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Hydrodynamic modeling of a continuous metal matrix composite fabrication process as a cylindrical array
Materials Science and Engineering A297 (2001) 132 – 137 www.elsevier.com/locate/msea
Hydrodynamic modeling of a continuous metal matrix composite fabrication process as a cylindrical array J.H. Nadler, J.A. Isaacs *, G.J. Kowalski Department of Mechanical, Industrial and Manufacturing Engineering, 334 Snell Engineering Center, Northeastern Uni6ersity, 360 Huntington A6e., Boston, MA 02115, USA Received 18 October 1999; received in revised form 9 June 2000
Abstract An alternative model for predicting adequate operating parameters of a continuous, metal matrix composite production process is developed. Predictions from the model are compared with a previous model and with experimental results. The fiber tow molten metal interaction is described as flow through a parallel, cylindrical fiber array (PCA) in the new model. The predicted critical pulling velocity, pulling force and the pressure gradient are self-consistent with experimentally measured parameters within 1%. The PCA model is expected to provide a useful design and operating tool for fabrication of metal matrix composite wire. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Metal matrix composite production process; MMC wire; Fiber-reinforced MMC; MMC process modelling
1. Introduction A continuous production technique to manufacture metal matrix composites (MMC) has been developed at Northeastern University [1,2]. A schematic of this technique is shown in Fig. 1. In this process a high-pressure melt is used to overcome the interfacial energies associated with a non-wetting ceramic fiber tow to produce MMC wire [3–7]. The fiber tow is quickly infiltrated in this process. This continuous fabrication technique has several advantages for MMC production over the batch-type process, powder metallurgy or diffusion bonding techniques, because continuous processing shows higher yields and lower costs than these other techniques. The shorter processing times and higher cooling rates achieved in the continuous process reduce or eliminate fiber degradation caused by chemical reactions with the melt. Previous work also determined a dependency of the ultimate tensile strength (UTS) of MMC continuous wire on processing speed [2]. Increases in UTS were qualitatively attributed to changes in the matrix microstructure including decreased grain * Corresponding author. Tel.: +1-617-3733989; fax: + 1-6173732921. E-mail address: [email protected] (J.A. Isaacs).
size and amount of second phase segregation. Grain size reduction was correlated with higher pulling velocities [8]. During continuous processing, there is a location along the length of the entrance orifice, where the temperature is equal to the solidus temperature [9]. The temperature gradient along this length ranges from the temperature of the melt (100°C above the liquidus temperature of the matrix metal), to ambient temperature. This location at the entrance orifice was initially intended to act as a ‘solidification gate’ providing a blocking mechanism to prevent the pressurized molten matrix metal from spraying out of the apparatus [1]. Infiltration was assumed to occur only after the fiber bundle had passed through this solidification gate. Initial experimental runs produced only a few meters of MMC wire due to fiber tow fracture caused by ‘solid choking’ in the entrance orifice. Williams et al. [9] developed a first-order, analytical hydrodynamic model that disproved the theory of the solidification gate. If the metal material solidifies within the entrance orifice, it will ‘choke’ the process and cause the fiber tow to break. Therefore to fabricate MMC wire successfully, the rate at which molten aluminum is pushed out of the vessel and into the entrance orifice must equal the rate at which the fiber bundle carries aluminum back up
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into the melt. This condition corresponds to a zero net mass flow rate along the length of the orifice, and implies that there is a critical minimum fiber tow pulling velocity for a given infiltration pressure. The critical pulling velocity was identified as the minimum speed for a given infiltration pressure at which the fiber bundle could be pulled through the system without choking the inlet orifice with molten metal. This analytical model qualitatively correlated with the initial experimental runs and led to operation procedures in which hundreds of meters of MMC wire were produced. While the analytical fluid model developed by Williams et al. [9] improved continuous infiltration processing, it did not quantitatively agree with the velocities and measured pulling forces of the experiments. This model treated the entrance orifice region as two concentric cylinders, representing the inner diameter of the orifice and the outer diameter of the fiber bundle. In the present analysis the physical mechanisms of the solid cylinder model will be applied to a parallel, cylindrical fiber array (PCA) model of the entrance orifice region. Comparison between the PCA model and experiment demonstrates that the PCA model quantitatively agrees with the experimental observations of the continuous MMC fabrication process. Fig. 1. High pressure infiltration apparatus is shown schematically, illustrating the simplified geometry of the entrance and middle orifices.
Fig. 2. Detail of entrance and middle orifices depicting the geometry of a representative cell surrounding fiber i, and the parameters used in model development.
2. Hydrodynamic modeling In the PCA model, the fiber tow is described as an array of parallel cylinders of reinforcement material surrounded by a flow of molten metal. The coordinate system is such that the origin lies at the center of the moving fiber tow at the top of the entrance orifice, where the pressure is at a maximum. Fig. 2 illustrates this simplified view where: Vf is fiber tow pulling velocity, Df is the diameter that encircles the fiber tow region, D0 is the inlet orifice diameter, lp is the length along which pressure gradient acts, Pm is the melt pressure and P is the atmospheric pressure. A representative cell surrounding a fiber, (i ), is also depicted in Fig. 2 and is used in the following development. The fiber tow cylindrical array travels in the negative z-direction with velocity Vf, while the molten metal moves in the positive z-direction as a result of the pressure gradient. As in the solid cylinder model [9], the net mass flow rate must equal zero to avoid the ‘choking’ condition. A model for the velocity profile in the molten metal is derived to allow calculation of the minimum critical pulling velocity and the pulling force. Both of these parameters are measured experimentally to test the model predictions. The velocity profile of the molten aluminum across the entrance orifice is developed by considering a single reinforcement fiber that is bounded by an imaginary cylindrical enclosure of parallel fibers (Fig. 2b), which
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represents an array of adjacent fibers to the fiber of interest. The development is based on a fiber that is located at the center of the fiber tow. This approach is appropriate because of the high packing density of the fibers in the infiltrated fiber tow and its near uniform distribution. The fiber and its imaginary cylindrical enclosure both move in the direction opposite to the pressure gradient with a velocity of Vf. The following simplifying assumptions were also made: Material properties do not significantly change in the temperature range of interest; The cylindrical fiber array is sufficiently longer than the entrance length of the developing flow; The geometric and pressure considerations allow for fully developed, laminar, and incompressible flow of the molten metal; The body force is neglected; Heat transfer effects are neglected. These assumptions are consistent with Poiseulle-Couette flow in a concentric annulus [10]. The fully developed flow assumption is justified because the Reynolds number for a typical pulling velocity of 0.127 m s − 1 is 2.59 and the estimated entrance length for the flow is 20 mm. This estimated entrance is much smaller than the dimensions of the system. The general solution to the conservation of momentum equations is: uz (r)=
C0r 2 +C1 ln r +C2 4m
(1)
where C1 and C2 are the integration constants. The axial component of conservation of momentum with a pressure gradient, C0, is:
The resulting velocity profile is: u(r)= Vf +
n
C0 2 (r 2p − r 2f ) ln(r/rf) r − r 2f + ln(rf/rp) 4m
(6)
As in the solid cylinder model, the condition for successful processing is zero net mass flow. With the PCA model, the zero net mass flow criterion is satisfied in each representative cell investigated. The mass flow rate criteria is given by: m= 0 = r
&
rp
u(2pr) dr
rf
= Vf +
(7a)
n
C0 (r 2p − r 2f ) r 2p ln(rp/rf)− (r 2p − r 2f )/2 2 ln(rf/rp) 4m
(7b)
The critical minimum velocity is obtained by rearranging Eq. (7b): Vf,crit = −
C0 (r 2p − r 2f ) r 2p ln(rp/rf)− (r 2p − r 2f )/2 4m 2 ln(rf/rp)
n
(8) Pulling velocities less than the critical velocity will allow aluminum to build up in the entrance orifice. The critical minimum pulling velocity is related to the system’s pressure gradient through Eq. (8), and solving for the pressure gradient in terms of the critical pulling velocity yields: C0 =
n
− 4mVf (r 2p − r 2f ) r 2p ln (rp/rf)− (r 2p − r 2f )/2 + 2 ln (rf/rp)
(9)
u(r = rp)=Vf
(3)
The pressure gradient is related to the fiber tow pulling force. The fiber tow pulling force, Fpull, is the force required to pull the fiber tow through the orifice, through the molten metal and the infiltrated fiber MMC wire out through the middle orifice. It is calculated using the viscous shear equation for the center fiber of the representative cell and multiplying it by the number of fibers.
u(r = rf)=Vf
(4)
Fpull = tAfiber = 2prflmeltNftr = r f
C0 =
dP m d du = r z dz r dr dr
(2)
In the PCA model, the boundary conditions are as follows:
where rf is the individual fiber radius ( 5 mm), and Vf is the process pulling velocity. The variable rp is the radial distance from the center of fiber of interest (fiber i ) to the surface of the imaginary cylindrical enclosure (Fig. 2). This distance is determined by assuming a uniform packing density of the fiber tow and that the fiber tow completely fills the entrance orifice. Using the geometry of Fig. 2b, the radius of the imaginary cylinder is related to the number of fibers and the inlet orifice diameter: rp =
'
2 0
D + rf 4Nf
(5)
where D0 is the total orifice diameter and Nf is the number of fibers in the bundle.
where
)
(10)
(u − C0 r 2 − r 2f t= −m = 2rf + p (r r = r f 4 rf ln (rf/rp)
(11)
Afiber represents the fiber surface area and lmelt is the length of the fiber tow in the molten metal. The pressure gradient, C0, is calculated from the measured pulling force by rearranging Eqs. (10) and (11). C0 =
4Fpull
r 2 − r 2f AfiberNf 2rf + p rf ln(rf/rp)
(12)
The preceding development provides two equations (Eqs. (9) and (12)) that relate three parameters: the critical pulling velocity, the pressure gradient and the
J.H. Nadler et al. / Materials Science and Engineering A297 (2001) 132–137 Table 1 Experimentally measured process parameters used to test the PCA model Symbol
Definition
Value
mal Pm lp Df D0 Vf Nf lmelt Rf Rp Fpull
Dynamic viscosity Infiltration pressure Pressure gradient length Solid cylinder diameter Entrance orifice diameter Pulling velocity Number of fibers in bundle Bundle length in melt Fiber radius Pore radius Pulling force
0.0012 N s m−2 2.4 MPa 0.038 m 0.00124 m 0.00132 m −0.127 m s−1 100 000 0.05 m 5 mm 9 mm −9 N
Fig. 3. Four velocity profiles, u(r), are calculated using the PCA model. Curve A represents the calculated velocity profile for a pulling velocity of 0.0127 m s − 1 and a pressure gradient, C0, of 9.6(107) N m − 2. Curve B uses 0.127 m s − 1 as the critical pulling velocity and Eq. (9) to calculate the pressure gradient, C0. Curve C uses a pressure gradient of 9.6(107) N m − 2 and fiber pulling velocity, Vf =0.127 m s − 1. Curve D uses a pressure gradient, C0, calculated using Eq. (12), with a pulling velocity Vf = 0.127 m s − 1 and a pulling force of 9 N.
fiber tow pulling force. The validity of this model is tested using experimental data from processing runs to determine its predictive capabilities.
3. Results The physical consistency and accuracy of the PCA model is tested using the preceding development and the observed values of the pulling velocity, fiber tow pulling force and the melt pressure (Vf, Fpull and Pm, respectively). The actual values of the pressure gradient, C0, and of the critical pulling velocity have not been determined for the continuous fabrication process. The process parameters used in this analysis are listed in Table 1. These parameters correspond to a successful production run using aluminum oxide fiber with a molten aluminum melt [8].
135
The velocity profile of the melt in the representative cell is plotted as a function of the radius in Fig. 3 for four different conditions that correspond to those observed in production runs. Curve A represents the calculated velocity profile for a pulling velocity of 0.0127 m s − 1 and a pressure gradient, C0, of 9.6(107) N m − 2. Unsuccessful production runs have been observed with a pulling velocity of 0.0127 m s − 1, as previously reported [9]. The majority of the area bounded by the zero velocity line and Curve A is in the positive velocity region, indicating a net mass flow in the positive z-direction into the orifice. A net mass flow rate into the entrance region results in the solid choking phenomenon, which is consistent with the unsuccessful run for the values of the input parameters. Curve B is determined using the PCA model with 0.127 m s − 1 as the critical pulling velocity in Eq. (9) to calculate the pressure gradient, C0, and again in Eq. (6) to calculate the velocity profile. This set of process parameters corresponds to conditions observed for a successful production run [8]. The velocity profile shown in Curve B has equal areas above and below the zero velocity line and hence, yields zero net mass flow. If the critical pulling velocity equals 0.127 m s − 1, the model predictions are consistent with experimental observation and results in continuous production. The velocity profile summarized by Curve C is calculated using a pressure gradient of 9.6(107) N m − 2 and fiber pulling velocity, Vf = 0.127 m s − 1. These parameter values do not limit the pulling velocity to its critical value as does the use of Eq. (8). This curve illustrates a net mass flow in the negative z-direction, towards the melt and would correspond to operating conditions for a successful run. Curve D corresponds to the velocity profile predicted with a pressure gradient, C0, calculated using Eq. (12), a pulling velocity Vf = 0.127 m s − 1 and a pulling force of 9 N. (This pulling force was chosen because it corresponds to the experimentally observed value for a successful run with a pulling velocity of 0.127 m s − 1.) The calculated velocity profile indicated by Curve D would also reflect a successful run. Curve D lies below the zero velocity line and indicates a mass flow rate in the negative z-direction, into the melt region. This mass flow rate would not lead to the choking conditions that Curve A reflects. The above calculations demonstrate that the PCA model has the capability to predict the qualitative physical behavior that is observed in the MMC fabrication process. At a low pulling velocity, the melt will exit the pressure vessel and solidify leading to fiber tow breakage. At sufficiently high pulling velocities, the melt is held within the vessel leading to a successful production run. The calculated velocity profiles (Curves A–D) are based on either a characteristic pressure gradient, pulling velocity or pulling force estimate. The trends
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summarized in Fig. 3 allow prediction of appropriate processing parameters for successful fabrication of MMC wire. The results of the PCA model, the solid cylinder model and the experiment are compared in Table 2. Three different methods of comparing the measured parameters (melt pressure, pulling velocity and pulling force) with the model predictions are reported. The experimentally measured parameters are shown in bold in Table 2. The observed pulling velocity is not necessarily the critical pulling velocity used in the model. In the first method, the pressure gradient is calculated assuming that the pressure varies linearly along the melt length, C0 = (Pmelt −P )/lp and the critical pulling velocity and pulling force are calculated parameters. In the second method, the pressure gradient is calculated using Eq. (9), and the observed velocity is used as the critical pulling velocity. The critical pulling force is the calculated parameter. In the third method the pressure gradient is calculated using Eq. (12) and the measured pulling force is an input variable. The critical pulling velocity is the calculated parameter. Results in Table 2 demonstrate that the PCA model is much more consistent, by at least an order of magnitude, than the solid cylinder model. Using methods 2 and 3 with the PCA model, the calculated critical pulling velocity and the pulling force are within 1% of their observed values. The calculations using method 1 are not in good agreement. However, the observed pressure gradient, Co, is estimated using the maximum possible length of the melt and may not be an accurate value for this parameter. The behavior predicted using methods 2 and 3 is consistent with Curve C in Fig. 3 and suggests that the measured pulling velocity is near its critical value.
4. Discussion Though the PCA model does not provide perfect agreement with the experimental results, its predictions
show much better quantitative agreement than the solid cylinder model. The PCA model provides much more consistency with the experimental results than does the solid cylinder model. As shown in Table 2, calculating the pulling velocity using the measured value of the pulling force or the pulling force using the measured value of pulling velocity results in predictions of the other observed parameters that are within 1%. This type of agreement is not observed for the solid cylinder model, which predicts values that are orders of magnitude different than the experimental measurements. The consistency of the PCA model and the experimental results provides strong evidence that the physical mechanisms used to develop it are correct and that the lack of better agreement with the experiment is due to other factors related to the specification of the input variables. For example, the calculated pressure gradient for methods 2 and 3 is an order of magnitude larger than that estimated in method 1. This discrepancy suggests that the pressure gradient is present in a short entrance region near the entrance orifice, not over the entire melt length as assumed in method 1. The calculations performed using methods 2 and 3 indicate that this entrance region is on the order of 0.004 m, not the 0.05 m length used in the method 1 calculation. There are also other factors that could explain the observed discrepancies in Table 2. These factors could explain the small, approximately 1% error observed in method 3 for the calculated critical pulling velocity. This calculated pulling velocity is greater than that observed by less than 1%. However, physically, this calculation would suggest that the observed successful run would not have occurred. Using the PCA model to predict operating conditions would have resulted in a conservative estimate of the pulling velocity and would have led to a successful production run. Other factors include variation in the number of fibers in the tow, the increase surface area of the individual fibers due to their surface roughness and its contribution to viscous drag, the variations in the packing density of the fiber tow as
Table 2 Comparison of predictions from the PCA and solid cylinder models with experimental resultsa Method 1 Pmelt−P C0 = lm
Method 2
Parallel, cylindrical array model C0 (Pa m−1) Vf,crit (m s−1) Fpull (N)
48 000 000 −0.0097 −0.67
631 903 420 −0.127 −8.916.24
638 016 631 −0.128 −9.0
Solid cylinder model C0 (Pa m−1) Vf,crit (m s−1) Fpull (N)
96 000 000 −21.789 −0.190
559 000 −0.127 −0.001
4 540 000 000 −1032.000 −9.000
a
Values in bold face represent experimental data.
C0 from Vf,crit (Eq. (9))
Method 3 C0 from Fpull (Eq. (12))
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it is being infiltrated, errors caused by the representative cell technique for calculating the shear stress and additional system level friction forces that are included in the experimentally measured pulling force. However, the close agreement between the calculations and the measured values support the assumption that these factors are second-order effects.
Acknowledgements
5. Conclusion
References
The predictions of the critical pulling velocity, pulling force and pressure gradient of the PCA model are compared with those of the solid cylinder model and with experimental results. This comparison provides strong evidence that the PCA model is a more accurate representation than the solid cylinder model and is in agreement with the observed behavior of the continuous metal matrix composite wire production. This evidence suggests that the physical mechanism responsible for the molten metal remaining in the crucible is a balance of the fluid drag forces on the infiltrated fiber tow and the pressure driven flow towards the inlet. The region of the melt over which the pressure driven flow occurs, is estimated to be an order of magnitude less than the melt length. The described PCA model provides a working design tool for setting the operating parameters in this manufacturing process.
[1] J.T. Blucher, US Patent Number 5736199, 7 April 1998. [2] S.L. Sampson, MS Thesis, Northeastern University, Boston, MA, September 1997. [3] J.A. Cornie, Y.-M. Chiang, D.R. Uhlman, A. Mortensen, J.M. Collins, Am. Ceram. Soc. Bull. 65 (1986) 293 – 304. [4] K.K. Chawla, Composite Materials, Springer, New York, 1987. [5] R.B. Bhagat, Metal Matrix Composites, vol. 1, Academic Press, Orlando, FL, 1990, pp. 43 – 82. [6] A. Mortensen, I. Jin, Int. Mater. Rev. 37 (3) (1992) 101– 128. [7] B. Ralph, H.C. Yuen, W.B. Lee, J. Mater. Proc. Tech. 63 (1997) 339 – 353. [8] J.H. Nadler, MS Thesis, Northeastern University, Boston, MA, June 1998. [9] M.G. Williams, J.A. Isaacs, J.H. Nadler, S.L. Sampson, G.J. Kowalski, J.T. Blucher, Analytically motivated improvements in continuous MMC wire fabrication, Mater. Sci. Eng., 1998. [10] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1960.
.
The authors would like to thank the Department of Mechanical, Industrial and Manufacturing Engineering of Northeastern University for support, and our colleague Dr Joseph T. Blucher for manufacturing the MMC wire samples.
Hydrodynamic modeling of a continuous metal matrix composite fabrication process as a cylindrical array J.H. Nadler, J.A. Isaacs *, G.J. Kowalski Department of Mechanical, Industrial and Manufacturing Engineering, 334 Snell Engineering Center, Northeastern Uni6ersity, 360 Huntington A6e., Boston, MA 02115, USA Received 18 October 1999; received in revised form 9 June 2000
Abstract An alternative model for predicting adequate operating parameters of a continuous, metal matrix composite production process is developed. Predictions from the model are compared with a previous model and with experimental results. The fiber tow molten metal interaction is described as flow through a parallel, cylindrical fiber array (PCA) in the new model. The predicted critical pulling velocity, pulling force and the pressure gradient are self-consistent with experimentally measured parameters within 1%. The PCA model is expected to provide a useful design and operating tool for fabrication of metal matrix composite wire. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Metal matrix composite production process; MMC wire; Fiber-reinforced MMC; MMC process modelling
1. Introduction A continuous production technique to manufacture metal matrix composites (MMC) has been developed at Northeastern University [1,2]. A schematic of this technique is shown in Fig. 1. In this process a high-pressure melt is used to overcome the interfacial energies associated with a non-wetting ceramic fiber tow to produce MMC wire [3–7]. The fiber tow is quickly infiltrated in this process. This continuous fabrication technique has several advantages for MMC production over the batch-type process, powder metallurgy or diffusion bonding techniques, because continuous processing shows higher yields and lower costs than these other techniques. The shorter processing times and higher cooling rates achieved in the continuous process reduce or eliminate fiber degradation caused by chemical reactions with the melt. Previous work also determined a dependency of the ultimate tensile strength (UTS) of MMC continuous wire on processing speed [2]. Increases in UTS were qualitatively attributed to changes in the matrix microstructure including decreased grain * Corresponding author. Tel.: +1-617-3733989; fax: + 1-6173732921. E-mail address: [email protected] (J.A. Isaacs).
size and amount of second phase segregation. Grain size reduction was correlated with higher pulling velocities [8]. During continuous processing, there is a location along the length of the entrance orifice, where the temperature is equal to the solidus temperature [9]. The temperature gradient along this length ranges from the temperature of the melt (100°C above the liquidus temperature of the matrix metal), to ambient temperature. This location at the entrance orifice was initially intended to act as a ‘solidification gate’ providing a blocking mechanism to prevent the pressurized molten matrix metal from spraying out of the apparatus [1]. Infiltration was assumed to occur only after the fiber bundle had passed through this solidification gate. Initial experimental runs produced only a few meters of MMC wire due to fiber tow fracture caused by ‘solid choking’ in the entrance orifice. Williams et al. [9] developed a first-order, analytical hydrodynamic model that disproved the theory of the solidification gate. If the metal material solidifies within the entrance orifice, it will ‘choke’ the process and cause the fiber tow to break. Therefore to fabricate MMC wire successfully, the rate at which molten aluminum is pushed out of the vessel and into the entrance orifice must equal the rate at which the fiber bundle carries aluminum back up
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into the melt. This condition corresponds to a zero net mass flow rate along the length of the orifice, and implies that there is a critical minimum fiber tow pulling velocity for a given infiltration pressure. The critical pulling velocity was identified as the minimum speed for a given infiltration pressure at which the fiber bundle could be pulled through the system without choking the inlet orifice with molten metal. This analytical model qualitatively correlated with the initial experimental runs and led to operation procedures in which hundreds of meters of MMC wire were produced. While the analytical fluid model developed by Williams et al. [9] improved continuous infiltration processing, it did not quantitatively agree with the velocities and measured pulling forces of the experiments. This model treated the entrance orifice region as two concentric cylinders, representing the inner diameter of the orifice and the outer diameter of the fiber bundle. In the present analysis the physical mechanisms of the solid cylinder model will be applied to a parallel, cylindrical fiber array (PCA) model of the entrance orifice region. Comparison between the PCA model and experiment demonstrates that the PCA model quantitatively agrees with the experimental observations of the continuous MMC fabrication process. Fig. 1. High pressure infiltration apparatus is shown schematically, illustrating the simplified geometry of the entrance and middle orifices.
Fig. 2. Detail of entrance and middle orifices depicting the geometry of a representative cell surrounding fiber i, and the parameters used in model development.
2. Hydrodynamic modeling In the PCA model, the fiber tow is described as an array of parallel cylinders of reinforcement material surrounded by a flow of molten metal. The coordinate system is such that the origin lies at the center of the moving fiber tow at the top of the entrance orifice, where the pressure is at a maximum. Fig. 2 illustrates this simplified view where: Vf is fiber tow pulling velocity, Df is the diameter that encircles the fiber tow region, D0 is the inlet orifice diameter, lp is the length along which pressure gradient acts, Pm is the melt pressure and P is the atmospheric pressure. A representative cell surrounding a fiber, (i ), is also depicted in Fig. 2 and is used in the following development. The fiber tow cylindrical array travels in the negative z-direction with velocity Vf, while the molten metal moves in the positive z-direction as a result of the pressure gradient. As in the solid cylinder model [9], the net mass flow rate must equal zero to avoid the ‘choking’ condition. A model for the velocity profile in the molten metal is derived to allow calculation of the minimum critical pulling velocity and the pulling force. Both of these parameters are measured experimentally to test the model predictions. The velocity profile of the molten aluminum across the entrance orifice is developed by considering a single reinforcement fiber that is bounded by an imaginary cylindrical enclosure of parallel fibers (Fig. 2b), which
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represents an array of adjacent fibers to the fiber of interest. The development is based on a fiber that is located at the center of the fiber tow. This approach is appropriate because of the high packing density of the fibers in the infiltrated fiber tow and its near uniform distribution. The fiber and its imaginary cylindrical enclosure both move in the direction opposite to the pressure gradient with a velocity of Vf. The following simplifying assumptions were also made: Material properties do not significantly change in the temperature range of interest; The cylindrical fiber array is sufficiently longer than the entrance length of the developing flow; The geometric and pressure considerations allow for fully developed, laminar, and incompressible flow of the molten metal; The body force is neglected; Heat transfer effects are neglected. These assumptions are consistent with Poiseulle-Couette flow in a concentric annulus [10]. The fully developed flow assumption is justified because the Reynolds number for a typical pulling velocity of 0.127 m s − 1 is 2.59 and the estimated entrance length for the flow is 20 mm. This estimated entrance is much smaller than the dimensions of the system. The general solution to the conservation of momentum equations is: uz (r)=
C0r 2 +C1 ln r +C2 4m
(1)
where C1 and C2 are the integration constants. The axial component of conservation of momentum with a pressure gradient, C0, is:
The resulting velocity profile is: u(r)= Vf +
n
C0 2 (r 2p − r 2f ) ln(r/rf) r − r 2f + ln(rf/rp) 4m
(6)
As in the solid cylinder model, the condition for successful processing is zero net mass flow. With the PCA model, the zero net mass flow criterion is satisfied in each representative cell investigated. The mass flow rate criteria is given by: m= 0 = r
&
rp
u(2pr) dr
rf
= Vf +
(7a)
n
C0 (r 2p − r 2f ) r 2p ln(rp/rf)− (r 2p − r 2f )/2 2 ln(rf/rp) 4m
(7b)
The critical minimum velocity is obtained by rearranging Eq. (7b): Vf,crit = −
C0 (r 2p − r 2f ) r 2p ln(rp/rf)− (r 2p − r 2f )/2 4m 2 ln(rf/rp)
n
(8) Pulling velocities less than the critical velocity will allow aluminum to build up in the entrance orifice. The critical minimum pulling velocity is related to the system’s pressure gradient through Eq. (8), and solving for the pressure gradient in terms of the critical pulling velocity yields: C0 =
n
− 4mVf (r 2p − r 2f ) r 2p ln (rp/rf)− (r 2p − r 2f )/2 + 2 ln (rf/rp)
(9)
u(r = rp)=Vf
(3)
The pressure gradient is related to the fiber tow pulling force. The fiber tow pulling force, Fpull, is the force required to pull the fiber tow through the orifice, through the molten metal and the infiltrated fiber MMC wire out through the middle orifice. It is calculated using the viscous shear equation for the center fiber of the representative cell and multiplying it by the number of fibers.
u(r = rf)=Vf
(4)
Fpull = tAfiber = 2prflmeltNftr = r f
C0 =
dP m d du = r z dz r dr dr
(2)
In the PCA model, the boundary conditions are as follows:
where rf is the individual fiber radius ( 5 mm), and Vf is the process pulling velocity. The variable rp is the radial distance from the center of fiber of interest (fiber i ) to the surface of the imaginary cylindrical enclosure (Fig. 2). This distance is determined by assuming a uniform packing density of the fiber tow and that the fiber tow completely fills the entrance orifice. Using the geometry of Fig. 2b, the radius of the imaginary cylinder is related to the number of fibers and the inlet orifice diameter: rp =
'
2 0
D + rf 4Nf
(5)
where D0 is the total orifice diameter and Nf is the number of fibers in the bundle.
where
)
(10)
(u − C0 r 2 − r 2f t= −m = 2rf + p (r r = r f 4 rf ln (rf/rp)
(11)
Afiber represents the fiber surface area and lmelt is the length of the fiber tow in the molten metal. The pressure gradient, C0, is calculated from the measured pulling force by rearranging Eqs. (10) and (11). C0 =
4Fpull
r 2 − r 2f AfiberNf 2rf + p rf ln(rf/rp)
(12)
The preceding development provides two equations (Eqs. (9) and (12)) that relate three parameters: the critical pulling velocity, the pressure gradient and the
J.H. Nadler et al. / Materials Science and Engineering A297 (2001) 132–137 Table 1 Experimentally measured process parameters used to test the PCA model Symbol
Definition
Value
mal Pm lp Df D0 Vf Nf lmelt Rf Rp Fpull
Dynamic viscosity Infiltration pressure Pressure gradient length Solid cylinder diameter Entrance orifice diameter Pulling velocity Number of fibers in bundle Bundle length in melt Fiber radius Pore radius Pulling force
0.0012 N s m−2 2.4 MPa 0.038 m 0.00124 m 0.00132 m −0.127 m s−1 100 000 0.05 m 5 mm 9 mm −9 N
Fig. 3. Four velocity profiles, u(r), are calculated using the PCA model. Curve A represents the calculated velocity profile for a pulling velocity of 0.0127 m s − 1 and a pressure gradient, C0, of 9.6(107) N m − 2. Curve B uses 0.127 m s − 1 as the critical pulling velocity and Eq. (9) to calculate the pressure gradient, C0. Curve C uses a pressure gradient of 9.6(107) N m − 2 and fiber pulling velocity, Vf =0.127 m s − 1. Curve D uses a pressure gradient, C0, calculated using Eq. (12), with a pulling velocity Vf = 0.127 m s − 1 and a pulling force of 9 N.
fiber tow pulling force. The validity of this model is tested using experimental data from processing runs to determine its predictive capabilities.
3. Results The physical consistency and accuracy of the PCA model is tested using the preceding development and the observed values of the pulling velocity, fiber tow pulling force and the melt pressure (Vf, Fpull and Pm, respectively). The actual values of the pressure gradient, C0, and of the critical pulling velocity have not been determined for the continuous fabrication process. The process parameters used in this analysis are listed in Table 1. These parameters correspond to a successful production run using aluminum oxide fiber with a molten aluminum melt [8].
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The velocity profile of the melt in the representative cell is plotted as a function of the radius in Fig. 3 for four different conditions that correspond to those observed in production runs. Curve A represents the calculated velocity profile for a pulling velocity of 0.0127 m s − 1 and a pressure gradient, C0, of 9.6(107) N m − 2. Unsuccessful production runs have been observed with a pulling velocity of 0.0127 m s − 1, as previously reported [9]. The majority of the area bounded by the zero velocity line and Curve A is in the positive velocity region, indicating a net mass flow in the positive z-direction into the orifice. A net mass flow rate into the entrance region results in the solid choking phenomenon, which is consistent with the unsuccessful run for the values of the input parameters. Curve B is determined using the PCA model with 0.127 m s − 1 as the critical pulling velocity in Eq. (9) to calculate the pressure gradient, C0, and again in Eq. (6) to calculate the velocity profile. This set of process parameters corresponds to conditions observed for a successful production run [8]. The velocity profile shown in Curve B has equal areas above and below the zero velocity line and hence, yields zero net mass flow. If the critical pulling velocity equals 0.127 m s − 1, the model predictions are consistent with experimental observation and results in continuous production. The velocity profile summarized by Curve C is calculated using a pressure gradient of 9.6(107) N m − 2 and fiber pulling velocity, Vf = 0.127 m s − 1. These parameter values do not limit the pulling velocity to its critical value as does the use of Eq. (8). This curve illustrates a net mass flow in the negative z-direction, towards the melt and would correspond to operating conditions for a successful run. Curve D corresponds to the velocity profile predicted with a pressure gradient, C0, calculated using Eq. (12), a pulling velocity Vf = 0.127 m s − 1 and a pulling force of 9 N. (This pulling force was chosen because it corresponds to the experimentally observed value for a successful run with a pulling velocity of 0.127 m s − 1.) The calculated velocity profile indicated by Curve D would also reflect a successful run. Curve D lies below the zero velocity line and indicates a mass flow rate in the negative z-direction, into the melt region. This mass flow rate would not lead to the choking conditions that Curve A reflects. The above calculations demonstrate that the PCA model has the capability to predict the qualitative physical behavior that is observed in the MMC fabrication process. At a low pulling velocity, the melt will exit the pressure vessel and solidify leading to fiber tow breakage. At sufficiently high pulling velocities, the melt is held within the vessel leading to a successful production run. The calculated velocity profiles (Curves A–D) are based on either a characteristic pressure gradient, pulling velocity or pulling force estimate. The trends
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summarized in Fig. 3 allow prediction of appropriate processing parameters for successful fabrication of MMC wire. The results of the PCA model, the solid cylinder model and the experiment are compared in Table 2. Three different methods of comparing the measured parameters (melt pressure, pulling velocity and pulling force) with the model predictions are reported. The experimentally measured parameters are shown in bold in Table 2. The observed pulling velocity is not necessarily the critical pulling velocity used in the model. In the first method, the pressure gradient is calculated assuming that the pressure varies linearly along the melt length, C0 = (Pmelt −P )/lp and the critical pulling velocity and pulling force are calculated parameters. In the second method, the pressure gradient is calculated using Eq. (9), and the observed velocity is used as the critical pulling velocity. The critical pulling force is the calculated parameter. In the third method the pressure gradient is calculated using Eq. (12) and the measured pulling force is an input variable. The critical pulling velocity is the calculated parameter. Results in Table 2 demonstrate that the PCA model is much more consistent, by at least an order of magnitude, than the solid cylinder model. Using methods 2 and 3 with the PCA model, the calculated critical pulling velocity and the pulling force are within 1% of their observed values. The calculations using method 1 are not in good agreement. However, the observed pressure gradient, Co, is estimated using the maximum possible length of the melt and may not be an accurate value for this parameter. The behavior predicted using methods 2 and 3 is consistent with Curve C in Fig. 3 and suggests that the measured pulling velocity is near its critical value.
4. Discussion Though the PCA model does not provide perfect agreement with the experimental results, its predictions
show much better quantitative agreement than the solid cylinder model. The PCA model provides much more consistency with the experimental results than does the solid cylinder model. As shown in Table 2, calculating the pulling velocity using the measured value of the pulling force or the pulling force using the measured value of pulling velocity results in predictions of the other observed parameters that are within 1%. This type of agreement is not observed for the solid cylinder model, which predicts values that are orders of magnitude different than the experimental measurements. The consistency of the PCA model and the experimental results provides strong evidence that the physical mechanisms used to develop it are correct and that the lack of better agreement with the experiment is due to other factors related to the specification of the input variables. For example, the calculated pressure gradient for methods 2 and 3 is an order of magnitude larger than that estimated in method 1. This discrepancy suggests that the pressure gradient is present in a short entrance region near the entrance orifice, not over the entire melt length as assumed in method 1. The calculations performed using methods 2 and 3 indicate that this entrance region is on the order of 0.004 m, not the 0.05 m length used in the method 1 calculation. There are also other factors that could explain the observed discrepancies in Table 2. These factors could explain the small, approximately 1% error observed in method 3 for the calculated critical pulling velocity. This calculated pulling velocity is greater than that observed by less than 1%. However, physically, this calculation would suggest that the observed successful run would not have occurred. Using the PCA model to predict operating conditions would have resulted in a conservative estimate of the pulling velocity and would have led to a successful production run. Other factors include variation in the number of fibers in the tow, the increase surface area of the individual fibers due to their surface roughness and its contribution to viscous drag, the variations in the packing density of the fiber tow as
Table 2 Comparison of predictions from the PCA and solid cylinder models with experimental resultsa Method 1 Pmelt−P C0 = lm
Method 2
Parallel, cylindrical array model C0 (Pa m−1) Vf,crit (m s−1) Fpull (N)
48 000 000 −0.0097 −0.67
631 903 420 −0.127 −8.916.24
638 016 631 −0.128 −9.0
Solid cylinder model C0 (Pa m−1) Vf,crit (m s−1) Fpull (N)
96 000 000 −21.789 −0.190
559 000 −0.127 −0.001
4 540 000 000 −1032.000 −9.000
a
Values in bold face represent experimental data.
C0 from Vf,crit (Eq. (9))
Method 3 C0 from Fpull (Eq. (12))
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it is being infiltrated, errors caused by the representative cell technique for calculating the shear stress and additional system level friction forces that are included in the experimentally measured pulling force. However, the close agreement between the calculations and the measured values support the assumption that these factors are second-order effects.
Acknowledgements
5. Conclusion
References
The predictions of the critical pulling velocity, pulling force and pressure gradient of the PCA model are compared with those of the solid cylinder model and with experimental results. This comparison provides strong evidence that the PCA model is a more accurate representation than the solid cylinder model and is in agreement with the observed behavior of the continuous metal matrix composite wire production. This evidence suggests that the physical mechanism responsible for the molten metal remaining in the crucible is a balance of the fluid drag forces on the infiltrated fiber tow and the pressure driven flow towards the inlet. The region of the melt over which the pressure driven flow occurs, is estimated to be an order of magnitude less than the melt length. The described PCA model provides a working design tool for setting the operating parameters in this manufacturing process.
[1] J.T. Blucher, US Patent Number 5736199, 7 April 1998. [2] S.L. Sampson, MS Thesis, Northeastern University, Boston, MA, September 1997. [3] J.A. Cornie, Y.-M. Chiang, D.R. Uhlman, A. Mortensen, J.M. Collins, Am. Ceram. Soc. Bull. 65 (1986) 293 – 304. [4] K.K. Chawla, Composite Materials, Springer, New York, 1987. [5] R.B. Bhagat, Metal Matrix Composites, vol. 1, Academic Press, Orlando, FL, 1990, pp. 43 – 82. [6] A. Mortensen, I. Jin, Int. Mater. Rev. 37 (3) (1992) 101– 128. [7] B. Ralph, H.C. Yuen, W.B. Lee, J. Mater. Proc. Tech. 63 (1997) 339 – 353. [8] J.H. Nadler, MS Thesis, Northeastern University, Boston, MA, June 1998. [9] M.G. Williams, J.A. Isaacs, J.H. Nadler, S.L. Sampson, G.J. Kowalski, J.T. Blucher, Analytically motivated improvements in continuous MMC wire fabrication, Mater. Sci. Eng., 1998. [10] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1960.
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The authors would like to thank the Department of Mechanical, Industrial and Manufacturing Engineering of Northeastern University for support, and our colleague Dr Joseph T. Blucher for manufacturing the MMC wire samples.