Mold-filling simulations for the injection molding of continuous fiber-reinforced polymer

Mold-Filling Simulations for the Injection Molding of Continuous Fiber-Reinforced Polymer

A. W. CHAN and S.-T. HWANG

Department of Chemical and Nuclear Engineering University of Cincinnati Cincinnati, Ohio 45221

Injection molding can be used to fabricate fiber-reinforced polymer composites by impregnating a continuous fabric mat preplaced in a mold cavity with a polymer resin. The mold-filling time is dependent on the flow and heat transfer behavior in the mold. A model is proposed that considers the non-Newtonian flow through the porous fabric mat and the heat transfer between mold, fabric mat, and flowing fluid. The model was simulated for the mold filling of a carbon fiber mat with a pseudoplastic polymer solution. The results from the simulation provide infor- mation for optimizing mold-filling parameters through proper selection of inlet fluid pressure, heat source temperature, and type of polymer-solvent system.

INTRODUCTION

iber-reinforced polymer composites are find- F ing wider usage in the aerospace industry and in commercial applications. A new breed of these high strength polymer composites uses carbon fibers as the reinforcing element. Con- tinuous carbon fiber reinforced composites are ideal for applications requiring higher temperatures, lower density, high modulus, good fa- tigue resistance, and chemical inertness. The injection-molding process is a n available technique for processing continuous fiber-reinforced polymer composites. An understanding of the role of fluid flow and heat transfer on mold filling is necessary in designing the mold and in optimizing the operating conditions for injection molding.

A substantial amount of theoretical and experimental work aimed at investigating the injection-molding process were reported in the literature. Recent studies in mold-filling simulations for thermoplastic melts were reported by Hieber and Shen (1) and by Van Wijngaar- den, et al. (2). Kamal and Lafleur (3) proposed a model that covers the mold filling, packing, and cooling stages in the injection molding of thermoplastic resins. The numerical modeling of mold filling with reactive thermoset resins was reported by Kamal and Ryan (4). Numerical simulations for the reaction injection-molding (RIM) process include the work of Domine and Gogos (5), Manzione (6). and Castro and Ma- cosko (7). A model for mold filling with ther-

moset and thermoplastic resins was reported by Frutiger (8).

Since the injection-molding process can be used to fabricate continuous fiber-reinforced polymer composites, it is of interest to investi- gate the effect of the operating parameters on the process. An experimental study on the isothermal filling of an aligned fiber bed with various fluids was reported by Williams, et al. (9). Gonzales and Macosko (10) reported their work on the filling of a mold packed with fiberglass mat by a nonreacting liquid, with heat transfer taking place between the fiberglass mat and the flowing liquid. The objective of this work is to develop a simple model for simulating the mold-filling step in the injection molding of continuous carbon fiber reinforced polymer. The coupled fluid flow and heat transfer phe- nomena during mold filling and the resulting effect on the molding performance are explored by studying the results of the computer simulation.

MATHEMATICAL MODEL Mold Arrangement

The mold configuration consists of a rectan- gular mold cavity packed with continuous reinforcing fibers in the form of a fabric mat. The fabric mat is covered by two mold plates heated at both top and bottom ends. A section view of the mold arrangement is shown in Fig. 1 . The mold cavity dimensions are: length, L; width, W and thickness, 2H. The coordinate axes are

POLYMER ENGINEERING AND SCIENCE, MID-MARCH, 1988, Vol. 28, NO. 5 333

A. W. C h a n a n d S.-T. H w a n g

shown in Fig. 1 with z = 0 being at the center of the mold cavity section. A polymer solution at constant temperature, T,,, is uniformly in- jected at one end of the mold (x = 0). At the start of mold filling, the mold plate temperature is T,, and the fabric is at the same temperature as the inlet fluid (i.e., Tro). During mold filling, heat transfer takes place between the mold plates, fabric mat, and the flowing polymer solution.

Assumptions The assumptions used in defining the model

are: 1.

2.

3.

4.

5.

6.

7.

8.

The polymer solution is in incompressible, laminar flow. There is no interaction between the polymer solution, fabric mat, and mold plates: hence, surface effects are neglected. Heat effect due to reaction is not included in the mold-filling model. The extent of reaction is small during the mold-filling stage. Heat generated from viscous dissipation is not considered in the model. The Brink- man number is assumed to be much smaller than the Graetz number (i.e., solution viscosity is much lower than melt viscosity). The mold cavity dimensions are large relative to fiber diameter and average pore size. One dimension of the cavity is small relative to the other two (i.e., a thin mold): hence, a two-dimensional model is appli- cable. The average pore size is large relative to molecular dimensions. The fabric mat is not shifted or deformed by the flowing polymer solution. Average properties of the fabric pore structure are used in the model.

Other assumptions are stated in subsequent sections as required.

Characterization of Porous Fabric Mat The fabric mat constitutes a porous medium

with a solid fiber phase and voids region. The voids region is formed from interconnecting pore spaces that constitute the effective flow channels of the fabric mat. A simple model that relates the permeability of the fabric mat to its pore structure is given by the representation of the fabric mat as a n equivalent system of

HEAT SOURCE

M O L D PLATE

MOLD C A V I T Y

Fig. 1. Mold cross section.

straight, parallel capillaries with constant diameter (1 1, 12). The capillary radius is related to the hydraulic radius defined by

cross-section area for flow wetted perimeter

volume available for flow total wetted surface

volume of voids/volume - of porous medium

wetted surface/volume of porous medium

Rh =

- - (11

-

The capillary radius is twice the hydraulic radius:

R, = 2 Rh. (2)

Polymer Solution Flow Through Capillary The flow of the polymer solution through the

fabric mat is represented by the viscometric flow of a pseudoplastic fluid through a system of capillaries with uniform diameter. The continuity, momentum, constitutive, and energy equations form a system of coupled partial dif- ferential equations. A simplification can be made when these equations are solved using the finite difference method. By keeping the grid sizes sufficiently small, the flow through a short section of a capillary is approximately isothermal. The continuity and momentum equations for flow through a capillary are solved sepa- rately from the energy equation (which is integrated over the mold cavity). The unsteady state terms in the continuity and momentum equations are neglected because of the compara- tively long duration of mold filling, The continuity and momentum equations in cylindrical coordinates for isothermal, laminar, and developed flow are:

d( r Vr) dr

-- - 0 (3)

(4) 1 a(r7,J _ - ap

dX r dr ’

Equation 3 is integrated with the no slip boundary condition V, = 0 at r = R, to yield the result V, = 0 for all r.

The power-law model is used to characterize the polymer solution viscosity. Since the only nonzero velocity gradient is d Vxldr, the power- law constitutive equation for the polymer solution becomes

334 POLYMER ENGINEERING AND SCIENCE, MID-MARCH, 1988, VOI. 28, NO. 5


The power-law consistency K is a function of temperature that follows an Arrhenius relation:

(6) K = m exp (AEIRT).

The velocity profile is obtained by integrating E q 4 with Eq 5. The average velocity obtained from this velocity profile is given by

where b = l/n.

Energy Equation

mat in the filled portion of the mold cavity is The energy equation for the fluid and fabric

In Equat ion 8, the density, heat capacity, and thermal conductivity were assumed to be constant. Axial conduction along fabric and fluid is neglected since the axial grid spacing used in the finite difference solution is much larger than the transverse grid spacing.

At a specified time and position in the mold cavity, the fluid temperature is approximately equal to the fabric temperature. Therefore, the new time values of Ts and TJare taken as being equal. This simplification is possible since carbon fibers have higher thermal conductivity and very small diameters. The higher thermal conductivity and large contact surface between fluid and fabric resulted in very rapid temperature equalization between the two phases. With this simplification, the heat transfer between fabric and fluid need not be considered.

The energy equation for the mold plate is

Axial conduction is neglected for the same reason as stated in the case of E q 8. The boundary conditions for E q s 8 and 9 are

(dT,/az) = 0 at z = 0 (10)

(dTf/az) = 0 at z = 0 (11)

The temperature and heat flow at the mold cavity to mold plate interface are assumed to be continuous.

TSIH = TfIH = TrnI, (12)

The heat source at the ends of the mold plates (i.e., at y = H + H m and y = H - H m ) is assumed as being maintained at a constant temperature.

NUMERICAL SIMULATION An explicit finite difference technique is used

to solve the system of equations consisting of E q s 7, 8, and 9. The solution generated is anal- ogous to the viewing of a series of photographs of the mold cross section taken at each time interval used in computation. Although an an- alytical solution to Eq 9 can be obtained, a numerical solution is used. The grid sizes can be varied over the region of computation so that smaller mesh sizes are used in regions where the temperature variations are large. The properties of the polymer solution, fabric mat, and mold are listed in Table 1.

Algorithm

briefly summarized below: The algorithm for the numerical solution is

1. The initial fluid velocity is calculated from E q 7 and the initial condition.

2. The new position of the front is calculated. 3. The new temperature distribution is cal-

culated from E q s 8 and 9 together with the initial and boundary conditions ( E q s 10 through 13).

4. The new pressure distribution in the filled portion of the mold cavity is computed by numerically integrating E q 7 using new time values of b and K, and previous time value of the pore velocity. The use of previous time velocity removes the require- ment for iteration. This is possible because the time increment is chosen very small and the computed velocity is very close to the previous time value.

5. The new velocity distribution is then calculated from E q 7 using new time values of the pressure gradient and power-law constants b and K . The average of the new and previous time velocity at each grid is then taken as the new time velocity.

Table 1. Mold-filling Parameters.

Polymer Solution Density (kg/m3) Heat capacity (J/kg-K) Thermal conductivity (w/m-k) Viscosity Parameters

n AE 0.12 14.7 kJ/mol

943 800

0.2

m 0.22 Pa.s

355-380 0.25 87.4 4.33 x 10- Fabric Mat Diameter (pm) 7 Density (kg/m3) 1,760 Heat capacity (J/kg-K) 1,000 Thermal conductivity (w/m-K) 0.86 Specific surface (m'/m3 fabric) Porosity 0.6 Mold Cavity Length (m) 0.25 Width (m) 0.05 Thickness (mm) 3.4 Mold Plate Thickness (mm) 12 Heat capacity (J/kg-K) 963 Thermal conductivity (w/m-K) 180

1 x 105

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A. W. Chan and S.-T. Hwang

6. The time increment is added to the previous time value and steps 2 through 6 are repeated until the mold filling is complete or the allotted mold-filling time is ex- ceeded.

Polymer Resin The model was simulated for a solution proc-

essable polyimide resin. The resin reaction is thermally activated, and it starts at about 380 K. Figure 2 is a plot of the resin viscosity to temperature behavior. Data for the plot were obtained from torsion impregnated cloth analy- sis (TICA) at an oscillation frequency of 10 rad/ s. Both the in-phase and out-of-phase viscosity components decrease from 300 to 380 K. From 380 to 4 0 0 K , both viscosity components increase slightly, indicating the presence of a reaction. The extent of this reaction is small below 400 K. For mold-filling duration of about 1 min and with mold temperatures below 400 K, the heat effect from the reaction is not signifi- cant and is therefore not included in the model. The heat of reaction is important during the curing stage.

Effects of Pressure, Porosity, and Viscosity on Fill Time

The results of the simulation for a few rep- resentative cases studied are shown in Figs . 3 through 8. Figures 3 through 5 summarize the effect of pressure, porosity, and viscosity on mold-filling time. In Fig. 3, the position of the flow front as a function of time are shown for three different inlet pressures (constant throughout the duration of mold filling). When the pressure gradient is sufficient, the mold is filled within a reasonable time (30 to 60 s). As the inlet pressure decreases, the flow front curve rapidly changes its curvature near the end of mold filling, which results in a rapidly increasing mold-filling time. When the inlet pressure is not sufficient to maintain the pressure gradient, the flow front curve quickly at-

TEMPERATURE <K)

Fig. 2. Viscosity-temperature plot f o r polymer solution (a, b = in-phase. out-of-phase component of viscosity. respectively).

Fig. 3. Position offluid front during mold filling lfor dij- ferent PJ (Tf, = 300 K; Tm, = 380 K; porosity = 0.6; Po = 500, 400. 300 kPafor a. b, c , respectively).

Fig. 4. Position offluid front during mold filling lfor d i f ferentfabric porosities) (T,o = 300 K ; Po = 400 kPa; Tm, = 380 K ; porosity = 0.65. 0.6, 0.55, 0.5for a. b, c , d . respectively).

8 I8 a 38 I I d

UOLO FILLING TIME <d Fig. 5. Pressure required f o r mold filling lfor dwerent polymer solutions) (T,” = 300 K ; Tm, = 380 K; porosity = 0.6).

tains a nearly horizontal shape and mold filling stops. The flow front “freezes” and stops ad- vancing before the mold is filled. This behavior is related to the highly pseudoplastic nature of

336 POLYMER ENGINEERING AND SCIENCE, MID-MARCH, 1988, YO/. 28, NO. 5


the polymer solution. As the mold is being filled, the fluid velocity decreases rapidly. This causes a rapid increase in the fluid viscosity because of the drop in fluid shear rate. For insufficient inlet pressures, the result is a “short shot”-the flow stops before mold filling is complete.

The effect of varying fabric mat porosities on mold-filling times is shown in Fig. 4 . For fabric mat porosities around 0.6, the mold-filling time is not greatly affected by changes in porosities. When the fabric mat porosity drops below 0.5, the flow front again “freezes” before the completion of mold filling. This conclusion holds when the inlet pressure is sufficiently high for mold filling at a porosity of 0.6.

The next parameter studied is the effect of varying polymer solution viscosities on the mold-filling time at different inlet pressures. The results are shown in Fig. 5. The polymer solution viscosity can be varied by changing the solvent, the amount of solvent, or the polyimide resin. Curve a in Fig. 5 refers to the reference polyimide solution with power-law constants as given in Table 1. Curves b, c, and d refer to polymer solutions with flow activation energy similar to that of curve a, but with increasingly larger values of the power-law constants rn and n as given in Table 2. The results show the heavy dependence of required inlet pressure on the power-law index n. Comparing curves b and c, a n increase of 50% in the power-law index R increases the required inlet pressure by more than 100%. whereas a n increase of 50% in rn increases the required inlet pressure by less than 50% over the reference value in curve a. By choosing the solvent and starting polyimide resin, the mold-filling time and required inlet pressures can be optimized.

The polymer solution viscosity is inherently coupled to the pressure gradient (affecting the fluid shear rate) and the fluid temperature. The effect of these two parameters on mold filling is considered below.

Temperature and Pressure Distribution The temperature profiles at 10% of mold inlet

to mold end distance at different stages of mold filling are shown in Fig. 6. A larger temperature variation occurs at the region near the mold wall surface, while at the mold cavity center, the temperature profile is relatively flat. The fluid temperature profiles quickly reach a nearly asymptotic shape as shown by curves b, c, and d. This behavior is due to the high thermal conductivity of carbon fibers; thus, heat is

Table 2. Viscosity Parameters for Fig. 5.

T, 3 300-355 K T, = 355-380 K

Cunre m n m n

a 0.22 Pa.s 0.12 4.33 x 10-”Pa.s 0.25 b 0.33 0.12 6.50 X lo-’’ 0.25 c 0.33 0.18 6.50 X lo-‘’ 0.38 d 0.44 0.18 8.66 X lo-’’ 0.38

Fig. 6. Temperature profiles at 10% .f mold length [TJ0 = 300 K; Po = 500 kPa; Tm, = 380 K ; porosity = 0.6; curve a, b, c, d : J o w front at 20%. 40%. 70%. 100% of mold length, respectively; Z”: dimensionless mold center to mold wall distance).

conducted faster from the heat source to the flowing fluid. Because of fast temperature rise in the fluid, it is of interest to maintain the mold plates at the optimum temperature where the polymer solution viscosity is minimum. Based on the premise of an isothermal heat source, the mold plate temperature variation is very small. This near isothermal behavior of the mold plate is due to its high thermal conductivity and large volume compared with the mold cavity (i.e., the mold plates are much thicker and, therefore, possess a higher total heat capacity than the mold cavity contents).

When the inlet pressure is insufficient for proper mold filling, the flow rapidly slows down. This causes a faster temperature rise in the polymer solution since less fluid at lower temperature enters the mold cavity. Hence, the temperature profiles in this case would shift to the right (i.e., toward curve d). The fluid quickly attains the temperature of the mold walls. The increase in viscosity caused by a rapid drop in fluid shear rate is greater than the decrease in viscosity caused by a temperature rise in the fluid. Hence, the net effect of insufficient inlet pressure is a “freezing” of the flow.

The pressure profiles at the completion of mold filling are shown in Fig. 7. The profiles show a rapid drop in pressure near the mold inlet followed by a nearly linear pressure drop beyond 30% of the mold length. The rapid pressure drop at the mold inlet is the result of a higher temperature gradient at this section. The large temperature variation causes a similarly large variation in fluid viscosity. Therefore, the larger variation in pressure near the inlet is required to maintain a uniform velocity (necessary for incompressible flow) in each capillary. Beyond 30% of the mold length, the temperature variation is very small, resulting in a near linear pressure profile. The pressure profile for a typ- ical case at different stages of mold filling is shown in Fig. 8. The pressure profile at an early

POLYMER ENGINEERING AND SCIENCE, MID-MARCH, 1988, Vol. 28, No. 5 337

A. W. Chan and S.-T. Hwang

CURRENT AND PROPOSED WORK The model is being modified at present to

include polymer resins that undergo a series of reactions and resins that undergo fast reaction under mold-filling conditions. The polyimide resins typically used for fabricating high performance polymer composites may undergo both physical and chemical changes during processing. A better understanding of these resins is required for forming more reliable and flexible kinetic and rheological models from measured data. Experimental measurements on these resins are currently being proposed to obtain kinetic and rheological data under conditions encountered in processing. More exten- sive models then can be developed for more accurate predictions for both mold-filling and curing stages.

ACKNOWLEDGMENTS This paper is based on work supported by the

General Electric Co. The authors are especially grateful to Dr. Howard Price and Dr. Shesh Srivatsa for their helpful suggestions and nu- merous instances of assistance during the course of this study.

n 0 0 t a W

3 u) In W

a a

X MOLD LENGTH

Fig. 7 . Pressure profile at end of mold filling (T, = 300 K; T m , = 380 K; porosity = 0.6; curve a, b, c: Po at 300. 400, 500 kPa. respectively).

X HOLD LENGTH

Fig. 8. Pressure profile at various s tages of mold filling [TJ0 = 300 K; Po = 500 kPa; Tm, = 380 K; porosity = 0.6; curve a. b, c:flow front at 35%, 70%, 100% of moZd length, respectively).

stage in mold filling is nearly linear, and it gradually develops into the profile observed in Fig. 7 at subsequent stages in mold filling.

Model Modifications

The model can be extended for application in cases where the polymer resin is much more reactive as in reaction injection molding (RIM) resins. The heat effect from reaction cannot be neglected in this case. The energy equation is modified by including a reaction term. An ad- ditional mass balance equation is required to calculate the extent of reaction. The polymer solution constitutive equation would also have to be modified to include the contribution from the chemical reaction of the resin. Finally, temperature-dependent properties may be required to describe the process accurately. These modifications would extend the model for use in cases of fast-reacting thermosets and short mold-filling times.

b =

AE = H = H , = k = K = L = m =

n = P = P = r = R = R, = Rh = s = t = T = v = (V> =

c, =

x = z = z* =

NOMENCLATURE inverse of power law index (1 /n). heat capacity. flow activation energy. half of mold cavity thickness. mold plate thickness. thermal conductivity. power-law consistency. length of mold cavity [or capillary). constant related to power-law consistency. power-law index. porosity of fabric mat. pressure. radial axis (for capillary). gas constant. radius of capillary. hydraulic radius. specific surface of fabric mat. time. temperature. velocity. average pore velocity. axial axis. transverse axis. dimensionless distance, measured from mold cavity centerline to mold plate/ mold cavity interface.

Subscripts f = fluid. m = mold plate. o = initial condition. r- = component in radial direction. S = fabric mat. x = component in x-direction.

338 POLYMER ENGINEERING AND SCIENCE, MID-MARCH, 1988, VO~. 28, NO. 5


Greek Letters 9 = non-Newtonian viscosity. f3 p = density. 7rx = shear stress (x-momentum in r-direc-

= angular axis in cylindrical coordinate.

tion).

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J. Non-Newt. Fluid Mech.. 11, 175 (1982). 3. M. R. Kamal a n d P. G. Lafleur, Polyrn. Eng. Sci.. 22,

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6. 7.

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M. R. Kamal and M. E. Ryan, Polyrn. Eng. Sci.. 20,859 (1 980). J. D. Domine and C. G. Gogos, Polym. Eng. Sci., 20, 847 (1 980). L. T. Manzione, Polym. Eng. Sci., 21. 1234 (1981). J. M. Castro and C. W. Macosko. AIChE J.. 28. 250 (1982). R. L. Frutiger, Polym. Eng. Sci., 26. 243 (1986). J. G. Williams, C. E. M. Morris, and B. C. Ennis, PoIym. Eng. Sci.. 14, 413 (1974). V. M. Gonzales and C. W. Macosko, Proc. Second In- ternational Conference on Reactive Processing of Pol- ymers, 384 (1982). A. E. Scheidegger, “The Physics of Flow Through Po- rous Media,” Ch. 6. University of Toronto Press, To- ronto (1 974). P. C. Carman. Trans. of Inst. of Chemical Engineers (London), 14, 150 (1937).

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Mold-filling simulations for the injection molding of continuous fiber-reinforced polymer