http://jcm.sagepub.com/ Materials Journal of Composite http://jcm.sagepub.com/content/26/9/1310 The online version of this article can be found at: DOI: 10.1177/002199839202600905 1992 26: 1310 Journal of Composite Materials Zhong Cai Analysis of Mold Filling in RTM Process Published by: http://www.sagepublications.com On behalf of: American Society for Composites can be found at: Journal of Composite Materials Additional services and information for http://jcm.sagepub.com/cgi/alerts Email Alerts: http://jcm.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: http://jcm.sagepub.com/content/26/9/1310.refs.html Citations: at UNIVERSITY OF WATERLOO on June 4, 2014 jcm.sagepub.com Downloaded from at UNIVERSITY OF WATERLOO on June 4, 2014 jcm.sagepub.com Downloaded from
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http://jcm.sagepub.com/Materials
Journal of Composite
http://jcm.sagepub.com/content/26/9/1310The online version of this article can be found at:
DOI: 10.1177/002199839202600905
1992 26: 1310Journal of Composite MaterialsZhong Cai
Analysis of Mold Filling in RTM Process
Published by:
http://www.sagepublications.com
On behalf of:
American Society for Composites
can be found at:Journal of Composite MaterialsAdditional services and information for
(Received November 9, 1990)(Revised August 15, 1991)
ABSTRACT : A simplified RTM process analysis is presented which is based on theone-dimensional resin flow model by using Darcy’s law for flow through porous media.Closed form solutions for the wet length, mold filling time, and pressure distributions arederived for different simplified mold shapes such as rectangular, trapezoidal, and circularsections. These results are then applied to the mold design and vent arrangement in RTMprocess. Two design principles are proposed from these derivations and confirmed by nu-merical application examples. One is to arrange the vents to assure that flow takes theshorter path whenever possible. The other is to use line gates and to inject the fluid fromthe larger side to the smaller side, or with rapid reduction of the filling volume. Illustrationexamples show that these arrangements can cut down the mold filling time or reduce therequired applied pressure. Especially in the case of RTM with high volume fraction offibrous preforms, where the process usually involves applying a constant inlet pressure tosqueeze the resin into the preform, the mold filling time or the applied pressure require-ment is lowered substantially by these arrangements.
INTRODUCTION
ESIN TRANSFER MOLDING (RTM) process is one of the newly developedRtechniques in manufacturing composite parts and structures. In RTM pro-cessing, fiber preforms or fiber mats are placed into the mold which is designedand constructed according to the shape of the finished parts. Then the mold isclosed and evacuated. The resin mixing and injection system is connected to themold, and the whole system is heated according to the cure kinetics of the resin.When the viscosity of the resin is lowered, the inlet gate is opened. Fluid stateresin is squeezed into the fiber preforms under the applied pressure. The flowprocess continues until the whole mold is filled. Then the pressure is maintainedor increased to fully consolidate the part, and heat is continuously supplied tocure the resin matrix. Finally the mold is removed and the finished part is ready.RTM process involves fluid flow, heat transfer, and cure reaction of the resin
material. In order to obtain a good and consistent quality of the finished parts,process design and control are critical. Research workers have been very active
Journal of COMPOSITE MATERIALS, Yol. 26, No. 911992
0021-9983/92/09 1310-29 $6.00/0@ 1992 Technomic Publishing Co., Inc.
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in studying and modeling RTM process in recent years. One of the most impor-tant issues in RTM is the resin flow, and Darcy’s law has been used in most ofthese efforts. Martin and Son [1] did some early studies to investigate and simu-late the mold filling process. They presented a simple relation to predict the moldfilling time, and also used a computer program, POLYFLOW, to simulate theflow. Gauvin, Chibani, and Lafontaine [2-5] studied the preform deformation inRTM and proposed empirical model. They also worked on verifying their resinflow model based on Darcy’s law. In their work the fiber mats were at a high po-rosity range ( > 80 % ) . Guceri, Coulter, and coworkers [6-11] studied the 2-DRTM flow process and developed a computer simulation program, TGMOLD,which includes all the physics involved, such as resin flow, heat transfer, and curereaction. The flow part in their model was also based on Darcy’s law. They in-cluded both pressure and flow rate inlet boundary conditions and used boundary-fitted coordinate system to transform irregular mold shapes into rectangular com-putational domains. Dave [12] used a unified approach to model the resin flow invarious composite manufacturing processes which include the RTM process withdiscussions of the effect of capillary forces. Lee, Castro, Tomlinson, and Straus[13] did modeling work on 2-D RTM process by transforming the problem intoan equivalent Hele-Shaw flow model for the injection molding, and then used theinjection molding software, TIMS, for the flow simulation. Lee and Brew [14] de-veloped mold filling and heat transfer models in their work. They derived formu-las for the rectangular flow with an edge injection and for the radial flow with acenter injection, and performed simulation experiments with a relatively low fibervolume fraction ( < 20 % ) . Molnar, Trevino, and Lee [15,16] reported their model-ing work on controlling the RTM process parameters. Their results showed thatwith a relatively constant flow rate applied, the pressure buildup increased withtime during the filling process. They also measured the permeability of the pre-form consisting of random and bidirectional mats at a porosity range of 0.65 to0.90. Lee, Liou, and coworkers [17-19] performed experiments on resin flow andpreform permeability in RTM processes and developed their simulation model.Other development work has also been done on RTM process. Chan and
Hwang [20] developed their mold filling model for the injection molding of thepolymer solution through continuous fiber mats. They considered the capillaryflow in the process. Hayward and Harris [21,22] studied effects of various processvariables on RTM part qualities. These variables included injection pressure,mold temperature, resin viscosity, and fiber volume fraction. They also foundthat vacuum assistance can substantially improve the product quality. They madeRTM plates with 0.55 fiber volume fraction. Begemann, Menges, and Michaeli[23] studied mold filling in RIM (Reaction Injection Molding) and SRIM (Struc-tural Reaction Injection Molding) with the comparison of RTM. They showed theeffects of different gate arrangements and found defects caused by the fiber move-ment. Hansen [24] developed a set of procedures to insure the quality of the RTMproducts. Based on his experience, he proposed design guidances for gate ar-rangements which can more effectively control the mold filling flow. Salem andcoworkers [25,26] successfully conducted experiments on RTM with thermo-plastic matrix material.
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One of the recent developments in RTM processing is the structural compositeparts, which involve complex geometry structures and high fiber content. Moldfilling becomes critical since the preform flow resistance is relatively high. Molddesign and vent arrangement are very important to the final product quality. Thedeveloped computer models can solve for 2-D or 3-D flow equations, but thecomputer simulation time and the user-computer interface effort could be cum-bersome. It will be a great help if some types of design guidance can be devel-oped based on simplified theories and calculations. This analysis is to take thisapproach and to develop certain design principles based on the simplified I-Dresin flow analysis. The advantage is the availability of the closed form solutionsfor simple mold shapes, such as tapered channel and circular sections. Differentinlet conditions can be explored as they represent the different mold filling char-acteristics. This will provide an insight of the resin flow problem and assist thedesign and optimization process.
SIMPLIFIED ONE-DIMENSIONAL FLOW MODEL
We first consider a very simple I-D composite structure, such as a long circularrod, as shown in Figure 1. We are going to use RTM and assume the inlet andoutlet are located at two ends respectively. As discussed in many papers about theconsolidation and resin flow in composite manufacturing [27-31], and in otherRTM modeling work summarized in the Introduction, this type of resin flow fol-lows Darcy’s law and can be written as [32]
where all symbols are defined in the Nomenclature. The minus sign takes into ac-
Figure 1. One-dimensional flow in FtTM process.
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count the different directions of q and the pressure gradient. If we assume thereis no preform movement during the flow process, the continuity condition is
We assume the inlet is at x = x,, = 0, and the outlet is at x = xf. The initial con-dition is x = 0 at t = 0. The boundary condition at the inlet xo = 0 could beeither a prescribed flow rate qo, or a prescribed pressure p,,. In practical if aprescribed pressure is set at the inlet, the flow process starts with a constant flowrate until this prescribed pressure is reached. Then the inlet condition becomesa constant pressure. This could also happen if a resin supply system with a con-stant flow rate is used, but the resin pressure buildup reaches the pressure limitof the system due to the high flow resistance. We will solve all these cases here.In the real process, the resin viscosity JL is usually a function of time, tempera-ture, and degree of cure of the resin material, and the preform permeability Scould vary at different locations or be a function of time if the resin flow pushesthe preform to move. In the discussion here, we assume both are constant for thesimplification of the solution. This is probably reasonable if the RTM processtime is much shorter than the resin pot life, and if woven or braided preformswith high fiber volume fraction are used.
In the following discussion, we will derive solutions for three different inletconditions: (a) a constant flow rate, (b) a constant pressure, and (c) a combinationof a constant flow rate and then a constant pressure.
(a) Prescribed flow rate qo at x = xo = 0. In this case p,, = p,, (t), but p =
p (x,t) . From Equation (2) we have
From Equation (1) we know that the pressure gradient is constant, so that thepressure distribution is linear. We assume that at the flow front the gage pressurep is zero so that we have
Therefore the resultant pressure buildup at the inlet is linearly proportional to xor t. In this simple case the mold filling time tf when x reaches xf is
In order to insure a constant flow rate qo all the time, the pressure applied atthe inlet must be kept increasing linearly with time. This can be realized only ifthe resin viscosity is very low, or the fiber volume fraction of the preform isrelatively low. For RTM process involving woven or braided preforms, usually it
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is difficult to keep these conditions. Therefore the constant pressure inlet condi-tion may be used.
(b) Prescribed po at x = Xo = 0. In this case, qo = qo (t) = q (t). Since thepressure gradient is constant over the wet length at a given time, we have
We still assume that at the flow front the gage pressure p is zero. By substitutingEquations (2) and (6) into Equation (1) and integrating, we can obtain
and
The filling time in this case is
This mold filling time calculation is similar to the results proposed in References[1] and [13], although in Reference [13] the Hele-Shaw flow model is used to ob-tain the equivalent permeability. Because in References [1] and [13] the perme-ability calculation is based on the average pore flow rate, there is an extra termof porosity in their expressions. Figure 2 shows the relation of x versus t. Figure3 shows the relation of q versus t. The process variables are set according to theexperimental setup. The permeability of the preform is calculated according toReference [33] for the unidirectional fiber assembly with fiber diameter df =12 X 10-6 (m), Kozeny constant kx = 0.5, and fiber volume fraction Vf = 0.5.The definition of the permeability in Reference [33] is the same as used here. Theresin viscosity is based on the measurement of the polyurethane material, whichvaries as a function of temperature and time. A representative reference value(1 Pa*s) is used in the calculation.From these expressions we can observe some interesting points. As we already
assumed, the resin viscosity and the permeability of the preform are treated asconstant. For a given resin material, its pot life is limited. Therefore the possiblewet length is determined by the applied pressure at the inlet. The wet length isproportional to the square root of the pressure value. In other words, if the wetlength needs to be doubled, the pressure required is four times higher if other pa-rameters are unchanged. From expression (7), it is also clear that in order to in-
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crease the wet length, we can either lower the resin viscosity, or extend the resinpot life, or increase the applied pressure. For a given product and process, theviscosity and pot life are set for a specific resin material, the applied pressure islimited by the operating equipment, and the fiber preforms are chosen accordingto the part performance. With all the information, we can use Equation (7) to es-timate the possible wet length we can achieve in the RTM process and to makesure the part or structure can be filled.Expression (9) clearly shows the relation between the filling length and the fill-
ing time. Since the filling length has the most significant effect on determining themold filling time, shorter flow path should always be considered as a priority inthe mold design and vent arrangements. This important concept will also be illus-trated later in the examples.The expression of the flow rate q shows that at the beginning of the process, q
reaches to infinity. This cannot happen in the real process. The actual limit of qis set by the resin supply system. The real flow process must have a short periodof time of constant flow rate when it starts. When the inlet pressure builds up andreaches the preset pressure value po , the inlet boundary condition then becomesa constant applied pressure. Only in the case that this constant flow rate periodis very short, can the flow process be treated as with a constant applied pressure.This will be discussed further in the examples.
(c) Solution of the practical applied conditions (a ramp pressure and then aconstant pressure, or a constant qo and then a constantpo at the inlet). We assumethat when t s t,, qo is prescribed, then after t > t1, po is prescribed. Therefore11 is defined as the transition time at which the inlet boundary condition changes.From the previous calculation, we have
After t1 is reached, we can still use expressions (1), (2), and (6) as in the constantapplied pressure case, but with initial conditions set at t = t1 and x = xl. There-fore we have
The mold filling time in this case becomes
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Figure 4. Comparison of the wet length versus time with different applied conditions.
Relations (10) and (11) show that if the preform permeability S is very small andthe resin viscosity > is high, t, and x, could be relatively small. Therefore theconstant inlet pressure condition can be used instead.
Figure 4 shows the comparison of x versus t by using the constant inlet pressureapproach and this calculation respectively. The process variables are chosen asbefore except for the preform permeability (3.95 x 10-9 m2 = 4,000 darcy),which is much higher and is taken from the data presented in Reference [34]. Inthis case the preform is a continuous, random glass fiber mat with fiber volumefraction Vf = 0.164. From the figure it is clear that if the mold filling time ismuch larger than (1, the difference between these two cases is not significant.
In some practical cases, the boundary condition at the inlet varies or fluctuatesover time. If the measurement of either the flow rate or the fluid pressure can betaken, we can still solve the problem by using numerical techniques. Same is truefor the case of varying resin viscosity, as long as the viscosity is a known functionof the process time.
This discussion of simplified I-D flow is the base of this analysis and will beused later in mold design examples. Following are the solutions of two othersimplified mold sections.
ONE-DIMENSIONAL CHANNEL FLOW WITH VARYING WIDTH
In this case, we assume that the mold shape is with some variations along itswidth, but it can still be simplified by the relation
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The mold is illustrated in Figure 5. We assume that the permeability of the pre-form is still homogeneous and the absolute value of the shape factor a is muchsmaller than 1. Therefore we assume that with a line source at the inlet side ofthe mold, the flow front remains parallel to the initial flow boundary. We havechecked the flow front movement in two cases (a = 0.25 and a = -0.25) by us-ing the TGMOLD program (from University of Delaware). The results showedthat the assumption about the flow front movement is reasonable except at the re-gion close to the wall. With these assumptions, the continuity condition becomes
To simplify the solution we assumes. = 0 in the following cases. By substituting
Equations (14) and (16) into Equation (1), we can obtain the following relation be-tween qo and po as
Here if po is prescribed, qo is a function of time or wet length, and vice versa. Wewill still consider three different inlet conditions as in the previous discussion.
(a) For a constant flow rate qo at the inlet, we can use the continuity conditions(2) and (16), and obtain
or
Substituting this into Equation (17), we know that the pressure buildup at the inletboundary is
The inlet pressure history is shown in Figure 6, where the values of the shape fac-tor a are chosen as 0.25 and -0.25 respectively. The dimensionless forms of p*and t* are used in the figure, since different process conditions can be includedinto one figure.
(b) For a prescribed constant pressure at the inlet, we can use relations (1),(2), (14), (16), and (17), and obtain
This can be integrated from the inlet x = xo = 0 and time t = 0, to x and t, andthe result is
This can be used to calculate the wet length for a given time, or to estimate the
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Figure 6. Inlet pressure versus time in channel flow with different shape factors andconstant inlet flow rate, shown in dimensionless form with p~ = p~/f~,~/S~ and f~ =~o/<7j.
filling time from the given mold filling length. The resin flow rate qo at the inletcan be derived as
The inlet flow rate history is shown in Figure 7 with different values of shape fac-tor a. The dimensionless forms of q* and 1: are used in the figure.
(c) For a combination of a constant flow rate and then a constant pressure, wefirst calculate t, and xl. By using Equations (17) and (18), we have
and
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By using the same technique as before, we can derive the relation between x andt as
The pressure buildup at the inlet before t, can be calculated according to Equa-tion (17), and the flow rate at the inlet after t can be obtained by using Equation(23). These are already plotted in the previous figures. The relations between thewet length and the filling time with different shape factors are plotted in Figure8, in which the absolute value of the shape factor a is relatively large (0.25). Thedimensionless forms of x* and tq* are used in the figure. The process variables are
Figure 7. Inlet flow rate versus time in channel flow with different shape factors and a con-stant applied pressure, shown in dimensionless form with q; = qol[(p.S)I(pho)], and tp =tl[(Ah 0 2)1(po S)].
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Figure 8. Wet length versus time in channel flow with different shape factors and with aconstant qo then a constant po at the inlet, shown in dimensionless form with x
chosen as in the previous figures. As time goes on, the effect of the shape factoron the wet length becomes significant. This result can be applied to the vent ar-rangements and mold design, as we will illustrate later in the examples.
ONE-DIMENSIONAL RADIAL FLOW IN A CIRCULAR SECTION
In this case we consider a mold with a circular shape section as shown inFigure 9. Flow is assumed only in the radial direction, either inward or outward.The flow equation in this case becomes [35]
We can also write this condition as
where qo and ro can be either at the inner radius or at the outer radius dependingon the flow direction, as shown in Figure 9. As usual we assume that both q andr are positive along the outward direction.
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If we neglect the high order term accounted for the effect of 0, the continuitycondition is similar to that used in the previous cases. For the outward flow case,
For the inward flow the flow direction has to be taken into account so that
Equation (1) of Darcy’s law can still be used here with the assumption that thepermeability is homogeneous, and the flow is only in the radial direction. Thevariable in the expression is r instead of x. We have checked the assumption onthe flow front movement by using the TGMOLD program. We solve the problemfor three inlet boundary conditions as mentioned earlier. The results are listedhere.
I
Figure 9. Circular shape mold sections with different flow arrangements.
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(a) A constant flow rate qo at the inlet. We solve for r and p,,. For the outwardflow where r > ro, we have
and for the inward flow where r < ro, we have
The inlet pressure histories for inward and outward flow are shown in Figures10 and 11 respectively. The dimensionless forms of p: and tq are used in thesefigures. Although we use the same scale factor ro lqo to obtain the dimensionlesstime tq , the values are different in these two opposite flow arrangements. One in-teresting point is the significant difference in the pressure buildup at the inlet ver-sus time in these two opposite flow arrangements.
(b) A constant applied pressure po at the inlet. In this case the expression be-tween r and t is the same for both inward and outward flow cases, and we have
where ro could be either the inside radius or the outside radius depending on theflow directions. The outward flow case is also discussed in Reference [13]. Simi-lar mold section with pressure flow is presented in Reference [36], which is thecase of RIM (Reaction Injection Molding) process. The solution of q. for the out-ward flow case (r > ro) can be expressed as
and the flow rate for the inward flow (r < ro) can be written as
These results are shown in Figures 12 and 13 respectively. The dimensionless
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Figure 10. Inlet pressure versus time in an outward flow (r > ro) of a circular section witha constant inlet flow rate, shown in dimensionless form with po* = pol(qoropls) and t* -tl(rolq,,).
.1
Figure 11. Inlet pressure versus time in an inward flow (r < ro) of a cirrcular section with aconstant inlet flow rate, shown in dimensionless form with po = po/(qoro’r/S) and t; =tl(rolqo).
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Figure 12. Inlet flow rate versus time in an outward flow of a circular section (r > ro) witha constant inlet pressure, shown in dimensionless form with q; = <7o~oSVf~o~. and
t; = tlI(r~o)l(Po S)I~
Figure 13. Inlet flow rate versus time in an inward flow of a circular section (r < r~) with aconstant inlet pressure, shown in dimensionless form with qo = <7o~fpo~~’oJ’/. and tP =tl(fr~o)l(Po S)J~
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forms of q,* and tp are used in these figures. There are also significant differencesin the flow rate versus time in these two opposite flow arrangements.
(c) A combination of a constant flow rate and then a constant pressure at theinlet. By using the same technique as before, we can obtain values of t, and r1 forthe case of r > ro and r < ro respectively. These results are
We can use the results of r with the constant qo boundary condition before t, hasbeen reached, which are given in expressions (30) and (32). Afterwards we canuse the results with the constant po boundary condition. The result is
where r, and t, are given in Equations (37) and (38). This expression can be usedto calculate the radial wet length for a given pot life of the resin, or to estimatethe filling time for a known structure. The expressions for qo after t, are the sameas in the previous cases of the constant applied pressure, which are given inEquations (35) and (36) respectively.As we can see from previous figures, if t, is much smaller than the total mold
filling time, there is no significant difference between the result of a constant inletpressure and this one. Otherwise, the calculation with a constant inlet pressurealways over-predicts the wet length at a given time.
Having gone through these derivations, we are going to apply these results tosome mold design examples to illustrate basic design concepts and guidances.
APPLICATIONS TO THE MOLD DESIGNAND VENT ARRANGEMENTS
We will use the results of these simplified calculations to illustrate two designconcepts through some numerical examples. One of the concepts is to arrange for
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the shorter flow paths. The other is to arrange the flow path such that the unfilledvolume is reduced rapidly. Also we will show that injecting with line gates is usu-ally better than with point sources.The first case we would like to discuss is a simple rectangular mold with the
ratio of length to width of 2 to 1. With a line inlet at one side and a line outletat the other side, there are two ways to arrange the vents. One is to let the flowadvance along the length and the other is along the width. With the same resinsupply system, the total flow rate Q would be the same in both arrangements. Inother words, the total amount of the resin flow (qoho8) remains the same in botharrangements. Apparently if the inlet condition is set as a constant flow rate, themold filling time in both cases must be the same. However, for the constant inletpressure case, the shorter filling length gives a great advantage regarding to themold filling time.
Table 1 summarizes the results of numerical examples for various process pa-rameters, which are taken either from the literature or from our experiment setupconditions. The main parameter being changed in these examples is the preformpermeability, which greatly influences the characteristics of the mold filling pro-cess. As discussed in the literature, the permeability of the preform is related tothe filament size, fiber volume fraction, porosity, and preform structures
[33,34,37,38]. In References [33,34,37] the permeability is based on the averageflow rate over the total cross section area. The axial permeability of a unidirec-tional fiber assembly ranges from 1 X 10-’3 to 2 x 10-12 (m2), and the trans-verse permeability ranges from 1 X 10-’4 to 2 x 10-’3 (m2), for fiber volumefraction 0.5 to 0.78 [33,37]. The transverse permeability of the random fiber matis about 2.96 x 10-9 to 3.95 x 10-9 (m2) (3,000-4,000 darcy) with fiber volumefraction of 0.164, and varies at different infiltration flow rates [34]. In Reference[38] the permeability definition is based on the average pore velocity. The inplanepermeability of the woven fabrics ranges from 2.5 X 10-’ to 3.1 X 10-9 (m2)(25-3166 darcy) with the porosity range 0.42 to 0.68. The permeability valuesselected in the following examples represent different mold filling flow character-istics.
In the calculation, we first use the given data of the prescribed qo and the maxi-mum po of the resin supply system to derive t, and xi . If t, and x, are very smallcompared with tf and xj respectively, the process can be approximated as a con-stant applied pressure one. On the other hand, if t, and xl are close to or largerthan tf or xf, the process can be thought as a constant applied flow rate one. Previ-ous derived formulas are then used accordingly. The main parameter in compari-son here is the estimated mold filling times in different cases. We should point outthat these estimated filling times are from the calculations only. There are casesthat these estimations deviate from the experimental data [1].
Clearly the arrangement with flow along the mold width, which is shortercompared to the mold length, requires much less mold filling time. The last casehere is the constant flow rate boundary condition, in which the filling times arethe same for both cases. When the process condition is close to the constant ap-plied pressure case, the saving in the filling time is significant. When the ratio oflength to width is higher, the difference in filling time in these two different cases
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is even greater. On the other hand if we fix the filling time in these two cases, therequired applied pressure at the inlet will be different. Therefore the shorter fill-ing length must be always considered as a priority in the mold design and ventarrangements. It is obvious that with any applied boundary conditions, the resultsof case 2 (shorter flow paths) are never worse than those of case 1.The second example here is a mold section with a trapezoid shape as shown in
Figure 5. Assuming that we have to arrange the resin flow path along the x axis,we have two options, either to fill the section from the larger side to the smallerside, or vice versa, as shown in the figure. In this case we assume that the totalflow rate, Q = ohoqo, does not vary. Again we choose xo = 0 to simplify the cal-culations. The results of two selected examples are listed in Table 2. We still com-pare the mold filling time tf of both arrangements.
In these cases, the arrangement with flow from the large side to the small sideshows advantage since less filling time is needed. If the mold filling times are setas the same, then the pressure requirement will be different in two cases. Thisdesign principle is illustrated further in the following circular shape mold case.The circular shape mold is shown in Figure 9 with only one quarter representa-
tion. We assume that the flow front moves uniformly in the radial direction,either inward or outward. The mold dimensions are shown in the figure. The twochoices here are the inward flow and the outward flow, also shown in the figure.Again we assume that the total flow rate, Q = br,,q,, is the same in both cases.The calculation results are listed in Table 3.These examples show clearly that the arrangement of flow inwards is much bet-
ter, which is the arrangement for the flow from the large side to the small side.From the previous figures of the flow rate history under the constant appliedpressure boundary condition, we can see the rate of the resin flow at the inletdecreases much faster with time in the inward flow case. Therefore, it is mucheasier to fill a mold if the unoccupied volume is reduced rapidly. From thecalculation it is also clear that if in both cases the filling time, which is relatedto the pot life of the resin system, is the same, the pressure requirement in case2 must be lower than that in case 1. This reduces the equipment setup cost. Thisconcept is important to the mold design and process control.
Figure 14 shows an example of two different vent arrangements for a rectan-
(a) Mold filling with a central inlet (b) Mold filling with a line gate at the perimeter
Figure 14. Comparison of mold filling with different flow arrangements.
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gular part. This was reported by Hansen in Reference [24]. One way to fill thepart is to have the inlet at the center and the outlet along the perimeter. His sug-gestion is to have the line inlet along the perimeter and the outlet at the center.He found that the latter case gives much better control of the mold filling and partquality. This agrees with the results shown above although the vent arrangementis only one of the factors which affect the RTM process. Also this is a 2-D con-figuration so that the exact solution is not possible by this calculation.From the examples we can see that the large difference in filling time happens
in the case of the pressure inlet condition. From the solutions of Equations (22)and (34) for the trapezoid and circular shape mold sections, we can obtain theratio of the filling times with these opposite vent arrangements under the constantinlet pressure condition. The ratio of the filling times is a function of the sideratio and can be written as
The value of y is always larger than 1. For the trapezoid mold section the relationholds only for the case that the absolute value of the shape factor a is muchsmaller than 1. This relation is plotted in Figure 15. This ratio reflects the savingsin filling time by rearranging flow directions. The higher the value of y, the moresavings in the mold filling time. For the rectangular mold example, with an inlet
Figure 15. Comparison of mold filling time with different vent arrangements, where F(y) =tb It. -
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hole at the center as one arrangement and with an outlet at the center as anotherarrangement, the ratio y is high if we can use the circular flow approximation.Therefore the effect by these opposite flow arrangements can be substantial if thepressure boundary condition holds. We can also suggest that in most cases a linesource is often better than a point source, since the point source can be treatedas the outward flow in a circular section with a very small r. at the inlet. In thereal design we have to include the pressure loss incurred in flow through gatesand channels which we omitted in this discussion.Mold filling without fiber preforms or with low fiber volume fraction preforms
can usually be treated as having a constant flow rate condition at the inlet. How-ever, when high fiber volume fraction preforms are involved, such as braided orwoven fiber preforms, pressure buildup at the inlet can be very high. Thereforethe constant inlet pressure condition is often used to insure the complete fillingof the structure. As we have shown in these examples, the vent arrangementsbecome very important under the pressure boundary condition.The discussions here are only limited to the one-dimensional resin flow with a
constant resin viscosity and a constant preform permeability. This is probablynever true in any practical RTM processes. With all the variations of the materialproperties, equipment performances, and operational conditions, the advantagesshown in these numerical examples could not always be guaranteed. However,the concepts presented here are very useful to the mold design and vent arrange-ments, as well as to the process control and final product quality. Even with com-plicated mold geometries, these concepts are still very helpful and can be used asthe design guidances. The main goal in this discussion is to offer design and pro-cess engineers some general rules and estimation methods without involvingcomplicated and time-consuming computer simulations.
CONCLUSIONS
A simplified resin flow calculation of RTM process is presented, which givesclosed form solutions of some simple mold sections with various inlet boundaryconditions. The calculation results reveal some basic concepts which can be usedin the mold design and vent arrangements.Two design principles are proposed and illustrated by several application ex-
amples. They can be summarized as follows: (1) inlets and outlets should be ar-ranged such that possible shorter flow paths can be achieved; (2) the resin flowdirection should be arranged from larger sides to smaller sides, or from outsideperimeter to the inside, which guarantees rapid reduction of the unoccupied vol-ume. When the real process is close to the constant applied pressure boundarycondition at the inlet, the saving in filling time is significant by these arrange-ments. Also line gates are better than point sources.A set of formulas and charts are derived for various one-dimensional resin flow
cases, which can be used for estimating the mold filling time, wet length, andoperating equipment requirements. This analysis provides design and process en-gineers a supplemental tool to the better use of the RTM process in compositemanufacturing.
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a = shape factor of a trapezoid mold sectionA = total cross section area (m2)d f = fiber diameter (m)h = mold width perpendicular to the flow (m)ha = mold width of the smaller side of a trapezoid section (m)hb = mold width of the larger side of a trapezoid section (m)ho = mold width at the inlet (m)kx = Kozeny constant for permeability calculationp = fluid pressure (gage pressure) (Pa)po = inlet gage pressure (Pa)p/ = dimensionless inlet pressure, p* = pol(qoho~,lS) for a trapezoid
section, /?* = po/(qorop,/S) for a circular sectionq = QlA, average flow rate (m/s)q,,= inlet flow rate (m/s)q* = dimensionless inlet flow rate, <?* = q.l[(p.S)I(Ikh,,)] for a trape-
zoid section, q: = q,l[(p,,S)I(Itr,,)] for a circular sectionQ = total amount of the flow (ml/s)r = radial wet length (m)ra = radial position of the smaller side of a circular section (m)rb = radial position of the larger side of a circular section (m)rf = radial position of the outlet (m)ro = radial position of the inlet (m)r1 = radial wet length at the transition time t, (m)S = permeability of the preform (m2)t = process time (s)ta = filling time from the smaller side to the larger side (s)tb = filling time from the larger side to the smaller side (s)tf = total filling time (s)t§ = dimensionless time for the pressure inlet condition, t* = tl[(Ith,,)l
(poS)J for a trapezoid section, tP = tl[(14r,2,)I(poS)] for a circularsection
t$ = dimensionless time for the flow rate inlet condition, tq = tl(holqo)for a trapezoid section, tq = tl(rolqo) for a circular section
t, = transition time for the inlet condition (s)t* = t1/(ho/qo), dimensionless transition timeVf = fiber volume fractionx = wet length (m)
x * = x/ho, dimensionless wet lengthxf = outlet position (m)xo = inlet position (m)xi = wet length at the transition time t1 (m)
xi * = xllho, dimensionless wet length at the transition time
y = side ratio of the mold, y = h6 /ha for a trapezoid section, y r, Ir.for a circular section
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The author gratefully acknowledges the technical support from Duncan Lawrieand Alex Berdichevsky of Lord Corporation. The author would also like to thankProf. S. Guceri of University of Delaware and J. Coulter of Lehigh University forproviding support of using TGMOLD programs.
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