Appl Compos Mater (2006) 13: 335–343 DOI 10.1007/s10443-006-9020-0
Mathematical Analysis of Resin Flow through Fibrous Porous Media Young Seok Song
Received: 16 May 2006 / Accepted: 31 May 2006 / Published online: 19 July 2006 # Springer Science + Business Media B.V. 2006
Abstract Resin flow through fiber preforms was analyzed mathematically. Closed form solutions for fiber volume fraction distribution and pressure field during resin infusion into fiber preforms were suggested, and a new effective permeability was defined. The effect of preform compressibility on the fiber volume fraction and pressure distributions in resinsaturated region was investigated analytically. The findings show that the compaction behavior of preforms has significant impact on the resin infusion process. The solutions derived analytically in this study can provide insight into a liquid composites molding (LCM) process. Key words permeability . resin infusion
1 Introduction Liquid composites molding (LCM) is a fascinating technique of manufacturing large and complex fiber-reinforced polymeric composites (FRC) used in automobile, aerospace, marine, and construction industries. Vacuum assisted resin transfer molding (VARTM), which is one of the most cost-effective variants in the LCM, has been extensively employed in the past two decades [1– 4]. In general, VARTM consists of lay-up of fiber preforms, impregnation of the preforms with resin, and resin curing. Since VARTM employs a singlesided tool and a room temperature resin unlike typical RTM, the process has great potential to form large scale composite parts with low cost fabrication [5–7]. Figure 1 describes the VARTM schematically. A great amount of efforts to understand resin infusion stage in the VARTM have been made [8–10], most of which are carried out based on numerical methods such as finite element method (FEM) and finite volume method (FVM) [11, 12]. However, analytic solutions for VARTM can be more useful since they provide insight into the process. For instance, Hsiao et al. showed the contribution of distribution media on the overall flow front Y. S. Song (*) Center for Composite Materials, University of Delaware, Newark, DE 19716, USA e-mail:
[email protected]
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Figure 1 Schematic diagram of a typical VARTM process
and a closed solution proposed by Lopatnikov et al. demonstrated the compressibility effect of fiber preform on the resin infusion process [13, 14]. However, their studies still contain some shortcomings: the compaction behavior of fiber preform and the influence of distribution media were not completely taken into account. In the current study, a new analytic solution to preform thickness in VARTM was proposed, which encompasses full compaction behavior of fiber preforms. Flow front advancement with respect to time was computed analytically by considering the contribution of distribution media. The main goal of this study is to investigate how the compaction behavior of preforms affects thickness and pressure distributions in the resin-saturated region. 2 Analytic Analysis 2.1 Governing Equations Resin flow through fibrous porous medium is governed by Darcy_s law on the assumption of the Newtonian, laminar, and incompressible fluids. Combining the continuity equation with Darcy_s law, the following equation is given. 1 @vf @ K @P ¼ ð1Þ vf @t @x μ @x where vf is the fiber volume fraction and K is the permeability of fiber preforms. Since the permeability is a function of fiber volume fraction, the following Kozeny–Carman model is adopted. 3 1 vf ð2Þ K¼k v2f where k is the Kozeny constant. Unlike in RTM, fiber volume fraction in VARTM is not constant across the entire mold due to the flexibility of vacuum bag, which leads to change
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in the permeability and fiber compaction pressure distributions. In the current study, the fiber compaction behavior is described by the following equation. b vf ¼ aPcomp
ð3Þ
Pcomp þ P ¼ 1
ð4Þ
where Pcomp and P denote the compaction pressure of preforms and the resin flow pressure, respectively. Their sum is equal to atmospheric pressure as shown in Equation 4. 2.2 Closed Form Solutions Substitution of Equations 2– 4 into Equation 1 gives the following equations. v 1=b f P ¼1 a @P 1 1=b1 @vf ¼ vf @x ab @x
1 @vf k @ ¼ vf @t abμ @x
1 vf v2f
ð5Þ
ð6Þ
3
1=b1 vf
@vf @x
! ð7Þ
By assuming the quasi-steady state of resin flow through fiber preforms, Equation 7 is rewritten as follows [14]. ! 3 1 vf @ 1=b1 @vf ¼0 ð8Þ vf @x @x v2f Integrating Equation 8 with respect to the flow advancement distance provides the following equation. ! bvf b 3b 1=b ¼ Aðt Þx þ BðtÞ ð9Þ vf 3b þ ð1 bÞvf 1þb ð1 2bÞv2f A and B are determined by imposing boundary conditions: vf ¼ vf 0 ¼ vmin at x = 0 and vf = vmax at x = L. The mass conservation principle is applied throughout the resin-filled region in order to obtain flow advancement with respect to time. Z 0
LðtÞ
1 vf hdx ¼
Z 0
t
1 vf 0 hð0; tÞV ð0; tÞdt
vf h ¼ vf 0 hð0; tÞ
ð10Þ ð11Þ
where h(x,t) and V(x,t) are the thickness of preforms and the resin, respectively. Equation (10) is cast by arranging variables as below. vf 0 1 vf 0
Z 0
L
1 vf dx ¼ vf
Z 0
t
V ð0; tÞdt
ð12Þ
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After combining Darcy_s law with Equations 2 and 6, the resin velocity is expressed as follows. ! 3 1=b1 3 1=b3 @vf vf k @vf K @P k 1 vf 1 v f vf ¼ ð13Þ ¼ V ðx; tÞ ¼ μab @x μ @x μ ab @x v2f At injection gate (x = 0), the resin velocity is
3 1=b3 @vf k V ð0; tÞ ¼ 1 vf 0 vf 0 μab @x x¼0
ð14Þ
From Equation 9, the following equations are obtained. 3 1=b3 @vf 1 vf vf ¼ AðtÞ @x 31=b AðtÞvf @vf ¼ 3 @x 1 vf x¼0 Equation 14 is re-cast as the following. V ð0; tÞ ¼
k Aðt Þ μab x¼0
ð15Þ
ð16Þ
ð17Þ
Substitution of Equation 17 into Equation 12 gives vf 0 1 vf 0
Z
L 0
1 vf dx ¼ vf
Z 0
t
k Aðt Þdt μab
ð18Þ
When the boundary condition at x = L is applied, Equation 9 becomes AðtÞLðtÞ þ Bðt Þ ¼ C
ð19Þ
Since a quasi-steady state is presumed in this study, the following equation is obtained. Z L vf 0 1 vf μab t ð20Þ dx ¼ LðtÞ 1 vf 0 k ðC BÞ 0 vf By changing an integration variable from flow distance to fiber volume fraction, Equation 20 is re-written as below. Z vmax 4 1=b4 vf 0 μab t 1 vf vf dvf ¼ ð21Þ 1 vf 0 k ðC BÞ2 vmin Lðt Þ2 Now, the following closed form solution for flow front advancement is obtained. " !#vmax vf 0 bvf μab b 4b 6b 1=b v 4b þ þ þ þ ð1 bÞvf 1 vf0 kðC BÞ2 f 1þb ð2b 1Þv2f ð1 3bÞv3f vmin
¼
t LðtÞ2 ð22Þ
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On the other hand, effective permeability can be defined by considering the flow front progression as follows. L2 ¼
Keff t μ
ð23Þ
Therefore, the effective permeability is expressed as vf 0 1 ¼ Keff 1 vf 0
ab k ðC BÞ2
" 1=b vf
bvf b 4b 6b 4b þ þ þ þ ð1 bÞvf 1þb ð2b 1Þv2f ð1 3bÞv3f
!#vmax vmin
ð24Þ This equation shows how the effective permeability is influenced by the compaction behavior of preforms. 3 Results and Discussion In the current study, parametric analyses of resin flow through porous media were carried out. In order to evaluate the effect of compaction behavior of preforms on the fiber volume fraction and pressure fields, fiber preforms with different compaction behavior were taken into account. The input parameters used in this study are listed Table I. In Equation 3, a and b are the fiber volume fraction at 1 Pa and the stiffening index standing for compressibility of preforms, respectively. Figure 2 presents the compaction behavior of preforms considered in this study. The completely different compaction behavior is shown according to b values. The fiber volume fraction distribution is displayed in Figure 3. The x-axis is nondimensionalized by using flow front advancement distance. As can be seen in the figure, the different fiber volume fraction fields are obtained. As the b value increases, the fiber volume fraction has a lower gradient near by the injection line of resin. Figure 4 exhibits the dimensionless pressure fields throughout the entire mold. It is shown that the increase in the b leads to more curved pressure field. Equation 9 indicates that the only compaction behavior determines the fiber volume fraction and resin pressure fields. It is interesting to see that shapes of these fields are not affected by permeability. Figure 5 shows the flow front advancement through fiber preforms with respect time. The different fiber volume fraction and pressure fields caused by the different compaction behavior yield the different flow front progression. Figure 6 demonstrates the effect of Table I Input parameters for mathematical modeling adopted in this study.
Values a b vmin vmax k (m2) μ (Pa·s) L (m)
0.1 0.1, 0.2, 0.3 0.3 0.6 1.0 × 10−8, 1.0 × 10−9, 1.0 × 10−10 0.4 1.0
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10000
Compaction pressure (Pa)
8000
b=0.1 b=0.2 b=0.3
6000
4000
2000
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fiber volume fraction Figure 2 Compaction behavior of preforms with different b values.
0.60
Fiber volume fraction
0.55
b=0.1 b=0.2 b=0.3
0.50
0.45
0.40
0.35
0.30 0.0
0.2
0.4
0.6
Dimensionless distance Figure 3 Fiber volume fraction profiles with respect to dimensionless distance
0.8
1.0
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1.0 b=0.1 b=0.2 b=0.3
Dimensionless pressure
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Dimensionless distance Figure 4 Resin pressure fields as a function of dimensionless distance
0.005 b=0.1 b=0.2 b=0.3
Flow advancement (m)
0.004
0.003
0.002
0.001
0.000 0
10
20
30
40
Time (min) Figure 5 Flow front advancement with respect to time
50
60
70
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0.010 k=10-8 k=10-9 k=10-10
Flow advancement (m)
0.008
0.006
0.004
0.002
0.000 0
10
20
30
40
50
60
70
Time (min) Figure 6 Flow advancement with respect to k when b = 0.2
permeability on the flow front advancement. It is found that more than the compaction behavior, the permeability has more significant impact on the flow front advancement. The current works shows that in LCM, the fiber volume fraction distribution, pressure field, and permeability field are interacted complicatedly.
4 Conclusions The analytical solutions for resin infusion process through porous media were proposed. The focus of this study is on understanding the effect of compaction behavior of fiber preforms on the LCM while resin is infused. It is found that in accordance with the compaction behavior of fiber preforms, the totally different fiber volume fraction and pressure fields during the resin infusion process are obtained, which also yields the different permeability distribution and flow front advancement. These findings indicate that the compaction behavior of fiber preforms has great influence on the resin infusion process and that it is crucial to understand the preform compaction behavior for analyzing fiber reinforced composite manufacturing.
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