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Fluid mechanics analysis of a two-dimensional pultrusion die inlet.

INTRODUCTION

Composite materials have made a significant impact in the field of materials science by offering an alternative to conventional materials. They are increasingly replacing conventional materials in the automotive, aerospace, construction, appliance, and medical industries. Some of the applications of composite materials include hydraulic and mechanical components such as brakes of aircrafts, helicopter rotor blades, antennas of telecommunication satellites, shafts, nozzles and ducts of engines, and medical devices such as X-ray equipment and prosthetics.

Pultrusion is a cost-effective process used primarily for manufacturing constant cross-section fiber reinforced composite materials which has gained in significance over the past few years (1). It is classified as a "wet" process as the fibers are impregnated with resin immediately before the resin cures. A schematic representation of the pultrusion process is shown in Fig. 1. A typical pultrusion machine (2) consists of fiber/resin creels, a resin bath, a shape preformer plate, a heated die, pullers, and a cut-off saw. The process consists of pulling fiber reinforcements continuously through a resin bath in the resin impregnating tank. The fibers are pulled by a hydraulically operated puller through a resin control unit to remove any excess resin, then through a shaping and forming guide system in the form of a shape preformer, and finally, through a heated die, where the product is heated and cured. The thermodynamic activity inside the pultrusion die bonds the resin and fibers into a product with the same cross-sectional profile as the die. Desired lengths of the composite are cut using the cut-off saw.

Some of the advantages of this process (3) over the other composite fabricating processes are that it is a continuous process with a high productivity and a low cost of labor. Also it is a versatile process that can handle products with complicated but constant cross-sections. The final product of a pultrusion process usually does not require further finishing.

Although the composite industry has accepted pultrusion as one of the primary composite manufacturing technologies, a lack of fundamental process understanding hinders pultrusion from entering more markets (4). Relatively little work has been done towards understanding pultrusion manufacturing as compared to some of the other methods of manufacturing composite materials. A fundamental study of the pultrusion process is important for developing a proper understanding of the process control parameters that influence it. Some of the important control parameters in a pultrusion process are fiber volume fraction, type of fiber reinforcement, resin system composition, resin viscosity, reaction kinetics of resin, die temperature profile, die shape and length, and pull speed.

The study of the fluid mechanics problem associated with the pultrusion process can be important in regards to product quality. Fluid mechanic analysis of the die inlet region of the pultrusion machine involves the determination of the velocity field and perhaps even more important, the pressure field in the die inlet. As compared to the rest of the die, the tapered die inlet region experiences a significant resin pressure rise during the pultrusion process. The magnitude of the resin pressure rise in the tapered portion can have a significant effect on the quality of product fabricated. This tapered region of the pultrusion die is about 6 mm in length, but its significance is far from trivial. The resin and fibers are squeezed into this portion of the die after exiting the preform plate resulting in the backflow of the excess resin entering the die and a corresponding significant pressure rise. Because there is a backflow of the resin in the tapered die inlet region, this area is referred to as the "resin backflow" region.

The fluid resin pressure rise in a pultrusion die should be sufficiently high. Good surface finish is made easier whenever resin pressure is large enough to suppress void formation. A large pressure rise is also important to achieve a good "wet out" (5); thus, a sufficiently high pressure rise in the pultrusion die is desirable. A mathematical model to predict the resin pressure rise as a function of the pultrusion die geometry is important as it can be utilized to design the contour of the pultrusion die inlet and obtain a quality product.

Limited work has been conducted to model the pressure rise in the tapered region of the pultrusion die. Quan (4) developed an asymptotic solution and presented simplistic results for the pultrusion process. He solved the coupled mass, linear momentum, energy and species equations. However, the pressure rise in the tapered inlet region was not dealt with in much detail. The tapered entrance model for a pultrusion die was probably attempted for the first time by Batch (6). His one-dimensional model was based on Darcy's law for porous media flow. He assumed that the flow in the direction transverse to the processing direction could be neglected. Batch presented some results for different die inlet contours. Gorthala (7) then developed a model for the tapered entrance region based on the hydrodynamic theory for lubrication. This was followed by Meyyappan (8), who attempted a two-dimensional model. His model was also based on the hydrodynamic theory for lubrication. Both Gorthala and Meyyappan had to assume a viscous annulus region thickness for the entrance in order to model the flow in the tapered region. Neither of the above mentioned researchers presented any extensive results for variation of pressure rise in the tapered entrance region as a function of the variation of the numerous pultrusion process variables listed above.

The present study provides a more detailed study of the phenomena of fluid resin pressure rise in the pultrusion die by developing a two-dimensional finite element model based on Darcy's law for porous media (9). As the resin and fiber system have been assumed to act like a flow through a porous media, Darcy's law has been found to describe the flow behavior in the backflow region with good accuracy as compared to the hydrodynamic theory used by both Meyyappan and Gorthala. The two-dimensional model also predicts a more realistic velocity profile then the unrealistic profile predicted by Batch's one-dimensional model.

STATEMENT OF PROBLEM

The finite element model developed in this study was employed to model the tapered inlet region of a pultrusion die. A two-dimensional model based on Darcy's law for porous media flow was utilized by assuming that the fiber/resin matrix being pulled through the die can be best approximated as a flow through a porous media. The schematic of the spatial computational domain is shown in Fig. 2. A Cartesian coordinate system based geometry was used in this finite element model. The inlet of the computational domain for this model is downstream of the preform plate section at an x-distance of 2L upstream from the point where the fiber/resin matrix contacts the tapered die inlet (intersection point). The x-distance between the intersection point and the beginning of the straight die region is L. Thus a total x-distance of 3L before the start of the straight portion of the die is included in the computational domain. The computational domain stretches downstream to the point where the resin cures and the liquid resin pressure rise decreases back to zero. This location where the pressure rise decreases back to zero is defined as the gelation location in this study. As a result of the curing/solidification of the liquid resin, its viscosity rises rapidly to high values resulting in a quick drop in liquid resin pressure. The combined overall length of the spatial computational domain for this finite element model is 67L as shown in Fig. 2. The fiber/resin matrix is compressed in the region between the intersection point and the straight portion of the pultrusion die accounting for the squeezing out of the liquid resin into the resin backflow region and resulting in a large pressure rise. In this region the thickness h(x) (distance from centerline to die wall) of the fiber/resin matrix is a function of x-distance. For the present study, the fiber/resin matrix in the straight portion of the die was set at a thickness ([h.sub.s]) of 3.18 mm (1/8[inches]). The thickness of the composite matrix at the preform plate exit ([h.sub.pp]) was nominally kept at 1.2 times [h.sub.s], but this thickness (preform plate size) can be varied to change the pressure rise in the die. For this study, a two-dimensional wedge shaped die inlet geometry is considered. The wedge as shown in Fig. 2 makes an angle [Alpha] with the horizontal axis. Other die inlet taper geometries such as circular and parabolic can also be modeled.

Since the inlet portion of the die is very short in length (and in this case is water cooled) the entire tapered inlet region of the pultrusion die can be assumed to have a constant viscosity. However, the spatial viscosity variation in the straight portion of the pultrusion die is predicted using a temperature and degree of cure dependent viscosity model obtained from the work of Gorthala (7). The values of temperature and degree of cure for different pull speeds to be substituted in the viscosity model were obtained from the thermal model of Chachad (10). Darcy momentum equations that yield the velocities in the x and y directions were substituted into the continuity equation to obtain a single equation with a single variable, namely pressure. This equation was then solved by a Galerkin's weighted residual based finite element method to obtain the pressure field. The velocity field was then obtained from the pressure field via Darcy's equations.

ANALYSIS

The composite matrix system being pultruded through the die acts like a flow through a porous media. This allows the use of Darcy's law to develop the governing equations (4, 9, 11) given as:

momentum equation in x-direction

u = U - [K.sub.11]/[Mu][Phi] [Delta]P/[Delta]x (1)

momentum equation in y-direction

v = - [K.sub.22]/[Mu][Phi] [Delta]P/[Delta]y (2)

continuity equation

[Delta]([Phi]u)/[Delta]x + [Delta]([Phi]v)/[Delta]y = 0 (3)

where P is the pressure, u is the resin velocity in the x-direction, v is the resin velocity in the y-direction, [K.sub.11] is the permeability in the x-direction, [K.sub.22] is the permeability in the y-direction, [Phi] is the porosity, [Mu] is the viscosity, and U is the pull speed of the pultrusion process (12). These governing equations can be mathematically simplified by substituting Eqs 1 and 2 into Eq 3 to obtain a single equation,

[Mathematical Expression Omitted] (4)

The porosity [Phi] of the fiber/resin matrix is considered to be a constant in the straight portion of the pultrusion die but varies in proportion to the tapered cross section of the inlet die region. It was assumed that the fiber volume fraction [V.sub.f](x) increases from a value of 0.52 (for [h.sub.pp] = 1.2 [h.sub.s]) at the computational domain inlet to 0.62 at the start of the straight portion of the pultrusion die where 0.62 is the fiber volume fraction of the finished product. The porosity of the composite matrix being pultruded can be expressed in terms of fiber volume as

[Phi](x) = (1 - [V.sub.f](x)) (5)

The value of the fiber volume in the all-liquid resin backflow region of Fig. 2 was taken to be 0.001. A value lower than this resulted in computational problems, but such a low value of fiber volume provided a good approximation of this all-liquid region.

The permeability of the fiber/resin system is dependent on the fiber/resin volume fraction and the fiber diameter of the composite. The permeability is calculated using the Kozeny-Carman equation (11), which is given as

[Mathematical Expression Omitted] (6)

where [D.sub.f] is the fiber diameter and C represents the Kozeny constant which accounts for pore nonuniformity (4) and is basically dependent on the structure of fiber collection in the fiber/resin system. The fiber diameter [D.sub.f] was selected to be 20 microns corresponding to E glass. A value of 1.4 was chosen for the Kozeny constant in this study (12). The permeability in the x-direction was assumed to be same as that in the y-direction ([K.sub.11] = [K.sub.22]) (13). According to Lundstrom and Gebart (13), the ratio [K.sub.22]/[K.sub.11] varies between 0.9 and 1.0 for the fiber volume fractions considered here. By varying this ratio between 0.9 and 1.0 the pressure results show a change of less than 1% for different pultrusion processing conditions.

The viscosity of the liquid resin is assumed to be constant in the tapered die inlet region on account of its short length. However, in the straight portion of the pultrusion die a variable viscosity model is used which assumes the viscosity to be a function of temperature and degree of cure. This variable viscosity model (7) is given as

[Mu] = [[Mu].sub.[infinity]] exp(E/RT + k[Psi]) (7)

where T represents the temperature and [Psi] the degree of cure; the axial values of T and [Psi] along the die were obtained from the work of Chachad (10). The symbol R represents the universal gas constant. The empirical constants corresponding to the resin system are E = 3.76 x [10.sup.4] J/mol, k = 20.0. The values used in this study are typical of resin systems such as EPON 9310 and EPON 9420. A value of [[Mu].sub.[infinity]] = 3.85 x [10.sup.-7] kg/m s yielding a value of [Mu] = 1.5 kg/m s at standard temperature and pressure (STP) was selected for this study.

The following boundary conditions are applied to the governing equations:

P = [P.sub.0] at x = 0 (8a)

tan [Alpha] = -v/u at y = h(x)(8b)

v = 0 at y = 0 (8c)

u = U at x = 67L (8d)

Here [P.sub.0] indicates atmospheric pressure at the die inlet, 67L is the length of the die region being modeled and [Alpha] is the angle of taper of the die inlet wan [ILLUSTRATION FOR FIGURE 2 OMITTED]. In order for the speed of the resin to be equal to the pull speed, the die is modeled up to the gelation location where the resin cures and bonds with the fibers resulting in the pressure rise falling back to zero. Since the die is assumed to be symmetrical about its centerline, only one half of it was modeled.

The boundary conditions must be applied in terms of pressure in order to be applicable to the modified governing equation (Eq 4). Thus, the velocity boundary conditions (Eqs 8b-8d) were transformed using Eqs 1 and 2 to be expressed in terms of pressure conditions. The first boundary condition is, of course, applicable since it is a direct boundary condition on pressure. Employing Eqs 1 and 2 we can transform Eq 8b as shown:

[Mathematical Expression Omitted] (9)

Equation 9 can be further modified to obtain a condition in terms of the pressure gradients as shown in Eq 10.

[K.sub.11]/[Mu] [Delta]P/[Delta]x sin [Alpha] + [K.sub.22]/[Mu] [Delta]P/[Delta]y cos [Alpha] = U[Phi] sin [Alpha] (10)

On the centerline the condition that the v component of velocity is zero can be translated using Eq 2 into the the condition [Delta]P/[Delta]y = 0. Equation 8d results in the pressure condition being [Delta]P/[Delta]x = 0. Thus, the transformed boundary conditions for the single equation (Eq 4) in terms of pressure are

P = [P.sub.0] at x = 0 (11a)

[K.sub.11]/[Mu] [Delta]P/[Delta]x sin [Alpha] + [K.sub.22]/[Mu] [Delta]P/[Delta]y cos [Alpha] = U[Phi] sin [Alpha] at y = h(x) (11b)

[Delta]P/[Delta]y = 0 at y = 0 (11c)

[Delta]P/[Delta]x = 0 at x = 67L (11d)

With the drastic increase in availability of powerful computing machines, the finite element method in one of its various forms has become a popular method for solving fluid flow problems. The finite element method can be used to model irregular and curved geometries with good accuracy. An added advantage of the finite element method is that there are a variety of ways in which to formulate the finite element equations. The finite element method involves mapping of the computational domain with an assemblage of discrete elements and then approximating the governing equations over each element (14). Discretization of the computational solution domain into appropriately shaped elements is the first step of a finite element solution. For the present study the computational domain was divided into 1250 eight-noded quadrilateral elements resulting in 3901 global nodal points. Figure 3 shows a typical finite element grid. A fine mesh was utilized to model the small tapered inlet region, with a sparse distribution of nodes in the straight portion of the pultrusion die. The second step of the finite element solution involves choosing the corresponding interpolation functions (15) and applying them over each element. If N is taken to represent the interpolation function for pressure P, then for 8 nodes per element, the approximate function for P is given as

[Mathematical Expression Omitted] (12)

The third step in the solution is to formulate the equations for each element; here the Galerkin's weighted residual technique (16) has been utilized to derive the finite element model. If w is taken to be the weighted function for pressure, then Eq 4 can be written for each element in the following form:

[Mathematical Expression Omitted] (13)

where [[Omega].sub.e] denotes the domain over which the equation is applied. Integrating Eq 13 by parts and then assembling the individual element equations the global matrix equations are obtained in the form

[K]{P} = {F} (14)

where [K] is the global stiffness matrix, {P} is the column vector of pressure variables, and {F} is the right hand side column vector. After applying the Dirchlet's boundary conditions the column vector of pressure variables is solved by inverting the global stiffness matrix. A banded matrix solution is used to minimize computation time.

RESULTS AND DISCUSSION

The fluid mechanics based finite element model developed in this study is used to predict the pressure rise in a pultrusion die generated from a change in the thickness of the fiber/resin matrix entering the tapered inlet die as a result of a variation in the preform plate size. It also predicts the pressure rise for a variation in the taper angle of the wedge shaped inlet die region. All the results in this section correspond to an E-glass/EPON 9420 resin matrix system. The fiber volume fraction [V.sub.f] is taken to be 0.62 along the straight portion of the die and, so also, for the finished product.

The preform plate section of the pultruder is positioned just upstream of the entrance to the die. The fibers coated with resin pass through the last preform plate before entering the tapered inlet region of the pultrusion die. Thus, the size of the last preform plate is equal to the thickness ([h.sub.pp]) of the fiber/resin matrix entering the computational domain inlet. This thickness is an important parameter that can influence the pressure rise in the pultrusion die. The pressure rise in the pultrusion die inlet as a result of preform plate size variation is shown in Fig. 4. The processing conditions for the different preform plate sizes were maintained at nominal values of U = 0.005 m/sec, [Mu] = 1.5 kg/m s, and [V.sub.f] = 0.62. The preform plate size ([h.sub.pp]) is varied as a factor of the fiber/resin matrix thickness in the straight die region ([h.sub.s]). Fig. 4 shows that variation in the preform plate size can have a strong impact on the magnitude of the pressure rise in the die inlet. As the size of the preform plate increases, the magnitude of the pressure rise in the die inlet increases. A preform plate size of [h.sub.pp] = 1.25 [h.sub.s] yields a high pressure rise while a size of [h.sub.pp] = 1.05 [h.sup.s] does not result in an appreciable pressure rise; here [h.sub.s] has been held constant at [h.sub.s] = 3.18 mm. As the size of the preform plate is increased, the thickness ([h.sub.pp]) of the fiber/resin matrix approaching the die inlet increases. This results in the intersection point of Fig. 2 being moved upstream. Thus, the compression of the fiber/resin matrix being pultruded is increased causing a higher pressure rise. Since the last preform plate is not a part of the die, its size can be modified without any structural changes to the pultrusion die. Hence, the size of the preform plate could be increased to some extent to obtain a higher pressure rise in the pultrusion die without affecting the other control parameters of the pultrusion process.

Another important parameter that can be varied to increase the pressure rise in the die is the wedge angle in the linearly contoured (wedge shaped) inlet region. The tapered inlet region can have different contours such as circular. parabolic and wedge shaped. The wedge shape was selected as the tapered inlet section contour for this study. The angle at which this wedge is inclined will influence the pressure rise in the die inlet. Several pressure rise profiles in the die inlet as a function of wedge angle [Alpha] are shown in Fig. 5. The nominal case processing conditions of U = 0.005 m/sec, [Mu] = 1.5 kg/m s, and [V.sub.f] = 0.62 were maintained while numerically modeling the different wedge angles. The wedge length of 3L was modeled for each case wherein L increased for smaller wedge angles. There is a decrease in the magnitude of the pressure rise with an increase in the wedge angle. A wedge angle of 16 [degrees] results in a comparatively smaller pressure rise while a wedge angle of 4 [degrees] yields a significantly higher pressure rise in the pultrusion die inlet. As the wedge angle is increased, the intersection point of Fig. 2 is moved closer to the straight portion of the die. This reduces the compression distance in the die inlet of the fiber/resin matrix being pultruded, thus resulting in a lower pressure rise. Thus, a suitably small wedge angle can be used to produce a higher pressure rise.

The centerline pressure data in each case is obtained for a coordinate system origin at an axial length of 3L upstream of the start of the straight portion of the die. A change in either preform plate size or wedge angle of a linearly contoured die inlet geometry results in a shift of the intersection point [ILLUSTRATION FOR FIGURE 2 OMITTED]. A variation in the intersection point leads to a change in the length 3L. Because of this the centerline pressure data for the non-nominal cases in Figs. 4 and 5 had to be shifted accordingly to correspond to the origin of the nominal case. This enabled the centerline pressures for the different cases to be properly compared at corresponding upstream distances from the start of the straight portion of the die.

A typical pressure distribution for the tapered entrance region of the die in the form of contours of constant pressure is shown in Fig. 6; the pressure contours correspond to a preform plate size of [h.sub.pp] = 1.25 [h.sub.s]. The region between the computational domain inlet and the intersection point experiences about a third of the total pressure rise while the remaining two-thirds of the total pressure rise occurs in the short section between the intersection point and the end of the tapered die inlet. This is true for all the different processing conditions for which a pressure distribution was predicted using this model.

Some preliminary experimental data were obtained from a trial experiment conducted on the Pultrusion Technology Incorporated (PTI) model Pulstar 804 pultruder in the composites laboratory facility located at the University of Mississippi. The centerline pressure variation along the pultrusion die cavity was measured by passing a Photonetics 1450 fiber-optic pressure probe of approximately 0.8 mm in diameter through the pultrusion die. The experimental data corresponds to a preform plate size ([h.sub.pp]) of about 1.2 times [h.sub.s] and a wedge angle ([Alpha]) of about 8 [degrees]. The centerline pressure profile predicted by the finite element model in the tapered die inlet region was qualitatively verified against these experimental data. Figure 7 shows the comparison of the numerically predicted and preliminary experimental data pressure profiles in the pultrusion die for a preform plate size of [h.sub.pp] = 1.2 [h.sub.s]. The predicted pressure profiles for three other preform plate sizes are also shown. As seen, the trends in each case are quite similar. There is a rapid and large pressure rise in the short tapered inlet region, after which it remains somewhat level along the straight portion of the die. Then, in each case, near the point of gelation the pressure rise quickly decreases back to zero. This is the gelation location where the pressure rise is forced to drop to zero due to the rapidly increasing resin viscosity. The gelation location does not change for different preform plate sizes indicating that the resin gels at about the same location irrespective of the thickness (preform plate size) of the fiber/resin matrix approaching the die inlet.

The comparison of the experimental and numerical centerline pressure data for several wedge angles is shown in Fig. 8. Once again there is qualitatively good agreement between the experimental data and numerical predictions. The pressure profiles for wedge angles of 4 [degrees] and 16 [degrees] are also shown in Fig. 8 to demonstrate how the pressure rise in the pultrusion die follows a similar trend for different wedge angles. The gelation location for all three wedge angles remains the same indicating that the wedge angle does not influence the point at which the liquid resin gels and the pressure rise decreases to zero.

The compression of the composite matrix being pultruded in the tapered inlet section of the die results in a backflow of some of the liquid resin. The amount of backflow in terms of mass of resin squeezed out from the die has been depicted in Fig. 9 for different preform plate sizes. The backflow in the tapered inlet region was calculated by integrating the resin mass flux [Rho]u the point where the u-velocity profile becomes negative to the tapered inlet die wall. The numerical predictions have been non-dimensionalized using the theoretical value of upstream mass backflow rate. This theoretical value of upstream mass backflow rate per unit width, verified in measurements performed at the composites laboratory of the University of Mississippi, is equal to [Mathematical Expression Omitted]) at the computational inflow boundary. The mass backflow rate of the liquid resin decreases as one moves downstream and reduces to zero before reaching the straight portion of the die. As seen in Fig. 9 the liquid resin mass backflow rate becomes zero at a point further downstream with decreasing preform plate size. As the preform plate size is reduced the point of fiber contact with the die inlet wall is pushed downstream resulting in the backflow continuing to a point farther downstream than for a larger preform plate size. Similar results are presented in Fig. 10 for a variation in the wedge angle, An increase in the wedge angle moves the intersection point farther downstream resulting in the point where the resin mass backflow rate goes to zero also being moved downstream. However, for each different wedge angle the resin mass backflow rate maintains the same trend, reducing gradually along the downstream direction and finally decreasing to zero completely before reaching the straight region of the pultrusion die.

Figures 11 and 12 show the u and v resin velocity profiles respectively at various x-locations to give a general idea of the flow field behavior of the resin in the die inlet. The backflow of resin in the inlet region is indicated by the negative u velocities. The magnitude of the negative velocities is comparatively higher than the positive velocities, which maintains the overall mass balance. Overall resin continuity per unit width across the die channel was verified by integrating the resin mass flux [Rho]u across the die channel at a variety of axial locations for each computational case as a quality control check. An epoxy resin density p equal to 1260 kg/[m.sup.3] (7) was used to calculate the theoretical value of resin mass flow per unit width given by [Mathematical Expression Omitted]. Here [[Phi].sub.s] is the porosity along the straight portion of the die, which is same as the porosity of the finished product. In regions other than the backflow region, the flow field predominately exhibits a forward velocity close to the pull speed indicating that the resin is pulled through the die, at a constant speed. Figure 12 shows that except for the resin backflow region the transverse flow of the liquid resin is small. The non-zero velocity components at the die wall correspond to the wall slip velocity which is characteristic of Darcy flow.

CONCLUSIONS

A two-dimensional finite element model based on Darcy's law for fluid flow through a porous media was developed. This model has been shown to be useful for predicting the pressure rise due to variations in preform plate size and the wedge angle. The pressure rise in the pultrusion die inlet was found to increase with an increase in the size of the preform plate ([h.sub.pp]). Also for linearly contoured inlet die regions a reduction in the wedge angle was found to increase the pressure rise in the pultrusion die inlet. Neither the size of the preform plate nor the wedge angle had any effect on the position of the gelation location.

This model provides the ability to select the appropriate preform plate size and wedge angle for obtaining a desirable pressure rise in the die inlet without changing the other processing conditions. An experimental effort to study the same phenomena would require manufacturing inlet sections with different wedge angles and preform plates of different sizes, which would be an expensive and time-consuming undertaking. The model can be useful in designing a pultrusion die. Finally, the numerical model provides an effective method to analyze the fluid mechanics problems associated with the pultrusion process.

ACKNOWLEDGMENTS

This research was supported by the National Science Foundation (Grant No. EPS - 9452857), the State of Mississippi, and the University of Mississippi.

NOMENCLATURE

[Alpha] = Angle the tapered die wall makes with the horizontal axis.

C = Kozeny constant.

[D.sub.f] = Fiber diameter, m.

e = Element number.

{F} = Right hand side column vector.

[h.sub.pp] = Thickness of the fiber/resin matrix existing the preform plate, m.

[h.sub.s] = Thickness of the fiber/resin matrix in the straight die region, m.

h(x) = Distance from die centerline to die wall in the die inlet region as a function of x-distance, m.

i = Local node number.

[K] = Global stiffness matrix.

[K.sub.11] = Permeability in the axial direction.

[K.sub.22] = Permeability in the transverse direction.

L = Axial distance between point of fiber contact with die wall and start of straight die region, m [ILLUSTRATION FOR FIGURE 2 OMITTED].

2L = Axial distance from start of computational domain to point of fiber contact with die wall, m [ILLUSTRATION FOR FIGURE 2 OMITTED].

67L = Axial length of spatial computational domain, m [ILLUSTRATION FOR FIGURE 2 OMITTED].

[Mathematical Expression Omitted] = Resin mass flux per unit width across the die channel, kg/[m.sup.2] s.

[Mathematical Expression Omitted] = Resin backflow mass flow per unit width in the resin backflow region, kg/[m.sup.2] s.

[Mu] = Viscosity of the liquid resin, kg/m s.

N = Interpolation function for pressure.

[Phi] = Porosity.

[[Phi].sub.s] = Porosity in the straight die region.

P = Pressure, N/[m.sup.2].

[P.sub.0] = Atmospheric pressure, N/[m.sup.2].

{P} = Column vector of pressure variables.

[Rho] = Density of the liquid resin, kg/[m.sup.3].

T = Temperature, [degrees] K.

[Psi] = Degree of cure.

u = Resin velocity in the axial direction, m/s.

U = Pull speed, m/s.

v = Resin velocity in the transverse direction, m/sec.

[V.sub.f] = Fiber volume fraction.

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Author:Sharma, D.; McCarty, T.A.; Roux, J.A.; Vaughan, J.G.
Publication:Polymer Engineering and Science
Date:Oct 1, 1998
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