshes of Eglass performance models for 2.97mm and 5.94mm thicknesses, respectively. The mesh of the Eglass preforms consisted of 3,642 and 5,376 elements, respectively.
Mesh independence was checked for the proposed model by measuring relative errors in the numerical results of the predicted infiltration time values. The computational mesh of the 2.97mm Eglass preform with 3,642 elements showed less than 4% difference with a finer mesh with 4,124 elements and was accepted. Similarly, mesh independency was evaluated for the 5.94mm Eglass preform model, and a relative error of less than 5% was achieved. A computational mesh with 5,376 elements was accepted after it was compared with a finer mesh with 5,921 elements.
An approximation was calculated for the pressure as a linear combination of basic functions :
, 
(9) 
where is the pressure and is a linear interpolation function [33].
The position of the flow front was tracked by assigning filling factors to the nodes. Initially, all nodes were empty (f = 0), except for the injection nodes. Here, pressure was calculated in fully filled nodes (f = 1). All partly filled nodes (0 < f < 1) resulted in the formation of pressure at the flow front, which was set to zero in all nodes that were not already filled (i.e., empty and front nodes).
The main boundary conditions for solving equations (6) include
where is the vector normal to the boundary, is the surface tension, is the contact angle, and is the hydraulic radius of the fiber bundle, defined as the crosssectional area normal to the flow divided by the perimeter applied to the fluid.
If the geometry of a unidirectional fiber bundle is considered, can be defined based on two flow directions: parallel to and perpendicular to the fiber. In this research, a case in which the flow is perpendicular to a unidirectional fiber bundle was chosen. This was because the impregnation of the 1,523 Eglass preforms used in the capillary pressure modeling occurred primarily in a transverse direction. Thus, for a flow perpendicular to a unidirectional fiber bundle, was determined to be
, 
(10) 
where is the characteristic particle diameter of the 1,523 Eglass fiber and is equal to 18 E6 m [29].
In this research, a constant capillary pressure was assumed using a static contact angle. In reality, this contact angle could increase in value as the viscous drag becomes significant, which would reduce the capillary pressure. Therefore, the capillary pressure generated during processing may be smaller than the values theoretically calculated here.
In molding processes, the evaluation of a free surface is an important factor in fluid injection modeling. Fig. 3 shows the free surface position in Zdirection with a comparison of the results obtained by Vafai and Srinivasan [1] for a 1D flow front pattern. The capillary pressure was modeled using a boundary condition modification and was evaluated using the fiber porosity, surface tension, and contact angle values for the Eglass–resin system.
In Fig. 3, denotes the location of the interface, is the horizontal extent of the preform, and denotes the Reynolds number, defined by permeability as . The model of RTM flow was first verified without the influence of capillary pressure. In this verification, two important factors were examined.


Figure 3. Free surface position in comparison with the results obtained by Vafai and Srinivasan [1] 
The first was the flow front pattern. It was desirable to have an accurate simulation of the flow front patterns during processing to determine the placement of resin–vacuum ports and to examine possible areas where void formation could occur. The second factor was the infiltration time, which was necessary for one obvious reason: Without a verification of the simulated infiltration time, an accurate prediction of the experimental processing time would not be available.
The flow front patterns during the VARTM process were verified by comparing the modeled and observed flow patterns during the infiltration of 2.97mm and 5.94mm Eglass preforms. Figs. 4 and 5 show the flow front pattern for Eglass preforms without considering capillary pressure, and these patterns are based on infiltration time.
Figs. 6 and 7 show the influence of capillary pressure on the flow front pattern and the infiltration time for the 2.97mm and 5.94mm Eglass preforms, respectively. From these patters, it can be stated that by considering capillary pressure effects, the infiltration times for two models have been reduced. Capillary effects are one of the most important causes of this reduction in infiltration time. Therefore, considering the capillary effects on the VARTM process, which is a method for manufacturing complex structures with various fiber laminates, is crucial. Misusing a proper runner distribution system in the infiltration process may cause a decrease in the vacuum pressure applied to the preform. However, the decrease in vacuum pressure leads to an increase in resin velocity. Slower injection speeds favor better bonding and wetting. Consequently, fingering appears at the flow front as a result of differing permeability. Fingering depends on the fluid rate, and the number of capillaries could easily show this dependency.
The effect of capillary pressure is necessary because the low pressure used for infiltration (~1 atm) produced small pressure gradients throughout the preform. Therefore, a lack of attention to capillary effects leads to these events and, finally, causes dry spot formation. Dry spots are a common void formation where a dry area results because of the inability of the resin to infiltrate a particular region of the preform. The strength and surface quality of parts manufactured by the VARTM process depends on void formation. Therefore, considering capillary effects could prevent undesirable construction defects in the manufacturing process.


Figure 4. Flow front pattern for a 2.97mm Eglass preform without considering capillary pressure 



Figure 5. Flow front pattern for a 5.94mm Eglass preform without considering capillary pressure 


Figure 6. Flow front pattern for a 2.97mm Eglass preform with capillary pressure consideration 



Figure 7. Flow front pattern for a 5.94mm Eglass preform with capillary pressure consideration 
Infiltration times give a much more quantitative measure for model verification than flow front pattern observations do. Table 1 shows the infiltration time for the Eglass–resin system in this study. In the table, the measured time by Stoll et al. [26] is compared with the results of the numerical solution in two cases, with and without considering capillary pressure. This table also presents the experimental results achieved by Stoll et al. [26] and Loos and Sayre [29]. These results show that the calculated infiltration time could be improved by considering capillary pressure effects. It is clear from Table 1 that the values from the numerical results that consider capillary pressure are much closer to the experimental results.
Table 2 presents the relative errors of the data introduced in Table 1. To determine the effects of capillary pressure, a comparison is made between experimental results (obtained by Stoll et al. [26] and Loos and Sayre [29]) and numerical results that consider capillary pressure. Numerical results without the capillary pressure consideration are also compared with the numerical results presented by Stoll et al. [26].
When capillary pressure is neglected, simulated infiltration times for both the 2.97mm and 5.94mm Eglass preforms were over the predicted experimental times by 8–9%. By considering the effects of capillary pressure on infiltration time for the 2.97mm Eglass preform, it can be seen from Table 2 that the model calculated infiltration time is now within 2% of the infiltration times of 332 seconds obtained by Stoll et al. and 330 seconds obtained by Loos and Sayre. When capillary pressure is considered for the 5.94mm Eglass preform, the simulated infiltration time is within 3% of the time of 585 seconds measured by Loos and Sayre, and it is within 0.7% of the time of 606 seconds measured by Stoll et al. According to the relative error values, it can be observed that there is an acceptable compatibility with other results.
Table 1. Infiltration time for an Eglass–resin system 


Fiber thickness (mm) 
Experimental results 
Numerical results 

Stoll et al. [26] 
Loos and Sayre 
Without capillary pressure 
With capillary pressure 

Stoll et al. [26] 
Present work 
Present work 

2.97 
332 
330 
357 
358 
338 

5.94 
606 
585 
629 
638 
602 

Table 2. Relative error of infiltration time results 


Fiber thickness (mm) 
2.97 
5.94 


Numerical results without capillary pressure 
Stoll et al. [26] 
1.8% 
0.7% 

Present work 


Experimental results 
Stoll et al. [26] 
0.3% 
1.4% 

Numerical results with capillary pressure 
Present work 

2.4% 
2.9% 

Experimental results 
Loos and Sayre [29] 

A simple approach to simulate resin infusion was presented. Models with and without capillary pressure were simulated of plug flows in the inplane direction. It can be seen from the results that the calculated infiltration time in capillary pressure consideration was within 2–3% of the measured infiltration time, in contrast to the 8–9% margin when capillary pressure was neglected. Based on these results, it is postulated that capillary pressure plays an important role in VARTM processing. Further modeling of capillary pressure can provide significant insight into this phenomenon. In this research effort, a constant capillary pressure was assumed using a static contact angle. In practice, this contact angle could increase in value as the viscous drag becomes significant, which would reduce the capillary pressure. Therefore, the capillary pressure generated during processing may be smaller than the values theoretically calculated here. Although the later statement may be true, these results show the capability of capillary pressure to reduce simulated infiltration times as much as 6% for the low pressure injection process.
Nomenclature
Superficial velocity 

Permeability 

Viscosity 

Pressure 

(Local) resin density 

Gravity vector 

Experimental coefficients in Eq. 4 

Porosity 

Areal weight or superficial density of the preform 

Density of the preform 

Thickness of the preform 

Element domain 

Element surface 

Pressure at each node 

Pressure due to gravity at each node 

Specified flux through the face of an element 

Linear interpolation function (shape function) 

Normal vector 

Surface tension 

Contact angle 

Hydraulic radius 

Characteristic particle diameter of fiber 

Location of the interface 

Horizontal extent of the preform 

Reynolds number defined by permeability 
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