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Enhancements of the simulation method on the edge effect in resin transfer molding processes
Materials Science and Engineering A 478 (2008) 384–389
Enhancements of the simulation method on the edge effect in resin transfer molding processes Junying Yang a,1 , Yu Xi Jia a,b,∗ , Sheng Sun a , Dong Jun Ma c , Tong Fei Shi b , Li Jia An b,∗∗ a School of Materials Science and Engineering, Shandong University, Jinan 250061, China State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China c School of Engineering Science, University of Science and Technology of China, Hefei 230027, China b
Received 18 November 2006; received in revised form 5 June 2007; accepted 12 July 2007
Abstract In resin transfer molding processes, small clearances exist between the fiber preform and the mold edges, which result in a preferential resin flow in the edge channel and then disrupt the flow patterns during the mold filling stage. A mathematical model including the effect of cavity thickness on resin flow was developed for flow behavior involving the interface between an edge channel and a porous medium. According to the mathematical analysis of momentum equations in a fully developed rectangular duct and formulations of the equivalent edge permeability, comparing with three-dimensional Navier–Stokes equations, the governing equations were modified in the edge channel. The volume of fluid (VOF) method was applied to track the flow front. A simple case is numerically simulated using the modified governing equations. The effects of edge channel width and cavity thickness on flow front and inlet pressure are analyzed, and the evolution characteristics of simulated results are in agreement with the experimental results. © 2007 Elsevier B.V. All rights reserved. Keywords: RTM; Edge effect; Numerical simulation; Modified governing equations
1. Introduction The resin transfer molding is an efficient and economical manufacturing process, which has gained popularity in the preparation of fiber-reinforced polymer–matrix composites. In RTM processes, a preform made of glass or carbon fabrics is cut into the desired shape and preplaced in the mold, then the mold is closed and a low viscosity resin is injected into the mold. Since it is difficult to precisely cut the fiber preform enough to fit the mold, sometimes a clearance may exist between the preform and the mold edges. This clearance can create a preferential flow path, called edge flow, which may
∗
Corresponding author at: School of Materials Science and Engineering, Shandong University, Jinan 250061, China. Tel.: +86 531 88395811; fax: +86 531 88395811. ∗∗ Corresponding author. Tel.: +86 531 88395811; fax: +86 531 88395811. E-mail addresses: jia [email protected] (Y.X. Jia), [email protected] (L.J. An). 1 Present address: School of Materials Science and Engineering, Dalian Jiaotong University, Dalian 116028, China. 0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.07.044
disrupt the flow patterns during the mold filling stage. Such flow perturbation is called the edge effect. The presence of the edge effect often results in incomplete wetting of the fiber preform, dry spot formation and other defects in the final composite part. In most work, the filling stage was considered as resin flow through a porous medium, which was described by Darcy’s law. Finite element method or finite difference method was adopted to solve the flow field [1–6]. However, the simulated results cannot reflect the real mold filling process because of not considering the edge effect. Wang et al. [7] reported that the permeability near the side wall was larger than that on the core region, and resulted in the edge effect near the side wall. Gupte and Advani [8,9] carried out a detailed experimental investigation of the flow by laser doppler anemometry (LDA) in the close vicinity of the permeable boundary of a porous medium. They found that the boundary layer in the porous medium was of the order of the thickness of the Hele-Shaw cell and not of the order of the permeability of the porous medium as long as the Hele-Shaw approximation was valid.
J. Yang et al. / Materials Science and Engineering A 478 (2008) 384–389
The equivalent permeability approach has been used frequently to simulate the edge flow in the past. Wu et al. [10] simulated the edge effect by the creeping motion of the resin with permeable boundary condition along the edge of the fiber mat. Results showed that the edge effect was very sensitive to the shape of the fiber mat. The edge effect was negligible as the dimension of the fiber mat was slightly larger than that of the mold cavity. Hammami et al. [11] presented a simple model of the edge channel to simulate the edge flow in a rectangular mold cavity with an edge channel along one side. In their studies, an equivalent permeability was used to describe the flow in the edge channel. Derivation of the equivalent edge permeability was based on the analysis of channel flow field between parallel plates. Hammami et al. [12] proposed an equivalent permeability model to integrate the cavity thickness. This model assumed that the flow was taking place in a cylinder, having the same cross-section as the channel. The effect of transverse flow from the edge channel to the preform was considered, developing a transverse flow factor to evaluate the amount of such flow. They suggested a limit value of the transverse flow factor, above which transverse flow should be taken into account in the equivalent permeability used. Ni et al. [13] established a simplified model for the multiregional flow. Both the fiber-free and the fibrous regions could be described by Darcy’s equation. They presented equivalent permeability formulas for various cross-sections such as pipe flow, flow between two parallel plates and rectangular channel flow. The slip velocity condition proposed by Beavers and Joseph [14] was applied for the steady state flow including a permeable boundary. Young and Lai [15] developed an approximate model to simulate the edge channeling flow. An equivalent permeability based on the size of the edge clearance was used to model the resistance of flow in the channel. The slip boundary condition was not considered during deriving the equivalent edge permeability due to its slight effect. Bickerton and Advani [16,17] studied a planar rectangular mold cavity, which had a single air cavity of known dimensions running along one side of the mold. The equivalent permeability magnitudes were based upon steady state, fully developed flow through a rectangular duct. Flow visualization experiments and simulations have been completed, recording flow front advancement and injection pressure histories, for two different preform types, and a variety of mold cavity thicknesses, preform volume fractions and air-channel widths. Several researchers adopted another approach, in which the Navier–Stokes equation was used to describe the flow in the edge channel. Costa et al. [18,19] presented the control volume based on finite element methods for flow problems dealing with combined free/porous medium. The Navier–Stokes equations and the Brinkman–Forchheimer-extended Darcy equations were used to model the free medium and the porous medium, respectively. The numerical approach avoided the well-known difficulties associated with linking the Navier–Stokes and the Darcy equations. But their studies were limited to steady state, two-dimensional fully developed the flow problems. The location and shape of the flow front at any time were not simulated. In this paper, a mathematical model including the effect of cavity thickness on resin flow is developed for flow behavior
385
Fig. 1. Schematic of the edge effect.
involving the interface between an edge channel and a porous medium. The volume of fluid (VOF) method is applied to track the flow front. The effects of edge channel width and cavity thickness on flow front and inlet pressure are analyzed. 2. Numerical simulation based on modified governing equations The edge effect is illustrated in Fig. 1, where the clearance d is the dimension between the mold wall and the fiber preform. The flow in the fiber preform and the edge channel are modelled as two-phase (resin and air) fluid flows through porous media and free fluid region, respectively. 2.1. Construction of mathematical models 2.1.1. Flow in the edge channel The conservation of mass is ∂u ∂v + =0 ∂x ∂y
(1)
where u and v are the velocity components in the x- and ydirection, respectively. In order to investigate the effect of cavity thickness on resin flow, three-dimensional Navier–Stokes equations are introduced. In the edge channel, the velocity component in the thickness direction is very small, so the z-momentum equation is neglected. Then, the three-dimensional momentum equations can be rewritten as 2 ∂(ρu) ∂(ρuu) ∂(ρυu) ∂p ∂ u ∂2 u ∂2 u + + = − +μ + + ∂t ∂x ∂y ∂x ∂x2 ∂y2 ∂z2 (2)
∂(ρυ) ∂(ρuυ) ∂(ρυυ) ∂p + + = − +μ ∂t ∂x ∂y ∂y
∂2 υ ∂2 υ ∂2 υ + + ∂x2 ∂y2 ∂z2
(3)
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J. Yang et al. / Materials Science and Engineering A 478 (2008) 384–389
where t denotes time, p the fluid pressure, ρ the fluid density and μ is the dynamic viscosity. In Ref. [26], by analyzing the steady state Stokes flow, the flow rate in a fully developed rectangular duct was obtained. According to the research, for the case of a well-defined edge channel within a mold, the analytical solutions for the flow rate were used in conjunction with Darcy’s law in one dimension, and then the expression for the equivalent permeability is derived [15]: 192h h2 πd Kme = 1 − 5 tan h , d≤h (4) 96 π d 2h 192d d2 πh 1 − 5 tan h , d>h (5) Kme = 96 π h 2d
The application of the momentum equations considering inertia terms is more extensive than that of Darcy’s law, especially for high filling velocity. For the sake of comparison, the momentum equations in the fiber preform still include the inertia effects. They take in the form of fiber preform [20]: 1 ∂(ρuu) ∂(ρuv) 1 ∂(ρu) + 2 + φ ∂t φ ∂x ∂y 2 f 2 ∂ u ∂ u μ ∂p − + u (13) + μeff =− 2 2 ∂x ∂x ∂y Kx
where h denotes the cavity thickness. In our work, when the x-direction is assumed as the main flow direction, according to the analytical solutions for the flow rate in a fully developed rectangular duct and the formulation of the equivalent edge permeability, comparing with Eq. (2), the x-momentum equation in the edge channel is modified as
(14)
where pf denotes the intrinsic phase-averaged pressure of the fluid, φ the porosity and K is the permeability of the fiber preform. μeff is an effective viscosity, μeff = μ/φ. The fractional volume function advection equation is
∂p ∂2 u μ ∂(ρu) ∂(ρuu) ∂(ρυu) + + =− +μ 2 − u ∂t ∂x ∂y ∂x ∂x Kme
u ∂f v ∂f ∂f + + =0 ∂t φ ∂x φ ∂y
(6)
When the x-direction is the main flow direction, the effect of the y-momentum equation is minor, so the viscous force in the thickness direction of the y-momentum equation could be omitted. Thus, Eq. (3) in the edge channel is simplified as 2 ∂(ρυ) ∂(ρuυ) ∂(ρυυ) ∂p ∂ υ ∂2 υ (7) + + =− +μ + 2 ∂t ∂x ∂y ∂y ∂x2 ∂y For each control volume, there are the formulations: μ = fμ1 + (1 − f )μ2
(8)
ρ = fρ1 + (1 − f )ρ2
(9)
where the subscripts denote the different fluids. The fractional volume function is defined as follows 1 for the point (x, y, t) inside fluid 1 f (x, y, t) = 0 for the point (x, y, t) inside fluid 2
(10)
where the fluid 1 denotes the resin and fluid 2 the air. Then, the interface is located within the control volumes where 0 < f < 1. The fractional volume function advection equation is ∂f ∂f ∂f +u +v =0 ∂t ∂x ∂y
(11)
1 ∂(ρv) 1 ∂(ρuv) ∂(ρvv) + 2 + φ ∂t φ ∂x ∂y ∂2 v ∂2 v μ ∂pf − + μeff + v =− ∂y ∂x2 ∂y2 Ky
where velocities within the reinforcement for the new flow front determination must be corrected by a factor of φ−1 . 2.1.3. Boundary conditions Schematic of boundary conditions is shown in Fig. 2. In this paper, the single-domain approach is adopted for the solution of the flow field of the fiber preform and edge channel, and the whole region is considered as a continuum region. The approach avoids the explicit formulation of the boundary conditions at the interface between the edge channel and the fiber preform. The following conditions for the continuity of velocity and normal stress are satisfied spontaneously: u = u (16) T · n = T · n where T is the stress tensor. The imposing of slip or no-slip boundary condition at the mold wall will result in the unrealistic interface predictions. The procedure described in Refs. [21,22] is adopted to overcome this problem. No resin is allowed to cross the mold wall, but the air is assumed to be free to leave the mold as pushed out by the advancing resin. Therefore, no-slip and traction free
where f denotes the volume fraction of resin phase. 2.1.2. Flow in the fiber preform In the fiber preform, the conservation of mass is ∂u ∂v =0 + ∂y ∂x
(12)
where u, v are phase-averaged velocity components in the x- and y-direction, respectively.
(15)
Fig. 2. Schematic of boundary conditions.
J. Yang et al. / Materials Science and Engineering A 478 (2008) 384–389
387
Fig. 3. Flow chart of the numerical simulation.
boundary conditions are switched dynamically according to the filling status there. for the filled mold wall (resin), Γwall,resin : u = 0,
when f ≥ fc
(17)
for the empty mold wall (air), Γwall,air : T · n = 0,
when f < fc
(18)
Fig. 4. Transient mold filling process sketches from the flow visualization experiment [13].
2.3. Simulated results and comparison with experiment Additional boundary conditions such as inlet and outlet conditions must be applied. 2.2. Simulation procedures The finite volume method (FVM) based on orthogonal grid is used to discretize the governing equations. To minimize storage requirements, a segregated solution strategy is favored with pressure and velocity coupled using the semi-implicit method for pressure linked equations (SIMPLE) algorithm. The staggered grid arrangement is applied in order to avoid the unrealistic behavior of the discretized momentum equations for spatially oscillation pressures like the ‘checkerboard’ pressure field [23]. The piecewise linear interface construction (PLIC–VOF) method is applied to track the flow front [24]. The flow chart of the numerical simulation is given in Fig. 3. The simulation algorithm is validated in Ref. [25].
An example is obtained from Ref. [13]. Their experimental results and our simulated results are shown in Figs. 4 and 5, respectively. For saving the computational time, we omit the right part of free fiber in the model.
Fig. 5. The simulated flow fronts based on modified governing equations.
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Fig. 6. Geometry of the filling model. Table 1 Input parameters related to the effect of edge channel width on flow Parameters
Numerical values
Resin viscosity, μ (Pa s) Flow rate, u (mm/s) Porosity, φ Permeability of the fiber preform, K (× 10−9 m2 ) Thickness of the cavity, h (mm) Air viscosity (× 10−5 Pa s)
0.1 5 0.81 2 4 [16] 2.1
It can be seen that the flow fronts in the upper and bottom edge channels are ahead of the bulk flow in the fiber preform region. With the filling process, the resin flows slowly into the fiber preform in the main flow direction. Since the pressure in the edge channels in y-direction is larger than that in the fiber preform, the resin is able to penetrate into the fiber preform through the permeable interface. The evolution characteristics of simulated results are in agreement with the experimental results by Ni et al. [13]. 3. Effect of key process parameters on flow patterns A simple model of the edge channel to simulate the edge flow in a rectangular mold cavity with an edge channel along one side is constructed in the cartesian coordinate system, as shown in Fig. 6. The condition at the entrance of the model is a constant flow rate of the resin. 3.1. Effect of edge channel width on flow patterns The filling parameters related to the effect of edge channel width on flow are listed in Table 1. Fig. 7 shows the effect of edge channel width on flow fronts. The filling times are 13.1 s
Fig. 8. Inlet pressure histories under various edge channel widths.
and 22.5 s and the edge channel widths are 2.0 mm, 5.0 mm and 8.0 mm, respectively. Because the viscous force in the edge channel is much less than that within the fiber preform, the edge flow is always ahead of the bulk flow within the fiber preform. The flow lead–lag distance between the bulk flow and the edge flow in the x-direction becomes larger, and the transverse flow is more obvious with the increase of edge channel width, such as 1–3 or 4–6 in Fig. 7. The reason is that the increase of edge channel width leads to the enhancement of the flow in the region. At the same edge channel width, the difference between the two flow fronts becomes larger with the filling process. The evolution characteristics of simulated results are in agreement with the work by Bickerton [16] and Hammami et al. [11]. The corresponding inlet pressure histories under various edge channel widths are presented in Fig. 8. After the flow fronts arrive at the fiber preform region, the inlet pressure almost linearly increases with the filling process since the total friction between the fluid and the fiber preform increases as the contacting area increases. It decreases with the increase of edge channel width. The evolution characteristics of simulated results agree with the work by Bickerton [16]. 3.2. Effect of cavity thickness on flow patterns The filling parameters related to the effect of cavity thickness on flow are listed in Table 2. Fig. 9 illustrates the effect of cavity thickness on flow fronts. The filling times are 13.0 s and 21.0 s and the cavity thicknesses are 2.5 mm, 5.0 mm and 7.5 mm, respectively. It can be seen that the flow lead–lag disTable 2 Input parameters related to the effect of cavity thickness on flow
Fig. 7. Effect of edge channel width on flow fronts.
Parameters
Numerical values
Resin viscosity, μ (Pa s) Flow rate, u (mm/s) Porosity, φ Permeability of the fiber preform, K (× 10−9 m2 ) Width of the edge channel, d (mm) Air viscosity (× 10−5 Pa s)
0.1 5 0.81 2 5 [16] 2.1
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Capillary pressure induced by surface tension can strongly augment transverse flow into a porous medium. The simulation work considering surface tension effects will be carried out, and the oblique channel will be treated in the future. Acknowledgements
Fig. 9. Effect of cavity thickness on flow fronts.
The authors would like to thank Prof. Wenbin Young for helpful discussions. This work is supported by the National Natural Science Foundation of China (50573079, 50390096 and 10602056) Programs and subsidized by the Special Funds for National Basic Research Program of China (2003CB615601). References
Fig. 10. Inlet pressure histories under various cavity thicknesses.
tance between the bulk flow and the edge flow in the x-direction becomes larger with the increase of cavity thickness. The reason is that the increase of cavity thickness leads to the decrease of viscous force in the edge channel region. The evolution characteristics of simulated results are in agreement with the work by Bickerton [16]. The corresponding inlet pressure histories under various cavity thicknesses are presented in Fig. 10. After the flow fronts arrive at the fiber preform region, the inlet pressure almost linearly increases with the filling process. It decreases with the increase of cavity thickness. The evolution characteristics of simulated results agree with the work by Bickerton [16]. 4. Conclusions A mathematical model including the effect of cavity thickness on resin flow was developed for resin flow involving the interface between an edge channel and a porous medium. The flow lead–lag distance between the bulk flow in the fiber preform and the edge flow in the edge channel region becomes larger with the increase of edge channel width and cavity thickness, and the transverse flow is more obvious with the increase of edge channel width. The evolution characteristics of simulated results agree with the experimental results.
[1] M.V. Bruschke, S.G. Advani, Polym. Compos. 11 (6) (1990) 398–405. [2] W.B. Young, K. Han, L.H. Fong, L.J. Lee, M.J. Liou, Polym. Compos. 12 (6) (1991) 391–403. [3] A.W. Chan, S.T. Hwang, Polym. Eng. Sci. 31 (15) (1991) 1149–1156. [4] M.K. Um, W.I. Lee, Proceedings of the 35th International SAMPE Symposium and Exhibition, vol. 35, 1990, pp. 1905–1916. [5] R. Gauvin, F. Trochu, Plast. Rubb. Comp. Proc. Appl. 19 (3) (1993) 151–157. [6] R.V. Mohan, N.D. Ngo, K.K. Tamma, Numer. Heat Transfer Part A 35 (8) (1999) 815–838. [7] T.J. Wang, C.H. Wu, L.J. Lee, Polym. Compos. 15 (4) (1994) 278–288. [8] S.K. Gupte, S.G. Advani, Exp. Fluids 22 (5) (1997) 408–422. [9] S.K. Gupte, S.G. Advani, Polym. Compos. 18 (1) (1997) 114–124. [10] C.J. Wu, L.W. Hourng, J.C. Liao, J. Reinf. Plast. Compos. 14 (7) (1995) 694–722. [11] A. Hammami, R. Gauvin, F. Trochu, O. Touret, P. Ferland, Appl. Compos. Mater. 5 (3) (1998) 161–173. [12] A. Hammami, R. Gauvin, F. Trochu, Composites Part A 29 (5–6) (1998) 603–609. [13] J. Ni, Y. Zhao, L.J. Lee, S. Nakamura, Polym. Compos. 18 (2) (1997) 254–269. [14] G.S. Beavers, D.D. Joseph, J. Fluid Mech. 30 (1967) 197–207. [15] W.B. Young, C.L. Lai, Composites Part A 28 (9–10) (1997) 817–822. [16] S. Bickerton, S.G. Advani, Compos. Sci. Technol. 59 (15) (1999) 2215–2229. [17] S. Bickerton, S.G. Advani, Proceedings of the ASME AMD-MD, June 1995. [18] V.A.F. Costa, M.S.A. Oliveira, A.C.M. Sousa, Comput. Struct. 82 (17–19) (2004) 1535–1541. [19] V.A.F. Costa, L.A. Oliveira, B.R. Baliga, A.C.M. Sousa, Numer. Heat Transfer Part A 45 (7) (2004) 675–697. [20] B. Goyeau, D. Lhuillier, D. Gobin, M.G. Velarde, Int. J. Heat Mass Transfer 46 (21) (2003) 4071–4081. [21] J.-F. H´etu, D.M. Gao, A. Garcia-Rejon, G. Salloum, Polym. Eng. Sci. 38 (2) (1998) 223–236. [22] R.Y. Chang, W.H. Yang, Int. J. Numer. Methods Fluids 37 (2) (2001) 125–148. [23] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics, World Publishing Corporation Beijing, Beijing, 2000, pp. 135–146. [24] D. Gueyffier, J. Li, A. Nadim, R. Scardovelli, S. Zaleski, J. Comput. Phys. 152 (2) (1999) 423–456. [25] J.Y. Yang, Y.X. Jia, S. Sun, D.J. Ma, T.F. Shi, L.J. An, Mater. Sci. Eng. A 435–436 (5) (2006) 515–520. [26] F.M. White, Viscous Fluid Flow, McGraw-Hill, 1974, p. 123.
Enhancements of the simulation method on the edge effect in resin transfer molding processes Junying Yang a,1 , Yu Xi Jia a,b,∗ , Sheng Sun a , Dong Jun Ma c , Tong Fei Shi b , Li Jia An b,∗∗ a School of Materials Science and Engineering, Shandong University, Jinan 250061, China State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China c School of Engineering Science, University of Science and Technology of China, Hefei 230027, China b
Received 18 November 2006; received in revised form 5 June 2007; accepted 12 July 2007
Abstract In resin transfer molding processes, small clearances exist between the fiber preform and the mold edges, which result in a preferential resin flow in the edge channel and then disrupt the flow patterns during the mold filling stage. A mathematical model including the effect of cavity thickness on resin flow was developed for flow behavior involving the interface between an edge channel and a porous medium. According to the mathematical analysis of momentum equations in a fully developed rectangular duct and formulations of the equivalent edge permeability, comparing with three-dimensional Navier–Stokes equations, the governing equations were modified in the edge channel. The volume of fluid (VOF) method was applied to track the flow front. A simple case is numerically simulated using the modified governing equations. The effects of edge channel width and cavity thickness on flow front and inlet pressure are analyzed, and the evolution characteristics of simulated results are in agreement with the experimental results. © 2007 Elsevier B.V. All rights reserved. Keywords: RTM; Edge effect; Numerical simulation; Modified governing equations
1. Introduction The resin transfer molding is an efficient and economical manufacturing process, which has gained popularity in the preparation of fiber-reinforced polymer–matrix composites. In RTM processes, a preform made of glass or carbon fabrics is cut into the desired shape and preplaced in the mold, then the mold is closed and a low viscosity resin is injected into the mold. Since it is difficult to precisely cut the fiber preform enough to fit the mold, sometimes a clearance may exist between the preform and the mold edges. This clearance can create a preferential flow path, called edge flow, which may
∗
Corresponding author at: School of Materials Science and Engineering, Shandong University, Jinan 250061, China. Tel.: +86 531 88395811; fax: +86 531 88395811. ∗∗ Corresponding author. Tel.: +86 531 88395811; fax: +86 531 88395811. E-mail addresses: jia [email protected] (Y.X. Jia), [email protected] (L.J. An). 1 Present address: School of Materials Science and Engineering, Dalian Jiaotong University, Dalian 116028, China. 0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.07.044
disrupt the flow patterns during the mold filling stage. Such flow perturbation is called the edge effect. The presence of the edge effect often results in incomplete wetting of the fiber preform, dry spot formation and other defects in the final composite part. In most work, the filling stage was considered as resin flow through a porous medium, which was described by Darcy’s law. Finite element method or finite difference method was adopted to solve the flow field [1–6]. However, the simulated results cannot reflect the real mold filling process because of not considering the edge effect. Wang et al. [7] reported that the permeability near the side wall was larger than that on the core region, and resulted in the edge effect near the side wall. Gupte and Advani [8,9] carried out a detailed experimental investigation of the flow by laser doppler anemometry (LDA) in the close vicinity of the permeable boundary of a porous medium. They found that the boundary layer in the porous medium was of the order of the thickness of the Hele-Shaw cell and not of the order of the permeability of the porous medium as long as the Hele-Shaw approximation was valid.
J. Yang et al. / Materials Science and Engineering A 478 (2008) 384–389
The equivalent permeability approach has been used frequently to simulate the edge flow in the past. Wu et al. [10] simulated the edge effect by the creeping motion of the resin with permeable boundary condition along the edge of the fiber mat. Results showed that the edge effect was very sensitive to the shape of the fiber mat. The edge effect was negligible as the dimension of the fiber mat was slightly larger than that of the mold cavity. Hammami et al. [11] presented a simple model of the edge channel to simulate the edge flow in a rectangular mold cavity with an edge channel along one side. In their studies, an equivalent permeability was used to describe the flow in the edge channel. Derivation of the equivalent edge permeability was based on the analysis of channel flow field between parallel plates. Hammami et al. [12] proposed an equivalent permeability model to integrate the cavity thickness. This model assumed that the flow was taking place in a cylinder, having the same cross-section as the channel. The effect of transverse flow from the edge channel to the preform was considered, developing a transverse flow factor to evaluate the amount of such flow. They suggested a limit value of the transverse flow factor, above which transverse flow should be taken into account in the equivalent permeability used. Ni et al. [13] established a simplified model for the multiregional flow. Both the fiber-free and the fibrous regions could be described by Darcy’s equation. They presented equivalent permeability formulas for various cross-sections such as pipe flow, flow between two parallel plates and rectangular channel flow. The slip velocity condition proposed by Beavers and Joseph [14] was applied for the steady state flow including a permeable boundary. Young and Lai [15] developed an approximate model to simulate the edge channeling flow. An equivalent permeability based on the size of the edge clearance was used to model the resistance of flow in the channel. The slip boundary condition was not considered during deriving the equivalent edge permeability due to its slight effect. Bickerton and Advani [16,17] studied a planar rectangular mold cavity, which had a single air cavity of known dimensions running along one side of the mold. The equivalent permeability magnitudes were based upon steady state, fully developed flow through a rectangular duct. Flow visualization experiments and simulations have been completed, recording flow front advancement and injection pressure histories, for two different preform types, and a variety of mold cavity thicknesses, preform volume fractions and air-channel widths. Several researchers adopted another approach, in which the Navier–Stokes equation was used to describe the flow in the edge channel. Costa et al. [18,19] presented the control volume based on finite element methods for flow problems dealing with combined free/porous medium. The Navier–Stokes equations and the Brinkman–Forchheimer-extended Darcy equations were used to model the free medium and the porous medium, respectively. The numerical approach avoided the well-known difficulties associated with linking the Navier–Stokes and the Darcy equations. But their studies were limited to steady state, two-dimensional fully developed the flow problems. The location and shape of the flow front at any time were not simulated. In this paper, a mathematical model including the effect of cavity thickness on resin flow is developed for flow behavior
385
Fig. 1. Schematic of the edge effect.
involving the interface between an edge channel and a porous medium. The volume of fluid (VOF) method is applied to track the flow front. The effects of edge channel width and cavity thickness on flow front and inlet pressure are analyzed. 2. Numerical simulation based on modified governing equations The edge effect is illustrated in Fig. 1, where the clearance d is the dimension between the mold wall and the fiber preform. The flow in the fiber preform and the edge channel are modelled as two-phase (resin and air) fluid flows through porous media and free fluid region, respectively. 2.1. Construction of mathematical models 2.1.1. Flow in the edge channel The conservation of mass is ∂u ∂v + =0 ∂x ∂y
(1)
where u and v are the velocity components in the x- and ydirection, respectively. In order to investigate the effect of cavity thickness on resin flow, three-dimensional Navier–Stokes equations are introduced. In the edge channel, the velocity component in the thickness direction is very small, so the z-momentum equation is neglected. Then, the three-dimensional momentum equations can be rewritten as 2 ∂(ρu) ∂(ρuu) ∂(ρυu) ∂p ∂ u ∂2 u ∂2 u + + = − +μ + + ∂t ∂x ∂y ∂x ∂x2 ∂y2 ∂z2 (2)
∂(ρυ) ∂(ρuυ) ∂(ρυυ) ∂p + + = − +μ ∂t ∂x ∂y ∂y
∂2 υ ∂2 υ ∂2 υ + + ∂x2 ∂y2 ∂z2
(3)
386
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where t denotes time, p the fluid pressure, ρ the fluid density and μ is the dynamic viscosity. In Ref. [26], by analyzing the steady state Stokes flow, the flow rate in a fully developed rectangular duct was obtained. According to the research, for the case of a well-defined edge channel within a mold, the analytical solutions for the flow rate were used in conjunction with Darcy’s law in one dimension, and then the expression for the equivalent permeability is derived [15]: 192h h2 πd Kme = 1 − 5 tan h , d≤h (4) 96 π d 2h 192d d2 πh 1 − 5 tan h , d>h (5) Kme = 96 π h 2d
The application of the momentum equations considering inertia terms is more extensive than that of Darcy’s law, especially for high filling velocity. For the sake of comparison, the momentum equations in the fiber preform still include the inertia effects. They take in the form of fiber preform [20]: 1 ∂(ρuu) ∂(ρuv) 1 ∂(ρu) + 2 + φ ∂t φ ∂x ∂y 2 f 2 ∂ u ∂ u μ ∂p − + u (13) + μeff =− 2 2 ∂x ∂x ∂y Kx
where h denotes the cavity thickness. In our work, when the x-direction is assumed as the main flow direction, according to the analytical solutions for the flow rate in a fully developed rectangular duct and the formulation of the equivalent edge permeability, comparing with Eq. (2), the x-momentum equation in the edge channel is modified as
(14)
where pf denotes the intrinsic phase-averaged pressure of the fluid, φ the porosity and K is the permeability of the fiber preform. μeff is an effective viscosity, μeff = μ/φ. The fractional volume function advection equation is
∂p ∂2 u μ ∂(ρu) ∂(ρuu) ∂(ρυu) + + =− +μ 2 − u ∂t ∂x ∂y ∂x ∂x Kme
u ∂f v ∂f ∂f + + =0 ∂t φ ∂x φ ∂y
(6)
When the x-direction is the main flow direction, the effect of the y-momentum equation is minor, so the viscous force in the thickness direction of the y-momentum equation could be omitted. Thus, Eq. (3) in the edge channel is simplified as 2 ∂(ρυ) ∂(ρuυ) ∂(ρυυ) ∂p ∂ υ ∂2 υ (7) + + =− +μ + 2 ∂t ∂x ∂y ∂y ∂x2 ∂y For each control volume, there are the formulations: μ = fμ1 + (1 − f )μ2
(8)
ρ = fρ1 + (1 − f )ρ2
(9)
where the subscripts denote the different fluids. The fractional volume function is defined as follows 1 for the point (x, y, t) inside fluid 1 f (x, y, t) = 0 for the point (x, y, t) inside fluid 2
(10)
where the fluid 1 denotes the resin and fluid 2 the air. Then, the interface is located within the control volumes where 0 < f < 1. The fractional volume function advection equation is ∂f ∂f ∂f +u +v =0 ∂t ∂x ∂y
(11)
1 ∂(ρv) 1 ∂(ρuv) ∂(ρvv) + 2 + φ ∂t φ ∂x ∂y ∂2 v ∂2 v μ ∂pf − + μeff + v =− ∂y ∂x2 ∂y2 Ky
where velocities within the reinforcement for the new flow front determination must be corrected by a factor of φ−1 . 2.1.3. Boundary conditions Schematic of boundary conditions is shown in Fig. 2. In this paper, the single-domain approach is adopted for the solution of the flow field of the fiber preform and edge channel, and the whole region is considered as a continuum region. The approach avoids the explicit formulation of the boundary conditions at the interface between the edge channel and the fiber preform. The following conditions for the continuity of velocity and normal stress are satisfied spontaneously: u = u (16) T · n = T · n where T is the stress tensor. The imposing of slip or no-slip boundary condition at the mold wall will result in the unrealistic interface predictions. The procedure described in Refs. [21,22] is adopted to overcome this problem. No resin is allowed to cross the mold wall, but the air is assumed to be free to leave the mold as pushed out by the advancing resin. Therefore, no-slip and traction free
where f denotes the volume fraction of resin phase. 2.1.2. Flow in the fiber preform In the fiber preform, the conservation of mass is ∂u ∂v =0 + ∂y ∂x
(12)
where u, v are phase-averaged velocity components in the x- and y-direction, respectively.
(15)
Fig. 2. Schematic of boundary conditions.
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Fig. 3. Flow chart of the numerical simulation.
boundary conditions are switched dynamically according to the filling status there. for the filled mold wall (resin), Γwall,resin : u = 0,
when f ≥ fc
(17)
for the empty mold wall (air), Γwall,air : T · n = 0,
when f < fc
(18)
Fig. 4. Transient mold filling process sketches from the flow visualization experiment [13].
2.3. Simulated results and comparison with experiment Additional boundary conditions such as inlet and outlet conditions must be applied. 2.2. Simulation procedures The finite volume method (FVM) based on orthogonal grid is used to discretize the governing equations. To minimize storage requirements, a segregated solution strategy is favored with pressure and velocity coupled using the semi-implicit method for pressure linked equations (SIMPLE) algorithm. The staggered grid arrangement is applied in order to avoid the unrealistic behavior of the discretized momentum equations for spatially oscillation pressures like the ‘checkerboard’ pressure field [23]. The piecewise linear interface construction (PLIC–VOF) method is applied to track the flow front [24]. The flow chart of the numerical simulation is given in Fig. 3. The simulation algorithm is validated in Ref. [25].
An example is obtained from Ref. [13]. Their experimental results and our simulated results are shown in Figs. 4 and 5, respectively. For saving the computational time, we omit the right part of free fiber in the model.
Fig. 5. The simulated flow fronts based on modified governing equations.
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Fig. 6. Geometry of the filling model. Table 1 Input parameters related to the effect of edge channel width on flow Parameters
Numerical values
Resin viscosity, μ (Pa s) Flow rate, u (mm/s) Porosity, φ Permeability of the fiber preform, K (× 10−9 m2 ) Thickness of the cavity, h (mm) Air viscosity (× 10−5 Pa s)
0.1 5 0.81 2 4 [16] 2.1
It can be seen that the flow fronts in the upper and bottom edge channels are ahead of the bulk flow in the fiber preform region. With the filling process, the resin flows slowly into the fiber preform in the main flow direction. Since the pressure in the edge channels in y-direction is larger than that in the fiber preform, the resin is able to penetrate into the fiber preform through the permeable interface. The evolution characteristics of simulated results are in agreement with the experimental results by Ni et al. [13]. 3. Effect of key process parameters on flow patterns A simple model of the edge channel to simulate the edge flow in a rectangular mold cavity with an edge channel along one side is constructed in the cartesian coordinate system, as shown in Fig. 6. The condition at the entrance of the model is a constant flow rate of the resin. 3.1. Effect of edge channel width on flow patterns The filling parameters related to the effect of edge channel width on flow are listed in Table 1. Fig. 7 shows the effect of edge channel width on flow fronts. The filling times are 13.1 s
Fig. 8. Inlet pressure histories under various edge channel widths.
and 22.5 s and the edge channel widths are 2.0 mm, 5.0 mm and 8.0 mm, respectively. Because the viscous force in the edge channel is much less than that within the fiber preform, the edge flow is always ahead of the bulk flow within the fiber preform. The flow lead–lag distance between the bulk flow and the edge flow in the x-direction becomes larger, and the transverse flow is more obvious with the increase of edge channel width, such as 1–3 or 4–6 in Fig. 7. The reason is that the increase of edge channel width leads to the enhancement of the flow in the region. At the same edge channel width, the difference between the two flow fronts becomes larger with the filling process. The evolution characteristics of simulated results are in agreement with the work by Bickerton [16] and Hammami et al. [11]. The corresponding inlet pressure histories under various edge channel widths are presented in Fig. 8. After the flow fronts arrive at the fiber preform region, the inlet pressure almost linearly increases with the filling process since the total friction between the fluid and the fiber preform increases as the contacting area increases. It decreases with the increase of edge channel width. The evolution characteristics of simulated results agree with the work by Bickerton [16]. 3.2. Effect of cavity thickness on flow patterns The filling parameters related to the effect of cavity thickness on flow are listed in Table 2. Fig. 9 illustrates the effect of cavity thickness on flow fronts. The filling times are 13.0 s and 21.0 s and the cavity thicknesses are 2.5 mm, 5.0 mm and 7.5 mm, respectively. It can be seen that the flow lead–lag disTable 2 Input parameters related to the effect of cavity thickness on flow
Fig. 7. Effect of edge channel width on flow fronts.
Parameters
Numerical values
Resin viscosity, μ (Pa s) Flow rate, u (mm/s) Porosity, φ Permeability of the fiber preform, K (× 10−9 m2 ) Width of the edge channel, d (mm) Air viscosity (× 10−5 Pa s)
0.1 5 0.81 2 5 [16] 2.1
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Capillary pressure induced by surface tension can strongly augment transverse flow into a porous medium. The simulation work considering surface tension effects will be carried out, and the oblique channel will be treated in the future. Acknowledgements
Fig. 9. Effect of cavity thickness on flow fronts.
The authors would like to thank Prof. Wenbin Young for helpful discussions. This work is supported by the National Natural Science Foundation of China (50573079, 50390096 and 10602056) Programs and subsidized by the Special Funds for National Basic Research Program of China (2003CB615601). References
Fig. 10. Inlet pressure histories under various cavity thicknesses.
tance between the bulk flow and the edge flow in the x-direction becomes larger with the increase of cavity thickness. The reason is that the increase of cavity thickness leads to the decrease of viscous force in the edge channel region. The evolution characteristics of simulated results are in agreement with the work by Bickerton [16]. The corresponding inlet pressure histories under various cavity thicknesses are presented in Fig. 10. After the flow fronts arrive at the fiber preform region, the inlet pressure almost linearly increases with the filling process. It decreases with the increase of cavity thickness. The evolution characteristics of simulated results agree with the work by Bickerton [16]. 4. Conclusions A mathematical model including the effect of cavity thickness on resin flow was developed for resin flow involving the interface between an edge channel and a porous medium. The flow lead–lag distance between the bulk flow in the fiber preform and the edge flow in the edge channel region becomes larger with the increase of edge channel width and cavity thickness, and the transverse flow is more obvious with the increase of edge channel width. The evolution characteristics of simulated results agree with the experimental results.
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