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Pressure distribution in resin transfer molding with a non-rigid fiber preform
Journal of Materials Processing Technology, 37 (1993) 363 371
363
Elsevier
Pressure distribution in resin transfer molding with a non-rigid fiber preform Haiqing Gong
School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263
Industrial
Summary
In mold filling processes of composite material manufacturing such as resin transfer molding (RTM), a polymer resin impregnates the fiber mat placed in a mold under a pressure gradient generated by pressure at gates or vacuum at outlets, or both. The flow characteristics in the fiber mat during the impregnation play a critical role in determining the mechanical properties of the final parts, as well as the cost effectiveness of the process. In this work, resin impregnation through a compressible fiber preform is modelled analytically using Darcy's law in an expanding flow domain of a rectangular shape with a uniform velocity profile at the injection gate. The model proposed in this work accommodates the effect of both the compressibility and relaxation of the fiber mat on the volume fraction of the resin. The compressible region of the fiber mat is confined phenomenologically to the front region of the resin impregnation where the impact of the resin on the fiber causes a higher volume fraction of the resin. A relaxation length is introduced in the resin front region to characterize the relaxation of the fiber mat under the impact of the resin. To analyse the pressure distribution, the Kozeny Carman relationship is used for integration of Darcy's law. The influences of the compressibility and relaxation of the preform on the pressure distribution are analysed, the result showing that the pressure level is lower when preform compressibility and relaxation are considered in the modelling. This conclusion is in a good qualitative agreement with existing experimental data.
1. I n t r o d u c t i o n D e v e l o p e d i n t h e 1970s, t h e b a s i c r e s i n t r a n s f e r m o l d i n g p r o c e s s h a s g r a d u ally, o v e r t h e l a s t five y e a r s , b e e n c o n s i d e r e d as a p o t e n t i a l l o w - c o s t m a n u f a c t u r i n g t e c h n o l o g y for f o r m i n g a d v a n c e d s t r u c t u r a l c o m p o s i t e p a r t s . H o w e v e r , r e l a t i v e l y l i t t l e k n o w l e d g e h a s b e e n o b t a i n e d r e g a r d i n g t h e effect of p r o c e s s i n g a n d m a t e r i a l c o n d i t i o n s o n o p t i m i z a t i o n of t h e p r o c e s s d e s i g n . M a n y difficult a n d c o m p l e x p r o b l e m s c o n t r i b u t e to t h e l a c k of a s c i e n t i f i c b a s e i n t h i s a r e a ,
Correspondence to: H. Gong, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263. 0924-0136/93/$06.00 (t2 1993 Elsevier Science Publishers B.V. All rights reserved.
364
H. Gong~Pressure distribution in resin transfer molding
such as characterization of fiber mats as porous media, resin/fiber interactLons kinematically, dynamically and thermally, fiber wetting, and fountain flow at microscopic level. So far, various numerical and experimental works have been conducted to understand resin impregnation in fiber preform [1 16], a good review on this subject being provided by Coulter and Guceri [2]. In the area of numerical simulations of heat and mass transfer during impregnation, Aoyagi et aL 13] have simulated successfully resin impregnation through an anisotropi( fiber preform in a complex mold using Darcy's law, where the influence of thermal transport and chemical reaction are included. A similar numerical modelling can be found also in the work of Coulter and Guceri [4,5]. Bruschke and Advani [6,7] applied a finite element/control volume approach to simulate resin flow in 3D shell-like domain using the modified Darcy law in an anisotropic porous media. Earlier numerical and experimental work in this area was conducted also by researchers such as Springer [8], Gutowski [9], Adams et al. [101~ for various applications. It has been recognized that the flow characteristics are influenced greatly ~;~ the fiber preform structure, which can be in part characterized by the perme ability tensor associated with Darcy's law. The inlet pressure during impregnation was found to increase with time [2,3,11] linearly or non-linearly depending on the structures of the preform, and the velocity profile is also related closely to the structures of the fiber preform [2,7]. The above two flow characteristics can be perceived easily qualitatively from the inspection of Darcy's law. To reveal more complicated process conditions, Gong [12] investigated the mflu~ ence of preform permeability variation and mold shape on the pressure field in resin transfer molding using Darcy's law. It was found that uneven prefbrm permeability can cause a significant change in the pressure field, especially i~ the case of periodically distributed permeability variation where the pressure distribution in the mold and the inlet pressure fluctuate drastically. It is found also th at the inlet pressure for resin impregnation in a converging channel increases non-linearly, the rate of such increase being proportional t(~ the degree of contraction. However, it is difficult to have accurate prediction of process characteristics due to various complications in a practical resin impregnation environment~o For example, unlike the problems of ground-water seepage, the woven', or braided preforms deform under the hydraulic pressure when interacting w~th the resin, which results in a variation in preform permeability. Some experimental evidence [11,13] shows also that, at high flow rate, the influence of the preform compressibility on the pressure drop causes a non-linear pressure growth with time at the inlet, and the pressure is lower than that predicted by Darcy's law. The a u t h o r found in this work that the lower pressure level can be predicted by taking the preform compressibility and relaxation into account. A model is proposed in this work to characterize the influence of a non-rigid fiber preform on the processing conditions of the RTM, in which both the preform compressibility and the relaxation under impact in the front region
H. Gong/Pressure distribution in resin transfer molding
365
are taken into account. Near the resin front during impregnation, the fiber preform is stretched under the impact of the resin front, which results in a higher volume fraction of the resin. Simultaneously, due to the elastic nature of the fiber preform, relaxation or recoil of the preform starts after resin impact, and the deformed preform is gradually restored or its resin volume fraction becomes lower after the resin front has passed by. Since the resin impregnation in this work is in steady-state, the above variations in resin volume fraction and permeability are confined to a region of a constant length which moves with the resin front. This length is determined by both the region influenced by the front impact and by the preform relaxation after the impact. The Kozeny-Carman relationship [1] is used to relate the preform structure to the permeability, being proven to be accurate for various fiber preforms and resins [14]. In this work, the fiber preform is constructed in such a way that the preform permeability in the longitudinal direction, kx, is variable only in that direction, i.e.,
5y the preform permeability in the transverse direction, k)., remaining constant. In the presence of the preform, the viscous force at the mold wall is small relative to the resin/preform interaction which is proportional to the magnitude of the velocity [15]; therefore, the macro-fountain flow at the front caused mainly by the viscous drag from the mold wall becomes insignificant and the flow of the impregnated resin becomes non-viscous. As a consequence, the shape of the flow front will be flatter due to the weak viscous drag at the wall. For example, if the preform is constructed in the way described above, a uniform velocity profile at the injection gate of a duct cavity can lead to the same uniform velocity profile in downstream region, and the flow front remains flat and the solution domain is rectangular. In general, the above flat-front assumption, even with a non-uniform velocity profile at the injection gate, provides a good approximation to the overall flow field if k), is constant and kx is a function of x and t only. Moreover, the assumption makes the analytical solution viable since the flow problem becomes one-dimensional. In addition, the resin impregnation is assumed to be isothermal and Newtonian in this work.
2. Modelling Consider a steady resin impregnation in the gap between two parallel plates distance 2b apart. As the flow front advances, the flow domain expands in the x direction, i.e., x e [0,a(t)], where t is the impregnation time, and the solution domain for this moving boundary flow problem in porous media is as shown in Fig. 1. It has been observed widely that anisotropic porous media exhibit an orthotropic behaviour, i.e., the porous flow domain has three mutually
366
H. Gong~Pressure distribution in resin transfer molding wall
1 .\,,-,.-,.,,.\.,.,,.-,,,-,,,..,,x,,.-.,,:,,:,,:,,.~\\\\\\\~\\\\\\~\\\\~ m
Y
'~ ~4
0
i:
!?
~J
a(t)
Fig. 1. Flow domain of 1D resin impregnation. o r t h o t r o p i c principle axes [2]. To facilitate the analyses in this work, the principle axes are chosen to coincide with the c o o r d i n a t e axes in the C a r t e s i a n system. Therefore, the p e r m e a b i l i t y t e n s o r describing the s t r u c t u r e of the fiber preform becomes
\<) where k_< and k,. r e p r e s e n t the preform permeabilities in the x and y directions respectively. In this work, the analysis is confined to the s i t u a t i o n t h a t the preform p e r m e a b i l i t y in the x direction, k,, is n o n - h o m o g e n e o u s in t h a t d i r e c t i o n only, In addition, the preform is assumed to be h o m o g e n e o u s in the t r a n s v e r s e direction, i.e., k), is constant. With the above preform structure, the resii) flo,a in the r e c t a n g u l a r cavity becomes one-dimensional if the velocity profile at th( ~ inlet is uniform, which leads to the following e q u a t i o n governed by Darcy',-~ law: 5p
/x
where U is the velocity c o m p o n e n t in the x direction, and i~ is the m o l e c u l a l viscosity of the resin which is set constant. In the f o r m u l a t i o n of Darcy's law (1), the viscous force is neglected since it is small compared to Ii~¢, r e s i n / p r e f o r m i n t e r a c t i o n , l l U / K ( x ) , so t h a t the Darcy flow requires ~i careful t r e a t m e n t for its b o u n d a r y conditions. It can also be proven t h a t Darcy's law, eqn. (1), does not lead to any f o u n t a i n flow at the macroscopic level (the streamlines do not exhibit a d i v e r t i n g p a t t e r n ) [16], since the no-slip b o u n d a r y c o n d i t i o n at the wall is invalid. Due to the absence of the viscous force and the n o n - h o m o g e n e o u s preform s t r u c t u r e defined in this work, a uniform inlet; v e l o c i t y leads to a flat flow front and a uniform velocity field in the entire r e c t a n g u l a r flow d o m a i n due to the c o n s e r v a t i o n of mass. Since the flow is symmetric a b o u t the c e n t e r l i n e y = 0, the solution d o m a i n becomes x~ [0, a (t)], ye[0, 1].
H. Gong~Pressure distribution in resin transfer molding
367
The pressure boundary condition at the front, x = a(t), is
(2)
p=O
Since the fiber preform functions as a non-rigid structure, it deforms under a pressure gradient when the resin interacts with the preform. Such deformation of the fiber preform results in a change of preform permeability. However, the transient variation of preform permeability can occur only in the presence of a transient pressure gradient or resin impact. The preform compressibility and relaxation in this work are defined in the context of the above transient permeability variation induced by the resin impact on the preform. The steady relations between permeability and pressure gradient mentioned in other works [3,5] do not fall into this category, since those works did not consider the elasticity or relaxation of the preform. In this regard, the 1D steady resin impregnation in this work can produce compressibility and relaxation effects only in the region of the resin front where a transient pressure gradient or resin impact is present. In the front region, the fiber preform is stretched forwards under impact from the resin front, which results in a higher volume fraction of the resin. Meanwhile, due to the elastic nature of the fiber preform, relaxation of the preform starts after resin impact: the deformed preform is graduately restored and the resin volume fraction becomes lower after the resin front has passed by. Since the resin impregnation in this work is steadystate, the above variations of resin volume fraction and permeability are confined to a fixed region of a constant characteristic length,/c, starting from the resin front, and this compressible region moves with the resin front, as shown in Fig. 2. Although the compressible region could extend to the entire flow domain, it is convenient analytically to divide the flow domain into two regions: the compressible region which is near the resin front where the resin volume fraction varies significantly, and the incompressible region where the resin volume fraction becomes nearly constant since the preform is fully relaxed after the impact of the resin front. The above characteristics can be represented by the following phenomenological model describing the variation of volume fraction of resin under a transient pressure gradient, as shown in Fig. 2, t)]/l~
{~bo+(~l--¢o)[x--xd
~b= q%0
within the compressible region within the incompressible region
(3) where ~bo is the volume fraction of resin in the incompressible region, 471 is the volume fraction of resin at the resin front, and Xd(t) is the axial position dividing the compressible and incompressible regions. Note that xd(t)= Ut in 1D impregnation, and thus the front position becomes a ( t ) = Ut + lc, where the characteristic length of the compressible region, Ic, remains constant during the impregnation.
H. Gong/Pressure distribution in resin transfer molding
368
Ut t
Oo
t
xa(t)
1
a(t)
incompressible compressible region region
Fig. 2. Variation of the volume fraction of the resin considering preform compressibility and relaxation.
The K o z e n y - C a r m a n r e l a t i o n [1] is used to r e l a t e the preform s t r u c t u r e to the permeability, K, i.e., ~3
K
r2
~i
( 1 - ~b)2 4k'
w h e r e r is the radius of the fibers, and k' is the K o z e n y c o n s t a n t r e p r e s e n t i n g the t o r t u o s i t y of the fiber a r c h i t e c t u r e which is assumed c o n s t a n t in this work. S u b s t i t u t i o n of the v o l u m e f r a c t i o n of resin, eqn. (3), into eqn. (4) yields Ko = K=
(1
~b~ r2 ~b0)24k'
O~x~x~(t)
[(~0 ~- (~bl - - ~ ) o ) ( X - - X d ( t ) ) / l c ] 3
K1
l"2
[ 1 - ~ b o - ( q ~ - - ¢ 0 ) ( x - x ~ ( t ) ) / l ¢ ] 24k'
xd(t)~xGa(t)
Therefore, i n t e g r a t i o n of D a r c y ' s law, eqn. (1), with the above p e r m e a b i l i t y and b o u n d a r y condition, eqn. (2), yields
fxa(t) l ~.v 1 p(x,t)----t~UJ,,, ~ d x - t ~ U ~ . , ( t ) KadX p(x,t)=-itU
i
x
for x
1
da(11 K 1
dx
for x>xa(t)
(6)
H. Gong/Pressure distribution in resin transfer molding
369
If the change of the volume fraction of the resin is small in the front region, the pressure distributions, eqns. (5) and (6), become
p(x, t)=p(x, t)[K_Ko--pUlc (1-C)/Ko
for X
(7)
p(x, t)=Cp(x, t)l~:-Ko for x>xa(t)
(8)
where C-
l q - q~l/q~O
2(~1/~0) a
represents the compressibility of the preform, and
p(x, t)IK=K,,=pU(x-- U t - lc) /Ko is the pressure distribution without considering the compressibility and relaxation induced by the transient pressure gradient. Coefficient C also represents the influence of the preform compressibility and relaxation on the pressure distribution in the compressible region, as shown in eqn. (8). The second term in eqn. (7) represents the influence of the preform compressibility and relaxation on the pressure distribution in the incompressible region. Since q~l > ~b0, C is less than unity. Therefore, eqns. (7) and (8) indicate that the pressure level is lower if the compressibility of the preform is considered, which is in a good qualitative agreement with existing experimental data [11,13]. It is further noted in eqns. (7) and (8) that as the preform compressibility increases, i.e., (J~l/(~0 increases, the coefficient C decreases, which results in lower pressure levels in both the compressible and incompressible regions than those with rigid preforms. The influence of the compressibility, q~l/~b0, on C is plotted in Fig. 3.
0.95
N. 0.9
0.85
0.8
0.75 0.7 L 1
--
I
. . . . . . . . . . . . . . .
1.06
i
1.12
1,18 $L/Oo
Fig. 3. Coefficient C versus compressibility.
1.24
1.3
370
H. Gong~Pressure distribution in resin transfer molding
It is also percievable, for a given qSo, t h a t ~b~ increases with the injection velocity U due to a g r e a t e r impact at the front: t h e r e f o r e the permeability is h i g h e r at a h i g h e r i n j e c t i o n rate, and the pressure drop becomes lower at a h i g h e r injection rate, as shown by Darcy's law, eqn. (1). The role of the r e l a x a t i o n p r o p e r t y of the preform can be c h a r a c t e r i s e d by the length ot the incompressible region, l,., which is p r o p o r t i o n a l to the r e l a x a t i o n time of the preform u n d e r a t r a n s i e n t load. The more flexible the preform, the longer the r e l a x a t i o n time and l~, which yields a lower pressure level, as shown in eqn (7:,.
3. C o n c l u s i o n s
A model for predicting the influence of the preform compressibility and r e l a x a t i o n on pressure distribution in a 1D resin t r a n s f e r molding is established in c o n j u n c t i o n with the Kozeny--Carman r e l a t i o n and D a r c y ' s law. The modelling of the preform compressibility is based on the moving resin front causing an impact on the preform and s t r e t c h i n g the fiber bundles forward: therefore, the volume f r a c t i o n of resin in the f r o n t region becomes higher. The modelling of the preform r e l a x a t i o n reflects the p h e n o m e n o n t h a t the fiber bundles recoil after the impact of the resin front. Therefore, the compressible region is confined p h e n o m e n o l o g i c a t l y to the front region of a c h a r a c t e r i s t i c length, l~, d e t e r m i n e d by the preform compressibility and the r e l a x a t i o n time of the preform. In the off-front region, the preform remains rigid due to the absence of the impact force and preform relaxation. T h e results show t h a t the pressure level is lower if a non-rigid preform is present, which is in a good q u a l i t a t i v e a g r e e m e n t with existing e x p e r i m e n t a l data. It is suggested also t h a t modelling w i t h o u t considering the non-rigidity of the preform results in a h i g h e r pressure level. It is f u r t h e r noted t h a t as the preform compressibility increases, the pressure level becomes m o n t o n i c a l t y lower. The v a l u e of the compressibility could be influenced greatly by the i n j e c t i o n velocity U, which results in a non-linear r e l a t i o n s h i p between thu pressure g r a d i e n t and the flow rate. The role of the r e l a x a t i o n p r o p e r t y of the preform can be c h a r a c t e r i z e d by the compressible length, l~, which is p r o p o r tional to the r e l a x a t i o n time of the preform u n d e r a t r a n s i e n t load.
4. R e f e r e n c e s
[1] P.C. Carman, Trans. Inst. Chem. Eng., 15 (1937) 150. [2] J.P. Coulter and S.I. Guceri, The Manufacturing Science of Composites, in: IL(i Gutowski (Ed.). Proc. A S M E WA, Vol. IV, Atlanta, Georgia, 1988. [3] H. Aoyagi, M. Uenoyama and S.I. Guceri. Accepted for publication in lnt. Pol~'m. Prn,< (1991). [41 d.P. Coulter and S.I. Guceri, J. Rein. Plast. Compos., 7 (1988) 200 219. [5] O.P. Coulter and S.I. Guceri, J. Comp. Sci. 7k,chnol., 35 (1989) 317 330. [6] M.V. Bruschke and S.G. Advani, Mechanics of Plastics and Plastic Composites, in: V.tK. Stokes fEd.), AMD.-Vol. 104, New York, 1989, 291 304.
H. Gong/Pressure distribution in resin transfer molding
371
M.V. Bruschke and S.G. Advani, Polym. Compos., 11 (1990) 398-405. G.S. Springer, J. Compos. Mater., 6 (1982) 400 410. T.G. Gutowski, SAMPE Q., (July, 1985) 58 64. K.L. Adams, B. Miller and L. Rebenfeld, Polym. Eng. Sci., 26 (1986) 1434 1441. J.A. Molnar, L. Trevino and L.J. Lee, Polym. Compos., 10 (1989) 414 423. H. Gong, J. Therm. Compos. Mater, 5 (1992) 76 88. L. Trevino, K. Rupel, W.B. Young, M.J. Liou and L.J. Lee, Polym. Compos., 12 (1991) 20 29. [14] R.C. Lam, J.L. Kardos, Polym. Eng. Sci., 31 (1991) 1064 1070. [15] R.A. Greenkorn, Flow Phenomena in Porous Media, Dekker, New York, 1983. [16} H. Gong, in: S.I. Guceri (Ed.), Proc. Int. Conf. Transport Phenomena in Processing, Hawaii, USA, March 1992, Technomic 1992.
[7] [8] [9] [10] [11] [12] [13]
363
Elsevier
Pressure distribution in resin transfer molding with a non-rigid fiber preform Haiqing Gong
School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263
Industrial
Summary
In mold filling processes of composite material manufacturing such as resin transfer molding (RTM), a polymer resin impregnates the fiber mat placed in a mold under a pressure gradient generated by pressure at gates or vacuum at outlets, or both. The flow characteristics in the fiber mat during the impregnation play a critical role in determining the mechanical properties of the final parts, as well as the cost effectiveness of the process. In this work, resin impregnation through a compressible fiber preform is modelled analytically using Darcy's law in an expanding flow domain of a rectangular shape with a uniform velocity profile at the injection gate. The model proposed in this work accommodates the effect of both the compressibility and relaxation of the fiber mat on the volume fraction of the resin. The compressible region of the fiber mat is confined phenomenologically to the front region of the resin impregnation where the impact of the resin on the fiber causes a higher volume fraction of the resin. A relaxation length is introduced in the resin front region to characterize the relaxation of the fiber mat under the impact of the resin. To analyse the pressure distribution, the Kozeny Carman relationship is used for integration of Darcy's law. The influences of the compressibility and relaxation of the preform on the pressure distribution are analysed, the result showing that the pressure level is lower when preform compressibility and relaxation are considered in the modelling. This conclusion is in a good qualitative agreement with existing experimental data.
1. I n t r o d u c t i o n D e v e l o p e d i n t h e 1970s, t h e b a s i c r e s i n t r a n s f e r m o l d i n g p r o c e s s h a s g r a d u ally, o v e r t h e l a s t five y e a r s , b e e n c o n s i d e r e d as a p o t e n t i a l l o w - c o s t m a n u f a c t u r i n g t e c h n o l o g y for f o r m i n g a d v a n c e d s t r u c t u r a l c o m p o s i t e p a r t s . H o w e v e r , r e l a t i v e l y l i t t l e k n o w l e d g e h a s b e e n o b t a i n e d r e g a r d i n g t h e effect of p r o c e s s i n g a n d m a t e r i a l c o n d i t i o n s o n o p t i m i z a t i o n of t h e p r o c e s s d e s i g n . M a n y difficult a n d c o m p l e x p r o b l e m s c o n t r i b u t e to t h e l a c k of a s c i e n t i f i c b a s e i n t h i s a r e a ,
Correspondence to: H. Gong, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263. 0924-0136/93/$06.00 (t2 1993 Elsevier Science Publishers B.V. All rights reserved.
364
H. Gong~Pressure distribution in resin transfer molding
such as characterization of fiber mats as porous media, resin/fiber interactLons kinematically, dynamically and thermally, fiber wetting, and fountain flow at microscopic level. So far, various numerical and experimental works have been conducted to understand resin impregnation in fiber preform [1 16], a good review on this subject being provided by Coulter and Guceri [2]. In the area of numerical simulations of heat and mass transfer during impregnation, Aoyagi et aL 13] have simulated successfully resin impregnation through an anisotropi( fiber preform in a complex mold using Darcy's law, where the influence of thermal transport and chemical reaction are included. A similar numerical modelling can be found also in the work of Coulter and Guceri [4,5]. Bruschke and Advani [6,7] applied a finite element/control volume approach to simulate resin flow in 3D shell-like domain using the modified Darcy law in an anisotropic porous media. Earlier numerical and experimental work in this area was conducted also by researchers such as Springer [8], Gutowski [9], Adams et al. [101~ for various applications. It has been recognized that the flow characteristics are influenced greatly ~;~ the fiber preform structure, which can be in part characterized by the perme ability tensor associated with Darcy's law. The inlet pressure during impregnation was found to increase with time [2,3,11] linearly or non-linearly depending on the structures of the preform, and the velocity profile is also related closely to the structures of the fiber preform [2,7]. The above two flow characteristics can be perceived easily qualitatively from the inspection of Darcy's law. To reveal more complicated process conditions, Gong [12] investigated the mflu~ ence of preform permeability variation and mold shape on the pressure field in resin transfer molding using Darcy's law. It was found that uneven prefbrm permeability can cause a significant change in the pressure field, especially i~ the case of periodically distributed permeability variation where the pressure distribution in the mold and the inlet pressure fluctuate drastically. It is found also th at the inlet pressure for resin impregnation in a converging channel increases non-linearly, the rate of such increase being proportional t(~ the degree of contraction. However, it is difficult to have accurate prediction of process characteristics due to various complications in a practical resin impregnation environment~o For example, unlike the problems of ground-water seepage, the woven', or braided preforms deform under the hydraulic pressure when interacting w~th the resin, which results in a variation in preform permeability. Some experimental evidence [11,13] shows also that, at high flow rate, the influence of the preform compressibility on the pressure drop causes a non-linear pressure growth with time at the inlet, and the pressure is lower than that predicted by Darcy's law. The a u t h o r found in this work that the lower pressure level can be predicted by taking the preform compressibility and relaxation into account. A model is proposed in this work to characterize the influence of a non-rigid fiber preform on the processing conditions of the RTM, in which both the preform compressibility and the relaxation under impact in the front region
H. Gong/Pressure distribution in resin transfer molding
365
are taken into account. Near the resin front during impregnation, the fiber preform is stretched under the impact of the resin front, which results in a higher volume fraction of the resin. Simultaneously, due to the elastic nature of the fiber preform, relaxation or recoil of the preform starts after resin impact, and the deformed preform is gradually restored or its resin volume fraction becomes lower after the resin front has passed by. Since the resin impregnation in this work is in steady-state, the above variations in resin volume fraction and permeability are confined to a region of a constant length which moves with the resin front. This length is determined by both the region influenced by the front impact and by the preform relaxation after the impact. The Kozeny-Carman relationship [1] is used to relate the preform structure to the permeability, being proven to be accurate for various fiber preforms and resins [14]. In this work, the fiber preform is constructed in such a way that the preform permeability in the longitudinal direction, kx, is variable only in that direction, i.e.,
5y the preform permeability in the transverse direction, k)., remaining constant. In the presence of the preform, the viscous force at the mold wall is small relative to the resin/preform interaction which is proportional to the magnitude of the velocity [15]; therefore, the macro-fountain flow at the front caused mainly by the viscous drag from the mold wall becomes insignificant and the flow of the impregnated resin becomes non-viscous. As a consequence, the shape of the flow front will be flatter due to the weak viscous drag at the wall. For example, if the preform is constructed in the way described above, a uniform velocity profile at the injection gate of a duct cavity can lead to the same uniform velocity profile in downstream region, and the flow front remains flat and the solution domain is rectangular. In general, the above flat-front assumption, even with a non-uniform velocity profile at the injection gate, provides a good approximation to the overall flow field if k), is constant and kx is a function of x and t only. Moreover, the assumption makes the analytical solution viable since the flow problem becomes one-dimensional. In addition, the resin impregnation is assumed to be isothermal and Newtonian in this work.
2. Modelling Consider a steady resin impregnation in the gap between two parallel plates distance 2b apart. As the flow front advances, the flow domain expands in the x direction, i.e., x e [0,a(t)], where t is the impregnation time, and the solution domain for this moving boundary flow problem in porous media is as shown in Fig. 1. It has been observed widely that anisotropic porous media exhibit an orthotropic behaviour, i.e., the porous flow domain has three mutually
366
H. Gong~Pressure distribution in resin transfer molding wall
1 .\,,-,.-,.,,.\.,.,,.-,,,-,,,..,,x,,.-.,,:,,:,,:,,.~\\\\\\\~\\\\\\~\\\\~ m
Y
'~ ~4
0
i:
!?
~J
a(t)
Fig. 1. Flow domain of 1D resin impregnation. o r t h o t r o p i c principle axes [2]. To facilitate the analyses in this work, the principle axes are chosen to coincide with the c o o r d i n a t e axes in the C a r t e s i a n system. Therefore, the p e r m e a b i l i t y t e n s o r describing the s t r u c t u r e of the fiber preform becomes
\<) where k_< and k,. r e p r e s e n t the preform permeabilities in the x and y directions respectively. In this work, the analysis is confined to the s i t u a t i o n t h a t the preform p e r m e a b i l i t y in the x direction, k,, is n o n - h o m o g e n e o u s in t h a t d i r e c t i o n only, In addition, the preform is assumed to be h o m o g e n e o u s in the t r a n s v e r s e direction, i.e., k), is constant. With the above preform structure, the resii) flo,a in the r e c t a n g u l a r cavity becomes one-dimensional if the velocity profile at th( ~ inlet is uniform, which leads to the following e q u a t i o n governed by Darcy',-~ law: 5p
/x
where U is the velocity c o m p o n e n t in the x direction, and i~ is the m o l e c u l a l viscosity of the resin which is set constant. In the f o r m u l a t i o n of Darcy's law (1), the viscous force is neglected since it is small compared to Ii~¢, r e s i n / p r e f o r m i n t e r a c t i o n , l l U / K ( x ) , so t h a t the Darcy flow requires ~i careful t r e a t m e n t for its b o u n d a r y conditions. It can also be proven t h a t Darcy's law, eqn. (1), does not lead to any f o u n t a i n flow at the macroscopic level (the streamlines do not exhibit a d i v e r t i n g p a t t e r n ) [16], since the no-slip b o u n d a r y c o n d i t i o n at the wall is invalid. Due to the absence of the viscous force and the n o n - h o m o g e n e o u s preform s t r u c t u r e defined in this work, a uniform inlet; v e l o c i t y leads to a flat flow front and a uniform velocity field in the entire r e c t a n g u l a r flow d o m a i n due to the c o n s e r v a t i o n of mass. Since the flow is symmetric a b o u t the c e n t e r l i n e y = 0, the solution d o m a i n becomes x~ [0, a (t)], ye[0, 1].
H. Gong~Pressure distribution in resin transfer molding
367
The pressure boundary condition at the front, x = a(t), is
(2)
p=O
Since the fiber preform functions as a non-rigid structure, it deforms under a pressure gradient when the resin interacts with the preform. Such deformation of the fiber preform results in a change of preform permeability. However, the transient variation of preform permeability can occur only in the presence of a transient pressure gradient or resin impact. The preform compressibility and relaxation in this work are defined in the context of the above transient permeability variation induced by the resin impact on the preform. The steady relations between permeability and pressure gradient mentioned in other works [3,5] do not fall into this category, since those works did not consider the elasticity or relaxation of the preform. In this regard, the 1D steady resin impregnation in this work can produce compressibility and relaxation effects only in the region of the resin front where a transient pressure gradient or resin impact is present. In the front region, the fiber preform is stretched forwards under impact from the resin front, which results in a higher volume fraction of the resin. Meanwhile, due to the elastic nature of the fiber preform, relaxation of the preform starts after resin impact: the deformed preform is graduately restored and the resin volume fraction becomes lower after the resin front has passed by. Since the resin impregnation in this work is steadystate, the above variations of resin volume fraction and permeability are confined to a fixed region of a constant characteristic length,/c, starting from the resin front, and this compressible region moves with the resin front, as shown in Fig. 2. Although the compressible region could extend to the entire flow domain, it is convenient analytically to divide the flow domain into two regions: the compressible region which is near the resin front where the resin volume fraction varies significantly, and the incompressible region where the resin volume fraction becomes nearly constant since the preform is fully relaxed after the impact of the resin front. The above characteristics can be represented by the following phenomenological model describing the variation of volume fraction of resin under a transient pressure gradient, as shown in Fig. 2, t)]/l~
{~bo+(~l--¢o)[x--xd
~b= q%0
within the compressible region within the incompressible region
(3) where ~bo is the volume fraction of resin in the incompressible region, 471 is the volume fraction of resin at the resin front, and Xd(t) is the axial position dividing the compressible and incompressible regions. Note that xd(t)= Ut in 1D impregnation, and thus the front position becomes a ( t ) = Ut + lc, where the characteristic length of the compressible region, Ic, remains constant during the impregnation.
H. Gong/Pressure distribution in resin transfer molding
368
Ut t
Oo
t
xa(t)
1
a(t)
incompressible compressible region region
Fig. 2. Variation of the volume fraction of the resin considering preform compressibility and relaxation.
The K o z e n y - C a r m a n r e l a t i o n [1] is used to r e l a t e the preform s t r u c t u r e to the permeability, K, i.e., ~3
K
r2
~i
( 1 - ~b)2 4k'
w h e r e r is the radius of the fibers, and k' is the K o z e n y c o n s t a n t r e p r e s e n t i n g the t o r t u o s i t y of the fiber a r c h i t e c t u r e which is assumed c o n s t a n t in this work. S u b s t i t u t i o n of the v o l u m e f r a c t i o n of resin, eqn. (3), into eqn. (4) yields Ko = K=
(1
~b~ r2 ~b0)24k'
O~x~x~(t)
[(~0 ~- (~bl - - ~ ) o ) ( X - - X d ( t ) ) / l c ] 3
K1
l"2
[ 1 - ~ b o - ( q ~ - - ¢ 0 ) ( x - x ~ ( t ) ) / l ¢ ] 24k'
xd(t)~xGa(t)
Therefore, i n t e g r a t i o n of D a r c y ' s law, eqn. (1), with the above p e r m e a b i l i t y and b o u n d a r y condition, eqn. (2), yields
fxa(t) l ~.v 1 p(x,t)----t~UJ,,, ~ d x - t ~ U ~ . , ( t ) KadX p(x,t)=-itU
i
x
for x
1
da(11 K 1
dx
for x>xa(t)
(6)
H. Gong/Pressure distribution in resin transfer molding
369
If the change of the volume fraction of the resin is small in the front region, the pressure distributions, eqns. (5) and (6), become
p(x, t)=p(x, t)[K_Ko--pUlc (1-C)/Ko
for X
(7)
p(x, t)=Cp(x, t)l~:-Ko for x>xa(t)
(8)
where C-
l q - q~l/q~O
2(~1/~0) a
represents the compressibility of the preform, and
p(x, t)IK=K,,=pU(x-- U t - lc) /Ko is the pressure distribution without considering the compressibility and relaxation induced by the transient pressure gradient. Coefficient C also represents the influence of the preform compressibility and relaxation on the pressure distribution in the compressible region, as shown in eqn. (8). The second term in eqn. (7) represents the influence of the preform compressibility and relaxation on the pressure distribution in the incompressible region. Since q~l > ~b0, C is less than unity. Therefore, eqns. (7) and (8) indicate that the pressure level is lower if the compressibility of the preform is considered, which is in a good qualitative agreement with existing experimental data [11,13]. It is further noted in eqns. (7) and (8) that as the preform compressibility increases, i.e., (J~l/(~0 increases, the coefficient C decreases, which results in lower pressure levels in both the compressible and incompressible regions than those with rigid preforms. The influence of the compressibility, q~l/~b0, on C is plotted in Fig. 3.
0.95
N. 0.9
0.85
0.8
0.75 0.7 L 1
--
I
. . . . . . . . . . . . . . .
1.06
i
1.12
1,18 $L/Oo
Fig. 3. Coefficient C versus compressibility.
1.24
1.3
370
H. Gong~Pressure distribution in resin transfer molding
It is also percievable, for a given qSo, t h a t ~b~ increases with the injection velocity U due to a g r e a t e r impact at the front: t h e r e f o r e the permeability is h i g h e r at a h i g h e r i n j e c t i o n rate, and the pressure drop becomes lower at a h i g h e r injection rate, as shown by Darcy's law, eqn. (1). The role of the r e l a x a t i o n p r o p e r t y of the preform can be c h a r a c t e r i s e d by the length ot the incompressible region, l,., which is p r o p o r t i o n a l to the r e l a x a t i o n time of the preform u n d e r a t r a n s i e n t load. The more flexible the preform, the longer the r e l a x a t i o n time and l~, which yields a lower pressure level, as shown in eqn (7:,.
3. C o n c l u s i o n s
A model for predicting the influence of the preform compressibility and r e l a x a t i o n on pressure distribution in a 1D resin t r a n s f e r molding is established in c o n j u n c t i o n with the Kozeny--Carman r e l a t i o n and D a r c y ' s law. The modelling of the preform compressibility is based on the moving resin front causing an impact on the preform and s t r e t c h i n g the fiber bundles forward: therefore, the volume f r a c t i o n of resin in the f r o n t region becomes higher. The modelling of the preform r e l a x a t i o n reflects the p h e n o m e n o n t h a t the fiber bundles recoil after the impact of the resin front. Therefore, the compressible region is confined p h e n o m e n o l o g i c a t l y to the front region of a c h a r a c t e r i s t i c length, l~, d e t e r m i n e d by the preform compressibility and the r e l a x a t i o n time of the preform. In the off-front region, the preform remains rigid due to the absence of the impact force and preform relaxation. T h e results show t h a t the pressure level is lower if a non-rigid preform is present, which is in a good q u a l i t a t i v e a g r e e m e n t with existing e x p e r i m e n t a l data. It is suggested also t h a t modelling w i t h o u t considering the non-rigidity of the preform results in a h i g h e r pressure level. It is f u r t h e r noted t h a t as the preform compressibility increases, the pressure level becomes m o n t o n i c a l t y lower. The v a l u e of the compressibility could be influenced greatly by the i n j e c t i o n velocity U, which results in a non-linear r e l a t i o n s h i p between thu pressure g r a d i e n t and the flow rate. The role of the r e l a x a t i o n p r o p e r t y of the preform can be c h a r a c t e r i z e d by the compressible length, l~, which is p r o p o r tional to the r e l a x a t i o n time of the preform u n d e r a t r a n s i e n t load.
4. R e f e r e n c e s
[1] P.C. Carman, Trans. Inst. Chem. Eng., 15 (1937) 150. [2] J.P. Coulter and S.I. Guceri, The Manufacturing Science of Composites, in: IL(i Gutowski (Ed.). Proc. A S M E WA, Vol. IV, Atlanta, Georgia, 1988. [3] H. Aoyagi, M. Uenoyama and S.I. Guceri. Accepted for publication in lnt. Pol~'m. Prn,< (1991). [41 d.P. Coulter and S.I. Guceri, J. Rein. Plast. Compos., 7 (1988) 200 219. [5] O.P. Coulter and S.I. Guceri, J. Comp. Sci. 7k,chnol., 35 (1989) 317 330. [6] M.V. Bruschke and S.G. Advani, Mechanics of Plastics and Plastic Composites, in: V.tK. Stokes fEd.), AMD.-Vol. 104, New York, 1989, 291 304.
H. Gong/Pressure distribution in resin transfer molding
371
M.V. Bruschke and S.G. Advani, Polym. Compos., 11 (1990) 398-405. G.S. Springer, J. Compos. Mater., 6 (1982) 400 410. T.G. Gutowski, SAMPE Q., (July, 1985) 58 64. K.L. Adams, B. Miller and L. Rebenfeld, Polym. Eng. Sci., 26 (1986) 1434 1441. J.A. Molnar, L. Trevino and L.J. Lee, Polym. Compos., 10 (1989) 414 423. H. Gong, J. Therm. Compos. Mater, 5 (1992) 76 88. L. Trevino, K. Rupel, W.B. Young, M.J. Liou and L.J. Lee, Polym. Compos., 12 (1991) 20 29. [14] R.C. Lam, J.L. Kardos, Polym. Eng. Sci., 31 (1991) 1064 1070. [15] R.A. Greenkorn, Flow Phenomena in Porous Media, Dekker, New York, 1983. [16} H. Gong, in: S.I. Guceri (Ed.), Proc. Int. Conf. Transport Phenomena in Processing, Hawaii, USA, March 1992, Technomic 1992.
[7] [8] [9] [10] [11] [12] [13]