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Simulation of molten metal through a unidirectional fibrous preform during MMC processing
Journal of Materials Processing Technology 209 (2009) 4337–4342
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Simulation of molten metal through a unidirectional fibrous preform during MMC processing Chih-Yuan Chang ∗ Department of Mechanical and Automation Engineering, Kao Yuan University, Lu-Chu 82101, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 7 January 2004 Received in revised form 7 August 2008 Accepted 5 November 2008 Keywords: Metal matrix composites Infiltration process Numerical simulation
a b s t r a c t The injection of molten metal through a fibrous preform is one of the techniques used to manufacture metal matrix composites (MMCs). The flow of liquid metal through a fibrous preform constitutes a moving boundary problem of fluid mechanics in a porous medium. In the present model, the isothermal infiltration process is considered. The numerical algorithm, based on the body-fitted finite element method (FEM), can precisely deal with the transient and irregular physical domains and calculate the flow behavior. Numerical results show that the infiltration pattern is strongly influenced by the nature of the preform. The injection flow rate under constant applied pressure decreases with the infiltration time. The effect of the preform orientation on infiltration pattern is also discussed. Similarity analysis shows that the infiltration time is inversely proportional to the ratio of applied pressure to liquid viscosity for the same type of preform. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Metal matrix composites (MMCs) can be produced via different methods (Chou et al., 1985). One of the methods for fabricating fiber-reinforced MMCs is by injecting molten metal into the interstitial spaces of the fiber preform. This process of preform infiltration has several advantages: it is relatively inexpensive, tooling is similar to the casting process, and it is possible to produce a net or near-net shape component that is difficult to machine. The phenomena governing preform infiltration are complex although infiltrated parts have found commercial applications. A number of physical, mechanical, and chemical phenomena interact, including flow of liquid metal in a porous medium, heat and mass transfer related to metal solidification and chemical interfacial reactions between the reinforcement and matrix. These phenomena have already been studied extensively from both theoretical and experimental viewpoints. Analytical and numerical solutions have been given and compared to experimental data, for unidirectional infiltration under constant applied pressure, including non-isothermal infiltration by a pure metal (Mortensen et al., 1989; Masur et al., 1989) or by a binary alloy (Mortensen and Michaud, 1990; Michaud and Mortensen, 1992); for non-isothermal infiltration by a pure metal, taking into account the influence of preform deformability (Sommer and Mortensen, 1996; Michaud et al., 1999); and for isothermal infiltration, taking into account
∗ Tel.: +886 7 6077028; fax: +886 7 6077112. E-mail address: [email protected]. 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.11.011
the influence of capillary phenomena (Mortensen and Wong, 1990; Michaud et al., 1994; Hu et al., 1999). For these cases, the boundary condition of constant applied pressure and the unidirectional configuration allowed the use of a single combined variable to describe time and position, thus yielding an analytical solution of the coupled phenomena. By using the one-dimensional solution, Aboulfatah et al. (1997) proposed an experimental procedure to measure the mobility of the molten metal through the fibrous preform as a function of the volume fraction of solidified metal. Xia et al. (1992) also proposed an approximate analytical solution for unidirectional saturated flow under a linearly increasing pressure function. More general cases have been treated as well. In these, analytical solutions can no longer be derived, and numerical tools such as finite element and finite difference methods are applied. Lacoste et al. (1991, 1993) proposed a model based on finite volume formulation to simulate the flow and the solidification of a liquid metal. Tong and Khan (1996) developed a numerical finite difference model: the mold walls were insulated, only heat exchange between fibers and metal was taken into account. Biswas et al. (1998) proposed a phenomenology model to describe fluid flow and transport phenomena for the case of unidirectional infiltration. From a finite element model initially developed in hydrogeology, Dopler et al. (2000) carried out a numerical simulation of the metal infiltration of a rigid fibrous preform with isothermal and low constant pressure. They could so follow the evolution of saturation according to the impregnation depth of the preform. Chang (2006) developed a two-dimensional numerical scheme to simulate isothermal infiltration through isotropic the preform under
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C.-Y. Chang / Journal of Materials Processing Technology 209 (2009) 4337–4342
Nomenclature f K P Pin Pinject Pfront Re Rf Ro rf1 , rf2 rin S t u, v Uc x,y
fiber volume fraction permeability pressure characteristic pressure injection pressure pressure along the flow front Reynolds number elliptical extent in the principal directions elliptical equivalent of the radius of the gate flow front in the principal directions radius of the inlet gate shape function infiltration time velocity components in x, y direction characteristic velocity coordinate components in the physical domain
Fig. 1. In-plane liquid flow through an anisotropic porous medium.
Greek letters ˛ ratio of y directional permeability to x directional permeability , coordinate components in the computational domain ˝, control functions in the body-fitted method molten metal density molten metal viscosity elliptical equivalent of the in-plane angle
1. The molten metal is an incompressible and Newtonian fluid. 2. The infiltration process can be regarded as a series of quasisteady state processes. 3. The infiltration process is isothermal. 4. The inertial force is negligible compared to the viscous force due to Re « 1.
Superscript * dimensionless parameter
Under the above assumptions, the momentum equation based on Darcy’s law can be written in matrix form as u
v
a time-dependent injection pressure. Ralph et al. (1997) summarized the variety of routes available for processing MMCs. Michaud and Mortensen (2001) and Lacoste et al. (2002) provided a general overview of the various infiltration processes and their associated terminology. The objective of this study is to carry out an investigation on liquid metal flowing through an anisotropic preform in MMC processing. The flow behavior of the infiltration process is twodimensional in the present model. The flow of liquid metal is a free moving boundary problem. A computational code is developed to solve the moving boundary problem by employing the body-fitted finite element method (FEM). Calculations are performed to predict the flow behavior of molten metal infiltrating fibrous preform under various conditions. Furthermore, similarity analysis is used to investigate the effect of the various parameters on the infiltration process. 2. Theoretical analysis 2.1. Governing equations This study applies a two-dimensional infiltration process of MMC since the thickness is much smaller than the other dimensions of the mold. The general case of in-plane flow of a liquid through the anisotropic porous preform is shown in Fig. 1. In the figure, rin is the radius of inlet gate, while rf1 and rf2 are the moving front resembling an ellipse characterized by maximum and minimum radial extents, respectively. is the preform orientation relative to the laboratory coordinate system. The definition of the orientation angle is the direction of maximum permeability in respect to the x-axis. For simplification, several assumptions are made as follows.
1 =−
Kxx Kyx
Kxy Kyy
⎡ ⎣
∂P ∂x ∂P ∂y
⎤ ⎦
(1)
where P, u and v represent the pressure and molten metal velocities in the x and y directions, respectively. is the viscosity of molten metal and K is the permeability tensor of the preform. The permeability is a second order tensor and measures the ease of flow. For the unidirectional preform, one can always find the two principal axes which will diagonalize the permeability. Here x- and y-axes are chosen to be the same as the principal preform ones. Thus, the non-diagonal term tensor is zero. Kxx and Kyy can denote Kx and Ky . The conservation of mass is ∂u ∂v + =0 ∂x ∂y
(2)
The governing equation can then be deduced by combining Eqs. (1) and (2) as ∂ ∂x
Kx ∂P ∂x
∂ + ∂y
Ky ∂P ∂y
=0
(3)
For a rectangular mold cavity, the physical domain is symmetric for x- and y-axes and only a quarter of the cavity needs to be concerned in the numerical simulation. The cavity is filled with the unidirectional preform having two different orientations. The corresponding boundary conditions applicable to Eq. (3) are shown in Fig. 2. In the figure, Pinject and Pfront denote the pressure at the inlet and the front, respectively. 2.2. Similarity analysis 1/2
In the infiltration process, the characteristic length (Kc ), the characteristic pressure (Pin ) and the characteristic velocity (Uc ) associated with process parameters are defined as Kc = (Kx Ky )1/2 ,
Pin = Pinject − Pfront ,
C.-Y. Chang / Journal of Materials Processing Technology 209 (2009) 4337–4342
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3.1. Galerkin’s method In the present FEM model, the weight residual method adopts the Galerkin’s method. The Galerkin’s residual equation for governing Eq. (3) can be derived from:
(e)
∂ ∂x
Kx ∂P ∂x
+
∂ ∂y
Ky ∂P ∂y
Sj dxdy = 0
j = 1, 2, . . . , 6 (10)
where Sj is the shape function or trial function (Burnett, 1988). The element type applied in the present simulation is a 2D and 6-nodal quadratic isoparametrical triangular element. The corresponding shape functions are: S1 (, ) = (1 − ( + ))(1 − 2( + )),
Fig. 2. Physical domain and boundary conditions.
S3 (, ) = (2 − 1), 1/2
Uc = [(Pinject − Pfront )/)]
(4)
where is the liquid density. The physical meaning of the characteristic pressure can be regarded as the effective infiltrating pressure. The dimensionless parameters are then listed as follows: 1/2
x∗ = x/Kc
1/2
y∗ = y/Kc
,
,
P ∗ = (P − Pfront )/(Pinjet − Pfront ),
u∗ = u/Uc , Kx∗ = Kx /Kc ,
Ky∗ = Ky /Kc
(5)
(6)
and ∗
u =
∂P ∗ −Re Kx∗ ∗
(7a)
∂P ∗ ∂y∗
(7b)
∂x
v∗ = −Re Ky∗
1/2 Uc Kc /
denotes the Reynolds number respectively, where Re = of the liquid metal flow. The combination of Eqs. (6) and (7) yields: ∂ ∂x∗
Kx∗
∂P ∗ ∂x∗
+
∂ ∂y∗
Ky∗
∂P ∗
∂y∗
=0
(8)
The boundary conditions are P∗ = 1 ∗
P =0 ∂P ∗ =0 ∂n∗
at the inlet gate
(9a)
along the flow front
(9b)
S6 (, ) = 4(1 − ( + ))
(11)
where and are the local coordinates for each element. Integrating Eq. (10) by parts and applying boundary conditions, the pressure distribution can be calculated numerically. Consequently, the velocity field can be obtained. 3.2. Nodal coordinates The nodal coordinates of the present FEM scheme are generated by the body-fitted method. A set of elliptic partial differential equations is used as follows (Thomas and Middlecoff, 1980): xx + yy = ˝(, )(x2 + y2 )
(12a)
xx + yy = (, )(2x + 2y )
(12b)
where ˝ and are the control functions. Interchanging the dependent variables (, ) with the independent variables (x, y) in the above equations yields: a(x + ˝x ) − 2bx + c(x + x ) = 0
(13a)
a(y + ˝y ) − 2by + c(y + y ) = 0
(13b)
where a = x2 + y2 , b = x x + y y and c = x2 + y2 . and are taken as coordinates in the computational domain. Based on the assumptions that the curvature along the geometric boundaries is locally zero and the grid lines are orthogonal to each other in the vicinity of geometric boundaries, the control functions are determined along the geometric boundaries by ˝=− =−
along the x, y-axis and side-wall
S4 (, ) = 4(1 − ( + )),
v∗ = v/Uc ,
Thus, the dimensionless forms of the continuity equation and Darcy’s law are ∂v∗ ∂u∗ + ∗ =0 ∗ ∂x ∂y
S5 (, ) = 4,
S2 (, ) = (2 − 1),
(9c)
The dimensionless governing Eq. (8) and boundary Eq. (9) are identical for flows with different injection pressures, molten metal viscosity and density. Consequently, their dimensionless pressure distributions are identical and the velocity profiles are similar, as long as their preform types are the same (Chang and Hourng, 1998). 3. Numerical procedure The body-fitted FEM, which is the combination of body-fitted grid generation with the FEM, is used to deal with irregular front surface of the molten metal in the infiltration process. The technique has advantages of uniformly spaced grids, easy application of surface boundary conditions, and modular technique.
x x + y y x2 + y2 x x + y y x2 + y2
(14a)
(14b)
For the interior points, the control function is interpolated from that on the geometric boundaries. The body-fitted grid generation is in finite difference method (FDM). A transformation of numbering of nodes between the bodyfitted FDM and FEM is necessary in the pre-process of the FEM. Fig. 3 illustrate the transformation mapping. In the numerical algorithm, the free surface is traced by multiplying the calculated velocities by the time interval. The details of the numerical procedure are described by the flow chart as shown in Fig. 4. 4. Results and discussion Materials used in the present study are similar to those reported in reference (Dopler et al., 2000). Molten aluminum with purity of
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C.-Y. Chang / Journal of Materials Processing Technology 209 (2009) 4337–4342
Fig. 3. A transformation of numbering of nodes between FDM and FEM. The numbering of grid points in FDM. The nodal numbering of a 6-nodal quadratic isoparametrical triangular element.
Fig. 4. Flow chart of the numerical simulation.
99.85% is injected at constant pressure into the mold cavity filled with a preform of alumina fibers. At 700 ◦ C, the viscosity of molten metal is 1.15 × 10−3 Pa-s. For the preform, volume fraction of the fiber is 0.24 and permeabilities along the directions of maximum and minimum flow velocities are 9.6 × 10−13 and 4 × 10−13 m2 , respectively. Fig. 5 shows the flow front of the molten metal at various times during the infiltration process. Two different orientations of unidirectional preform fill the cavity. One section of the preform with = 0◦ covers approximately 60% of the mold and the other section of the preform with = 90◦ covers the rest. For the preform with = 0◦ , the direction of maximum permeability is parallel to the xaxis as mentioned above. Owing to the nature of the unidirectional preform, the moving flow front exhibits ellipse-like shape before it encounters other orientation of preform or mold walls. For such a simple case, Eq. (3) can be transformed into an equivalent isotropic system. Thus, the differential equation governing the movement of the flow front becomes (Adams et al., 1988): Kx Pin dRf = 2 dt (1 − f )rin
˛ 1−˛
1 (Rf − Ro )(cos h2 Rf − cos2 )
The analytical solution of Eq. (15) is (Rf − Ro ) +
sinh(2R ) f
4
+
Rf 2
+
cos2 (Rf − Ro )2 2
˛ Ro2 − Rf2 cosh(2Ro ) − cosh(2Rf ) + = T∗ 8 4 1−˛
where T* defined as Kx Pin t/(1 − f )Ro2 is a dimensionless time. The analytical flow front is also denoted by a dashed line as shown in
(15)
where Rf is an elliptical extent, is the elliptical equivalent of the in-plane angle, Ro is an elliptical equivalent of the radius of the gate, f is the volume fraction of the fiber in the preform, and ˛ = Ky /Kx . The corresponding initial condition is: Rf = Ro
at t = 0
(16)
(17)
Fig. 5. Motion of flow fronts of molten metal inside fibrous preform.
C.-Y. Chang / Journal of Materials Processing Technology 209 (2009) 4337–4342
4341
Fig. 6. The velocity vector field at the infiltration time of 14.97 s.
Fig. 5. The predicted and analytical flow fronts correspond well. After the infiltration front meets the other orientation of preform, the elliptic shape of front starts to deform. Finally, a unidirectional flow is formed due to the confinement of side walls. A proper time increment can reasonably ensure that the transient infiltration process is approximated by a series of quasi-steady-state processes. The different dimensionless time increment, dT* , is adopted to reflect the variation of interfacial pressure gradients encountered. Near the inlet gate, where 0 < T* < 20 and the pressure gradient is the largest, the dimensionless time increment is less than 0.005. In the remainder, the dimensionless time increment is 0.05. If the dimensionless time increment is chosen to be larger than the time increment as mentioned above, the deviation between the exact solution (17) and the numerical solution is possibly greater than 1%. The velocity distribution at the time of 14.97 s, as shown in Fig. 6, illustrates the transition from a radial flow into a unidirectional flow. The infiltration velocity decreases very rapidly since the friction between the fluid and fibrous preform increases as their contacting area increases. The injection flow rates decrease with the infiltration time as shown in Fig. 7. As expected, the injection flow rates in preform orientation of (0◦ , 90◦ ) and (90◦ , 0◦ ) are identical in the early time of the infiltration process. After that, a higher injection flow rate is found for the case of preform orientation of (0◦ , 90◦ ) since the ellipse-like flow pattern remains. Both injection flow rates are close
Fig. 8. The relation of infiltration time versus the ratio of effective inlet pressure to molten metal viscosity.
at the end of the infiltration process because the frictions between the fluid and fibrous preform for both cases are equal. Thus, a shorter infiltration process is evident for the case of preform orientation of (0◦ , 90◦ ). Moreover, a rapid drop of the injection flow rate occurs at the instant when the molten metal first meets the side wall. This is because the flow pattern starts to transform at the instant. The trend is particularly conspicuous for the case of preform orientation of (0◦ , 90◦ ) due to severe transformation of flow pattern. Fig. 8 shows the relation between infiltration time and the ratio of effective inlet pressure to molten metal viscosity. A reduction in infiltration time can be achieved by high injection pressure and low molten metal viscosity. This is because using high injection pressure can speed up molten metal velocity and low viscosity can reduce the flow resistance in the preform. All data can be exactly fitted by hyperbolic curves as the similarity described. Thus, for pure molten aluminum at temperature of 660 ◦ C ( = 1.35 × 10−3 Pa-s), preform orientation of (0◦ , 90◦ ), and effective inlet pressure of 1.7 MPa, the predicted infiltration time is about 15.51 s. 5. Conclusions The objective of this study is to use numerical simulation to investigate the molten metal through the unidirectional fibrous preform during MMC processing. The infiltration behavior of molten metal is governed by Darcy’s law. The conclusions are summarized as follows:
Fig. 7. The variation of injection flow rate during the infiltration process.
1. The moving flow front exhibits ellipse-like shape in the early time of the infiltration process owing to the nature of the unidirectional preform. After the infiltration front meets the side walls, the fluid is gradually transformed from a radial flow to a unidirectional flow due to the confinement of side walls. 2. The infiltration process in MMCs is modeled as a series of quasisteady state processes. In order to reflect the large pressure gradient near the inlet gate, a small dimensionless time increment should be adopted to reduce the numerical deviation. 3. The injection flow rate decreases with the infiltration time. A shorter infiltration process is found for the case of preform orientation of (0◦ , 90◦ ) because of the higher injection flow rate. In addition, a rapid drop of the injection flow rate occurs at the instant when the flow pattern starts to transform. 4. A high injection pressure and low molten metal viscosity can reduce the infiltration time. The similarity analysis shows that
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the infiltration time is inversely proportional to the ratio of applied pressure to liquid viscosity for the same type of preform. References Aboulfatah, M., Danis, M., Lacoste, E., 1997. Mobility of molten metal through a fibrous preform during MMC processing. J. Mater. Synth. Process. 5, 459–466. Adams, K.L., Russel, W.B., Rebenfeld, L., 1988. Radial penetration of a viscous liquid into a planar anisotropic porous medium. Int. J. Multiphase Flow 14, 203–215. Biswas, D.K., Gatica, J.E., Tewari, S.N., 1998. Dynamic analysis of unidirectional pressure infiltration of porous preforms by pure metals. Metall. Mater. Trans. A 29, 377–385. Burnett, D.S., 1988. Finite Element Analysis. Addison-Wesley Publishing Company, Singapore, pp. 583–608. Chang, C.Y., 2006. Numerical simulation of the pressure infiltration of fibrous preforms during MMC processing. Adv. Compos. Mater. 15, 287–300. Chang, C.Y., Hourng, L.W., 1998. Numerical simulation for the transverse impregnation in resin transfer molding. J. Reinf. Plast. Compos. 17, 165–182. Chou, T.W., Kelly, A., Okura, A., 1985. Fiber-reinforced metal-matrix composites. Composites 16, 187–206. Dopler, T., Modaressi, A., Michaud, V., 2000. Simulation of metal-matrix composite isothermal infiltration processing. Metall. Mater. Trans. B 31, 225–234. Hu, L., Yang, Y., Luo, S., Xu, X., 1999. Investigation on the kinetics of infiltration of liquid aluminium into alumina fibrous preform. J. Mater. Process. Technol. 94, 230–272. Lacoste, E., Aboulfatah, M., Danis, M., Girot, F., 1993. Numerical simulation of the infiltration of fibrous preforms by a pure metal. Metall. Mater. Trans. A 24, 2667–2678. Lacoste, E., Danis, M., Girot, F., Quenisset, J.M., 1991. Numerical simulation of the injection moulding of thin parts by liquid metal infiltration of fibrous preforms. Mater. Sci. Eng. A: Struct. 135, 45–49.
Lacoste, E., Mantaux, O., Danis, M., 2002. Numerical simulation of metal matrix composites and polymer matrix composites processing by infiltration: a review. Composites A 33, 1605–1614. Masur, L.J., Mortensen, A., Cornie, J.A., Flemings, M.C., 1989. Infiltration of fibrous preforms by a pure metal. Part II. Experiment. Metall. Trans. A 20, 2549– 2557. Michaud, V.J., Comption, L.M., Mortensen, A., 1994. Capillarity in isothermal infiltration of alumina fiber preforms with aluminum. Metall. Mater. Trans. A 25, 2145–2152. Michaud, V.J., Mortensen, A., 1992. Infiltration of fiber preforms by a binary alloy: part II. Further theory and experiments. Metall. Trans. A 23, 2263–2280. Michaud, V., Mortensen, A., 2001. Infiltration processing of fibre reinforced composites: governing phenomena. Composites A 32, 981–996. Michaud, V.J., Sommer, J.L., Mortensen, A., 1999. Infiltration of fibrous preforms by a pure metal: part V. Influence of preform compressibility. Metall. Mater. Trans. A 30, 471–482. Mortensen, A., Masur, L.J., Cornie, J.A., Flemings, M.C., 1989. Infiltration of fibrous preforms by a pure metal: part I. Theory. Metall. Trans. A 20, 2535–2547. Mortensen, A., Michaud, V.J., 1990. Infiltration of fiber preforms by a binary alloy: part I. Theory. Metall. Trans. A 21, 2059–2072. Mortensen, A., Wong, T., 1990. Infiltration of fibrous preforms by a pure metal: part III. Capillary phenomena. Metall. Trans. A 21, 2257–2263. Ralph, B., Yuen, H.C., Lee, W.B., 1997. The processing of metal matrix composites—an overview. J. Mater. Process. Technol. 63, 339–353. Sommer, J.L., Mortensen, A., 1996. Forced unidirectional infiltration of deformable porous media. J. Fluid Mech. 311, 193–217. Thomas, P.D., Middlecoff, J.F., 1980. Direct control of the grid point distribution in meshes generated by elliptic equation. AIAA J. 18, 652–656. Tong, X., Khan, J.A., 1996. Infiltration and solidification/remelting of a pure metal. J. Heat Transfer 118, 173–180. Xia, Z., Zhou, Y., Mao, Z., Shang, B., 1992. Fabrication of fiber-reinforced metal-matrix composites by variable pressure infiltration. Metall. Mater. Trans. B 23, 295–302.
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Simulation of molten metal through a unidirectional fibrous preform during MMC processing Chih-Yuan Chang ∗ Department of Mechanical and Automation Engineering, Kao Yuan University, Lu-Chu 82101, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 7 January 2004 Received in revised form 7 August 2008 Accepted 5 November 2008 Keywords: Metal matrix composites Infiltration process Numerical simulation
a b s t r a c t The injection of molten metal through a fibrous preform is one of the techniques used to manufacture metal matrix composites (MMCs). The flow of liquid metal through a fibrous preform constitutes a moving boundary problem of fluid mechanics in a porous medium. In the present model, the isothermal infiltration process is considered. The numerical algorithm, based on the body-fitted finite element method (FEM), can precisely deal with the transient and irregular physical domains and calculate the flow behavior. Numerical results show that the infiltration pattern is strongly influenced by the nature of the preform. The injection flow rate under constant applied pressure decreases with the infiltration time. The effect of the preform orientation on infiltration pattern is also discussed. Similarity analysis shows that the infiltration time is inversely proportional to the ratio of applied pressure to liquid viscosity for the same type of preform. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Metal matrix composites (MMCs) can be produced via different methods (Chou et al., 1985). One of the methods for fabricating fiber-reinforced MMCs is by injecting molten metal into the interstitial spaces of the fiber preform. This process of preform infiltration has several advantages: it is relatively inexpensive, tooling is similar to the casting process, and it is possible to produce a net or near-net shape component that is difficult to machine. The phenomena governing preform infiltration are complex although infiltrated parts have found commercial applications. A number of physical, mechanical, and chemical phenomena interact, including flow of liquid metal in a porous medium, heat and mass transfer related to metal solidification and chemical interfacial reactions between the reinforcement and matrix. These phenomena have already been studied extensively from both theoretical and experimental viewpoints. Analytical and numerical solutions have been given and compared to experimental data, for unidirectional infiltration under constant applied pressure, including non-isothermal infiltration by a pure metal (Mortensen et al., 1989; Masur et al., 1989) or by a binary alloy (Mortensen and Michaud, 1990; Michaud and Mortensen, 1992); for non-isothermal infiltration by a pure metal, taking into account the influence of preform deformability (Sommer and Mortensen, 1996; Michaud et al., 1999); and for isothermal infiltration, taking into account
∗ Tel.: +886 7 6077028; fax: +886 7 6077112. E-mail address: [email protected]. 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.11.011
the influence of capillary phenomena (Mortensen and Wong, 1990; Michaud et al., 1994; Hu et al., 1999). For these cases, the boundary condition of constant applied pressure and the unidirectional configuration allowed the use of a single combined variable to describe time and position, thus yielding an analytical solution of the coupled phenomena. By using the one-dimensional solution, Aboulfatah et al. (1997) proposed an experimental procedure to measure the mobility of the molten metal through the fibrous preform as a function of the volume fraction of solidified metal. Xia et al. (1992) also proposed an approximate analytical solution for unidirectional saturated flow under a linearly increasing pressure function. More general cases have been treated as well. In these, analytical solutions can no longer be derived, and numerical tools such as finite element and finite difference methods are applied. Lacoste et al. (1991, 1993) proposed a model based on finite volume formulation to simulate the flow and the solidification of a liquid metal. Tong and Khan (1996) developed a numerical finite difference model: the mold walls were insulated, only heat exchange between fibers and metal was taken into account. Biswas et al. (1998) proposed a phenomenology model to describe fluid flow and transport phenomena for the case of unidirectional infiltration. From a finite element model initially developed in hydrogeology, Dopler et al. (2000) carried out a numerical simulation of the metal infiltration of a rigid fibrous preform with isothermal and low constant pressure. They could so follow the evolution of saturation according to the impregnation depth of the preform. Chang (2006) developed a two-dimensional numerical scheme to simulate isothermal infiltration through isotropic the preform under
4338
C.-Y. Chang / Journal of Materials Processing Technology 209 (2009) 4337–4342
Nomenclature f K P Pin Pinject Pfront Re Rf Ro rf1 , rf2 rin S t u, v Uc x,y
fiber volume fraction permeability pressure characteristic pressure injection pressure pressure along the flow front Reynolds number elliptical extent in the principal directions elliptical equivalent of the radius of the gate flow front in the principal directions radius of the inlet gate shape function infiltration time velocity components in x, y direction characteristic velocity coordinate components in the physical domain
Fig. 1. In-plane liquid flow through an anisotropic porous medium.
Greek letters ˛ ratio of y directional permeability to x directional permeability , coordinate components in the computational domain ˝, control functions in the body-fitted method molten metal density molten metal viscosity elliptical equivalent of the in-plane angle
1. The molten metal is an incompressible and Newtonian fluid. 2. The infiltration process can be regarded as a series of quasisteady state processes. 3. The infiltration process is isothermal. 4. The inertial force is negligible compared to the viscous force due to Re « 1.
Superscript * dimensionless parameter
Under the above assumptions, the momentum equation based on Darcy’s law can be written in matrix form as u
v
a time-dependent injection pressure. Ralph et al. (1997) summarized the variety of routes available for processing MMCs. Michaud and Mortensen (2001) and Lacoste et al. (2002) provided a general overview of the various infiltration processes and their associated terminology. The objective of this study is to carry out an investigation on liquid metal flowing through an anisotropic preform in MMC processing. The flow behavior of the infiltration process is twodimensional in the present model. The flow of liquid metal is a free moving boundary problem. A computational code is developed to solve the moving boundary problem by employing the body-fitted finite element method (FEM). Calculations are performed to predict the flow behavior of molten metal infiltrating fibrous preform under various conditions. Furthermore, similarity analysis is used to investigate the effect of the various parameters on the infiltration process. 2. Theoretical analysis 2.1. Governing equations This study applies a two-dimensional infiltration process of MMC since the thickness is much smaller than the other dimensions of the mold. The general case of in-plane flow of a liquid through the anisotropic porous preform is shown in Fig. 1. In the figure, rin is the radius of inlet gate, while rf1 and rf2 are the moving front resembling an ellipse characterized by maximum and minimum radial extents, respectively. is the preform orientation relative to the laboratory coordinate system. The definition of the orientation angle is the direction of maximum permeability in respect to the x-axis. For simplification, several assumptions are made as follows.
1 =−
Kxx Kyx
Kxy Kyy
⎡ ⎣
∂P ∂x ∂P ∂y
⎤ ⎦
(1)
where P, u and v represent the pressure and molten metal velocities in the x and y directions, respectively. is the viscosity of molten metal and K is the permeability tensor of the preform. The permeability is a second order tensor and measures the ease of flow. For the unidirectional preform, one can always find the two principal axes which will diagonalize the permeability. Here x- and y-axes are chosen to be the same as the principal preform ones. Thus, the non-diagonal term tensor is zero. Kxx and Kyy can denote Kx and Ky . The conservation of mass is ∂u ∂v + =0 ∂x ∂y
(2)
The governing equation can then be deduced by combining Eqs. (1) and (2) as ∂ ∂x
Kx ∂P ∂x
∂ + ∂y
Ky ∂P ∂y
=0
(3)
For a rectangular mold cavity, the physical domain is symmetric for x- and y-axes and only a quarter of the cavity needs to be concerned in the numerical simulation. The cavity is filled with the unidirectional preform having two different orientations. The corresponding boundary conditions applicable to Eq. (3) are shown in Fig. 2. In the figure, Pinject and Pfront denote the pressure at the inlet and the front, respectively. 2.2. Similarity analysis 1/2
In the infiltration process, the characteristic length (Kc ), the characteristic pressure (Pin ) and the characteristic velocity (Uc ) associated with process parameters are defined as Kc = (Kx Ky )1/2 ,
Pin = Pinject − Pfront ,
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3.1. Galerkin’s method In the present FEM model, the weight residual method adopts the Galerkin’s method. The Galerkin’s residual equation for governing Eq. (3) can be derived from:
(e)
∂ ∂x
Kx ∂P ∂x
+
∂ ∂y
Ky ∂P ∂y
Sj dxdy = 0
j = 1, 2, . . . , 6 (10)
where Sj is the shape function or trial function (Burnett, 1988). The element type applied in the present simulation is a 2D and 6-nodal quadratic isoparametrical triangular element. The corresponding shape functions are: S1 (, ) = (1 − ( + ))(1 − 2( + )),
Fig. 2. Physical domain and boundary conditions.
S3 (, ) = (2 − 1), 1/2
Uc = [(Pinject − Pfront )/)]
(4)
where is the liquid density. The physical meaning of the characteristic pressure can be regarded as the effective infiltrating pressure. The dimensionless parameters are then listed as follows: 1/2
x∗ = x/Kc
1/2
y∗ = y/Kc
,
,
P ∗ = (P − Pfront )/(Pinjet − Pfront ),
u∗ = u/Uc , Kx∗ = Kx /Kc ,
Ky∗ = Ky /Kc
(5)
(6)
and ∗
u =
∂P ∗ −Re Kx∗ ∗
(7a)
∂P ∗ ∂y∗
(7b)
∂x
v∗ = −Re Ky∗
1/2 Uc Kc /
denotes the Reynolds number respectively, where Re = of the liquid metal flow. The combination of Eqs. (6) and (7) yields: ∂ ∂x∗
Kx∗
∂P ∗ ∂x∗
+
∂ ∂y∗
Ky∗
∂P ∗
∂y∗
=0
(8)
The boundary conditions are P∗ = 1 ∗
P =0 ∂P ∗ =0 ∂n∗
at the inlet gate
(9a)
along the flow front
(9b)
S6 (, ) = 4(1 − ( + ))
(11)
where and are the local coordinates for each element. Integrating Eq. (10) by parts and applying boundary conditions, the pressure distribution can be calculated numerically. Consequently, the velocity field can be obtained. 3.2. Nodal coordinates The nodal coordinates of the present FEM scheme are generated by the body-fitted method. A set of elliptic partial differential equations is used as follows (Thomas and Middlecoff, 1980): xx + yy = ˝(, )(x2 + y2 )
(12a)
xx + yy = (, )(2x + 2y )
(12b)
where ˝ and are the control functions. Interchanging the dependent variables (, ) with the independent variables (x, y) in the above equations yields: a(x + ˝x ) − 2bx + c(x + x ) = 0
(13a)
a(y + ˝y ) − 2by + c(y + y ) = 0
(13b)
where a = x2 + y2 , b = x x + y y and c = x2 + y2 . and are taken as coordinates in the computational domain. Based on the assumptions that the curvature along the geometric boundaries is locally zero and the grid lines are orthogonal to each other in the vicinity of geometric boundaries, the control functions are determined along the geometric boundaries by ˝=− =−
along the x, y-axis and side-wall
S4 (, ) = 4(1 − ( + )),
v∗ = v/Uc ,
Thus, the dimensionless forms of the continuity equation and Darcy’s law are ∂v∗ ∂u∗ + ∗ =0 ∗ ∂x ∂y
S5 (, ) = 4,
S2 (, ) = (2 − 1),
(9c)
The dimensionless governing Eq. (8) and boundary Eq. (9) are identical for flows with different injection pressures, molten metal viscosity and density. Consequently, their dimensionless pressure distributions are identical and the velocity profiles are similar, as long as their preform types are the same (Chang and Hourng, 1998). 3. Numerical procedure The body-fitted FEM, which is the combination of body-fitted grid generation with the FEM, is used to deal with irregular front surface of the molten metal in the infiltration process. The technique has advantages of uniformly spaced grids, easy application of surface boundary conditions, and modular technique.
x x + y y x2 + y2 x x + y y x2 + y2
(14a)
(14b)
For the interior points, the control function is interpolated from that on the geometric boundaries. The body-fitted grid generation is in finite difference method (FDM). A transformation of numbering of nodes between the bodyfitted FDM and FEM is necessary in the pre-process of the FEM. Fig. 3 illustrate the transformation mapping. In the numerical algorithm, the free surface is traced by multiplying the calculated velocities by the time interval. The details of the numerical procedure are described by the flow chart as shown in Fig. 4. 4. Results and discussion Materials used in the present study are similar to those reported in reference (Dopler et al., 2000). Molten aluminum with purity of
4340
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Fig. 3. A transformation of numbering of nodes between FDM and FEM. The numbering of grid points in FDM. The nodal numbering of a 6-nodal quadratic isoparametrical triangular element.
Fig. 4. Flow chart of the numerical simulation.
99.85% is injected at constant pressure into the mold cavity filled with a preform of alumina fibers. At 700 ◦ C, the viscosity of molten metal is 1.15 × 10−3 Pa-s. For the preform, volume fraction of the fiber is 0.24 and permeabilities along the directions of maximum and minimum flow velocities are 9.6 × 10−13 and 4 × 10−13 m2 , respectively. Fig. 5 shows the flow front of the molten metal at various times during the infiltration process. Two different orientations of unidirectional preform fill the cavity. One section of the preform with = 0◦ covers approximately 60% of the mold and the other section of the preform with = 90◦ covers the rest. For the preform with = 0◦ , the direction of maximum permeability is parallel to the xaxis as mentioned above. Owing to the nature of the unidirectional preform, the moving flow front exhibits ellipse-like shape before it encounters other orientation of preform or mold walls. For such a simple case, Eq. (3) can be transformed into an equivalent isotropic system. Thus, the differential equation governing the movement of the flow front becomes (Adams et al., 1988): Kx Pin dRf = 2 dt (1 − f )rin
˛ 1−˛
1 (Rf − Ro )(cos h2 Rf − cos2 )
The analytical solution of Eq. (15) is (Rf − Ro ) +
sinh(2R ) f
4
+
Rf 2
+
cos2 (Rf − Ro )2 2
˛ Ro2 − Rf2 cosh(2Ro ) − cosh(2Rf ) + = T∗ 8 4 1−˛
where T* defined as Kx Pin t/(1 − f )Ro2 is a dimensionless time. The analytical flow front is also denoted by a dashed line as shown in
(15)
where Rf is an elliptical extent, is the elliptical equivalent of the in-plane angle, Ro is an elliptical equivalent of the radius of the gate, f is the volume fraction of the fiber in the preform, and ˛ = Ky /Kx . The corresponding initial condition is: Rf = Ro
at t = 0
(16)
(17)
Fig. 5. Motion of flow fronts of molten metal inside fibrous preform.
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Fig. 6. The velocity vector field at the infiltration time of 14.97 s.
Fig. 5. The predicted and analytical flow fronts correspond well. After the infiltration front meets the other orientation of preform, the elliptic shape of front starts to deform. Finally, a unidirectional flow is formed due to the confinement of side walls. A proper time increment can reasonably ensure that the transient infiltration process is approximated by a series of quasi-steady-state processes. The different dimensionless time increment, dT* , is adopted to reflect the variation of interfacial pressure gradients encountered. Near the inlet gate, where 0 < T* < 20 and the pressure gradient is the largest, the dimensionless time increment is less than 0.005. In the remainder, the dimensionless time increment is 0.05. If the dimensionless time increment is chosen to be larger than the time increment as mentioned above, the deviation between the exact solution (17) and the numerical solution is possibly greater than 1%. The velocity distribution at the time of 14.97 s, as shown in Fig. 6, illustrates the transition from a radial flow into a unidirectional flow. The infiltration velocity decreases very rapidly since the friction between the fluid and fibrous preform increases as their contacting area increases. The injection flow rates decrease with the infiltration time as shown in Fig. 7. As expected, the injection flow rates in preform orientation of (0◦ , 90◦ ) and (90◦ , 0◦ ) are identical in the early time of the infiltration process. After that, a higher injection flow rate is found for the case of preform orientation of (0◦ , 90◦ ) since the ellipse-like flow pattern remains. Both injection flow rates are close
Fig. 8. The relation of infiltration time versus the ratio of effective inlet pressure to molten metal viscosity.
at the end of the infiltration process because the frictions between the fluid and fibrous preform for both cases are equal. Thus, a shorter infiltration process is evident for the case of preform orientation of (0◦ , 90◦ ). Moreover, a rapid drop of the injection flow rate occurs at the instant when the molten metal first meets the side wall. This is because the flow pattern starts to transform at the instant. The trend is particularly conspicuous for the case of preform orientation of (0◦ , 90◦ ) due to severe transformation of flow pattern. Fig. 8 shows the relation between infiltration time and the ratio of effective inlet pressure to molten metal viscosity. A reduction in infiltration time can be achieved by high injection pressure and low molten metal viscosity. This is because using high injection pressure can speed up molten metal velocity and low viscosity can reduce the flow resistance in the preform. All data can be exactly fitted by hyperbolic curves as the similarity described. Thus, for pure molten aluminum at temperature of 660 ◦ C ( = 1.35 × 10−3 Pa-s), preform orientation of (0◦ , 90◦ ), and effective inlet pressure of 1.7 MPa, the predicted infiltration time is about 15.51 s. 5. Conclusions The objective of this study is to use numerical simulation to investigate the molten metal through the unidirectional fibrous preform during MMC processing. The infiltration behavior of molten metal is governed by Darcy’s law. The conclusions are summarized as follows:
Fig. 7. The variation of injection flow rate during the infiltration process.
1. The moving flow front exhibits ellipse-like shape in the early time of the infiltration process owing to the nature of the unidirectional preform. After the infiltration front meets the side walls, the fluid is gradually transformed from a radial flow to a unidirectional flow due to the confinement of side walls. 2. The infiltration process in MMCs is modeled as a series of quasisteady state processes. In order to reflect the large pressure gradient near the inlet gate, a small dimensionless time increment should be adopted to reduce the numerical deviation. 3. The injection flow rate decreases with the infiltration time. A shorter infiltration process is found for the case of preform orientation of (0◦ , 90◦ ) because of the higher injection flow rate. In addition, a rapid drop of the injection flow rate occurs at the instant when the flow pattern starts to transform. 4. A high injection pressure and low molten metal viscosity can reduce the infiltration time. The similarity analysis shows that
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