Optimization of cure kinetics parameter estimation for structural reaction injection molding/resin transfer molding

Optimization of Cure Kinetics Parameter Estimation for Structural Reaction Injection

M o I d i n g/Res i n Transfer Molding

ROBERT J. DUH

Automotive Systems Group Johnson Controls, Inc. Plymouth, MI 481 70

SUSAN MANTELL

Department of Mechanical Engineering University of Minnesota Minneapolis, M N 55455

JEFFREY H. VOGEL

SciMed Li$e S y s t e m Minneapolis, MN 55369

ROBEm S . MAIER

Minnesota Supercomputer Center University of Minnesota Minneapolis, M N 55455

A numerical method is proposed for polymer kinetic parameter estimation of either Structural Reaction Injection Molding (SRIM) or Resin Transfer Molding (m). The method simulates either radial flow or axial flow of reactive resins through a fiber preform inside a mold cavity. This method considers a non-isothermal envi- ronment with different inlet boundary conditions. Based on the molding conditions, this method can find the best values of chemical kinetic parameters by comparing the simulated temperature history and the experimental temperature history. Since the kinetic parameters are estimated with the real molding conditions, the simulations using these parameter values can have better agreement with molding data than those parameters which are obtained from idealized conditions such as Differ- ential Scanning Calorimeter [DSC). The optimization approach was verified by estimating kinetics parameters for RTM data available in the literature. Temperatures predicted by the optimized kinetics parameters are compared with experimental data for two different molding conditions: injection of a thermally activated resin into a radial mold under constant pressure flow, and injection of a mix activated resin into a radial mold under constant volume. In both cases, the optimized kinetics parameters fit the temperature data well.

1 INTRODUCTION final products. Understanding and modeling chemical reactions are key issues for SRIM/RTM processes. Correct polymer kinetic model and parameter values are necessary for the study of SRIM/FVM. Typically, parameters are determined by performing isothermal

RIM and EiTM are becoming widely used by indus- try because of their advantages: no intermediate

products are required, low processing temperature, reduced clamping pressure, and good properties of

S

730 POLYMER COMPOSITES, DECEMBER 2001, Vol. 22, No. 6

Optimization of Cure Kinetics Parameter Estimation

7

Material Molding & process thermal Kinetic parameters properties model

Improved parameters error <

Experimental temperature

I

Parameter J

Simulation

initial J parameters

> estimates (kinetics, heat transfer. fluid flow)

,___-__-__________________________I

or adiabatic experiments with resin only. These labora- tory fitted parameters often do not adequately represent the chemical reaction during real molding conditions.

We propose a n empirical approach to determine kinetic parameters (Flg. 1). In this approach, an experi- ment with a simple mold geometq under constant flow rate or constant pressure is modeled. Temperature data from experiments are used to numerically determine the kinetics parameters. Parameters found in this way may more accurately reflect the thermo-chemical behavior of the fiber/resin system under injection conditions as compared to a classical DSC approach.

This study has three components: a numerical simulation model, an experimental verification, and a regression analysis. Since many simulations are required for the regression, an efficient, accurate modeling strategy is desired. This numerical model has an option to choose either a one-dimensional or two-dimensional heat transfer calculation to meet the different molding conditions. Experiments were conducted to validate the simulation model. Two kinetic models were incorporated for either chain-wise or step-wise reactive materials. The model was integrated with a MINPACK nonlinear least-squares routine to perform parameter estimation) (1). Parameter estimations have been compared with work by Macosko and coworkers (2-4).

2 PREVIOUS WORK

In resin transfer molding and structural reaction injection molding, fiber is placed inside the mold cavity. This pre-placed fiber has a significant effect on the resin flow and chemical reaction. As a result, the properties of the final product are affected by the pre- placed fiber. Research has focused on the resin flow during molding and on chemical kinetics of the poly- mers themselves. However, what is needed is an approach which integrates both effects.

Research in resin flow (mold filling) has included both experimental and numerical approaches. The focus of mold filling studies has been to describe the interaction between the fibers and resin flow. Darcy’s law (5) has provided the foundation for these experimental and numerical studies (2, 6-10).

Polyurethane, polyester, and polyureas are common resins used for RTM and SRIM. When a certain temperature or mixing concentration is reached, polymer- ization occurs and the resin begins to change from liquid to solid. This change from liquid to solid is typically either thermally activated or mix activated. For thermally activated materials, step wise kinetics models have been proposed (4, 11, 12). For mix activated materials, chain-wise kinetics models have been proposed (3). Within these kinetics models, several kinetics parameters must be determined. Differential Scanning Calorimetry (DSC) is the main tool used to obtain kinetic parameter values. DSC tests are run under either isothermal or adiabatic temperature conditions and a particular kinetic model is fit to the data. The kinetic parameters are varied until the model accurately predicts the relationship between the temperature, conversion (degree of cure) and conversion rate obtained via the DSC for that particular resin. The primary shortcoming of the DSC approach to estimating kinetics parameters is that the DSC is run on the neat resin only and does not include the effects of the fiber preform on the kinetic behavior of the resin.

The current study of parameter estimation is focus- ing on the methodology for gathering data to be used in the estimation. Recall that the kinetics parameter values obtained from DSC tests do not include the af- fects of mold wall, fiber, and high processing temperature. Although better accuracy has been obtained with adiabatic DSC runs than isothermal runs, this method cannot be applied to thermally activated

POLYMER COMPOSITES, DECEMBER 2001, Vol. 22, No. 6 731

Robert J. Duk Susan Mantell, Jefiey H. Vogel, and Robert S. Maier

resins (13). To tailor kinetics parameters to RTM/ SRIM studies, it is proposed that data be obtained in controlled RTM/SRIM experiments.

3 PROBLEM STATEMENT

The goal of this study is to find kinetic parameter values that accurately describe the chemical reaction occurring in FU'M/SRIM processes. Using these kinetic parameters, the numerical predictions of temperature during processing will be compared with experimental temperature data. Experimental temperature data, mold geometry, injection type, material properties, and molding conditions are required for this parameter estimation.

Two studies published in the literature will provide temperature data during molding: 1) constant pressure flow of a thermally activated resin (3). and 2) constant volumetric flow of a mix activated resin (13). For these selected studies, a nonlinear data fitting technique was used to fit the kinetics data.

4 APPROACH

In estimating parameters for a particular resin system, existing data from similar processing conditions are used to estimate kinetic parameters. Initial kinetics and viscosity models must be selected. These models must capture the temperature, cure and viscosity behavior of that particular resin system. Once the material models have been identified, mold filling is simulated numerically. A finite element numerical model of mold filling was developed. This model is run simultaneously with an optimization algorithm. Given the molding conditions, the resin cure temperatures are calculated and compared with existing data. The optimiA.ion algorithm adjusts the kinetic parameters to minimize the difference between the numerical pre- diction of temperature and the experimental temperature data.

4.1 Kinetics Model

The urethane system studied by Chen et aL (13) is a thermally activated material. Therefore, a stepwise kinetics model was selected.

& = (K, + K,a"l)(l - a)"'

where

K, = A, exp(- 5) K2 = Az exp (-$)

dc is the degree of conversion rate of the polymer. The value of a is between 0 (no reaction) and 1 (fully re- acted). K,, and & are reaction rate constants, A, and A, are kinetics parameters, n, and n, are reaction kinetics exponents, and El and E, are activation energy

for each reaction. A,, 4, E, , q, n,, and n, are the parameters to be estimated in this study. This kinetic model is included in two widely distributed RTM/ SRTM simulation packages, CSET (14) and UMS (15).

The mix activated resin system studied by Gonzdez-Romero (3), is considered a chain-wise material. The chain-wise model uses a system of higher order differential equations to mathematically describe fast chemical reactions. For the styrene- dimethacrylate (SDM) copolymer used in the Gor~zdez- Romero study, a 3rd order system of differential equations is used to model the reaction. There are three species in this chemical system: the monomer, the radicals, and the inhibitor. The species balance is given by following set of equations, where a is the conversion of the monomer, R* is the conversion of the radicals, and Z* is the conversion of the inhibitor: Monomex

aa a t _ - - k , R * [ l - a ]

Radicals aR* kz -- - 1 - k t -[1 - Z * ] R *

a t kp Inhibitor

a z* kz __- - kx-[l - Z * ] R * a t kP

(4)

k, is the inhibition rate constant and kp is the propa- gation rate constant. t, and k, can be described by the kinetics parameters A, B,, and & and the activation ener@, Ex as follows:

tz = B1 exp(- 7) BZ

A, B, , B2, E, and kJ kp are the parameters to be esti - mated in this work.

In both models, the heat generation rate (i is given by the following equation:

q = v,H,& (7) where u, is the volume fraction of resin, H, is the chemical heat of reaction.

4.2 Viscosity Model

Viscosity is an important material property in SRIM/ RTM processes. As soon as the reaction begins, heat evolves, and the conversion and viscosity fields are changed. The viscosity of the resin initially decreases due to the hotter mold temperature, but it then increases as a result of the chemical reaction (13). The reaction must be fast enough to achieve a high pro- ductivity, yet the viscosity must remain low enough during the filling stage to permit complete filling of the cavities.

732 POLYMER COMPOSITES, DECEMBER 200 1, Vol. 22, No. 6


center-gate (radial mold)

Mold

Lr Fig. 2. Numerical grid scheme. y is in the thickness direction. r is in thejlowfront direction.

In general, viscosity is a function of temperature T, conversion a, and shear rate + (3):

P = P(T, a, +) (8) Temperature influences viscosity in two opposing ways. Raising the temperature will decrease viscosity at a given conversion, and increase the reaction rate. On the other hand, the viscosity will increase with the reaction rate.

Castro and Macosko (4) have found the following empirical relation which satisfies these viscosity char- acteristics:

where a is the conversion, and ag is the conversion at gel point. 4, Ep, A, and B are constants. This equation adequately predicts divergence of the viscosity at the gel point. This equation has been used for the polyurethane reaction injection molding (RIM) process (4) and is empirical in nature. Therefore, this viscosity model is incorporated in the parameter estimation.

4.3 Simulation Model

Temperatures during molding were predicted by a two-dimensional finite difference model. The radial mold geometry was modeled by a set of axis-symmetric elements. The grid configuration had 5 elements through the thickness direction and 12 elements in the radial direction (Q. 2). The temperature can vary in the thickness direction y or the radial direction r. When the resin reaches the gelation point, ag. the viscosity of the polymer increases rapidly and, as a result, the flow is stopped. If ag is reached during the filling stage, the filling process will cease, resulting in

an incompletely molded part. During parameter estimation, this condition is checked to flag unsuitable estimated parameter values.

For the injection simulation model, we assume:

no pressure difference through the thickness, a p a Y

thermal conductivity k, heat capacity C, and density p are independent of temperature (1 7);

the resin melt is assumed homogeneous; and

the temperature distribution is axis-symmetric,

_ - - 0 (16);

- 0 (Rg. 3). aT _ - ae

The two-dimensional model for a radial mold for either constant pressure injection or constant volumetric flow rate injection is developed as follows. The con- tinuity equation describes fluid velocity in the mold:

where v, and v, are the velocity components in the r- and e- direction. The influence of the porous fiber bed on the fluid velocity is described by Darcy's law:

where p is viscosity, Pis pressure, and K~ K ~ , K , ~ and K,, are permeability components. The thermal behavior of the system is described by the energy equation. Conservation of enera is applied to the liquid phase (the liquid resin):

POLYMER COMPOSITES, DECEMBER 2001, Yo/. 22, No. 6 733

Robert J . Duh, Susan Mantell, Jefley H . Vogel, and Robert S . Maier

radial flow

gate I

@ pr c, (Z + u r z j =

Y

Flg. 3. Mold geometry and coordinate systems.

r

(12) and the solid phase (the fiber preform):

aTf a t ( 1 - @) pfcf- =

During the curing stage, the resin and fiber are treated as a homogeneous material:

where p is density, C is heat capacity, T is temperature, t is time, a is the reactant conversion, @ is the fill factor (varies from 0 to l), and ffr is the resin heat conductivity. The conversion rate 0; is a function of time and temperature which is given by the kinetics model. The subscripts r and f apply to the resin and fiber respectively.

The initial conditions required to solve the energy equation are the initial resin temperature Tr, mold temperature T,, and the fiber temperature 5:

TA~==o = Tinject (151

It is assumed that the resin has not begun to cure prior to injection,

(YIt=o = 0 (18)

For the boundary conditions, if the mold wall temperature is prescribed,

If convection is prescribed, then

When symmetry about the midplane of the mold is considered, the boundary condition at the midplane is

Conditions for the heat flux at the mold center (radial case) or mold edge and at the flow front must be spec- ified. It is assumed that there is no heat flux at either of these boundaries:

where '-,are the flow front locations as a function of time for the radial and axial cases, respectively.

Case 1: Constant Pressure Iqiection

For constant pressure injection, the flow front velocity in the radial direction may be obtained from Darcy's law:

ap ae Note, for axisymmetric conditions - = 0. Equation

23 can not be directly applied to interpret experimental data, however, because the radial pressure gradient aP/ar is not constant (18). A mass balance on the region of the mold filled with fluid is used to obtain the radial pressure gradient.



2ml ur, = 2nr2ur, (24) where r1 and r, are two radii within the filled portion of the mold, and url and ur2 are the fluid velocities at r1 and r,, respectively. Combining the mass balance with Darcy’s law results in a pressure gradient that varies with the reciprocal of the radial position (18):

where r,, is the radius of the injection port at the center of the mold, rfiM is the radius of the flow front.

After integrating this equation from ro to rhnt) the pressure can be expressed as a function of r.

(26)

where C is constant which can be obtained from the boundary conditions. The pressure at the inlet port is the inlet pressure Pw Therefore, C is zero. The pressure field can be calculated from

r 2 Font

Substituting the pressure field (Eq 27) into Darq’s law (Eq 23). The velocity at the flow front is

Since the flow front velocity is the derivative of the flow front position, we can re-write the equation as

(30)

In the computer model, the integrated term is discretized as

‘r...r

/r@apdr = l r ’ p d r + lrypdr + - - . + lr, kclr (30)

In the constant pressure injection case, the flow front position (i.e., radial velocity) can be determined only by calculating the pressure gradient. Consequently, the flow front position, pressure field, viscosities, velocity field, heat transfer and chemical reaction must be solved simultaneously. An Eulerian framework was selected so as to include the velocity terms. The con- tinuity equation (Eq 10). Darcy’s Law (Eq 11). and the energy equations (Eq 12, 13, 14) were discretized

r, r,

following a finite difference approach. A differential/ algebraic system solver provided by Brown et al. (19) is incorporated to solve this system of finite difference equations (20).

Case 2: Constant Volumetric Flow Rate Injection

For constant volumetric flow into the mold, an arm lytic solution for the flow field was obtained. Based on conservation of volume, the fluid velocity in the radial direction is

where Qis the flow rate and H is the mold thickness. The radial velocity for the constant volumetric flow case does not depend on the pressure gradient. The energy equations (Eqs 12, 13, 14), need only be solved, since the velocity can be explicitly calculated. This system of equations was discretized in space (central differences) and time (backward Euler). The discretized set of equations was solved on the commercially available software package provided by Brown et al. (19).

4.4 Optimization Technique

Several methods can be applied to minimize the difference between the estimated and experimental temperature history data. The least squares method, which can be traced back to the German mathemati- cian Gauss (1777-1855). can provide a best fit to the observed data points (21). The least squares method, was selected.

In this study, a set of experimental temperature data ?e at one sensor location was recorded. N data points were recorded at that sensor location during the molding time. If temperature is assumed to be a function of the kinetics parameters, @, then the simulated temperature history at that sensor location?s, is a function of these parameters, such that fs = T~.). [For example, there are six parameters in a step-wise resin kinetics model (22),@ = (Al, 4, El, Q, n,, R&I By the least square method (23). the residual vector E is:

+ - - E = T,- T,

Since the simulated temperature dataTS is assumed to be only a function of the kinetics parameters$, different simulated temperature data can be obtained using different kinetics parameter values. The optimization problem here is to minimize the difference between the simulated and experimental temperature data by changing the kinetics parameter values.

The normf(@) is the function to be minimized. The minimization can expressed as (24):

rn?[j(i;) = i~ El (33) P

whereET is the transpose of the residual vector. A nonlinear least squares method was used to minimize the norm.


Robert J. Duh, Susan Mantell, Jefiey H. Vogel, and Robert S . Maier

For nonlinear least squares minimization, the Lev- enberg-Marquardt method (25, 26) is both robust and efficient, and has become a standard for nonlinear least squares curve fitting (1). The directions and step sizes necessary to change the parameter values can be determined given the first and second partial derivatives. The norm f i s a function of parameter vector$: and is expressed as:

f =f& + f t t A f i (34)

Since f' = 0 when freaches a minimum, the change in the parameter vector can be calculated by

L - I

The norm can be expressed as

f = C ( f S - F e y Thus the first partial derivative of the norm is

(35)

(37)

the second partial derivatives term can be calculated as

Since Ts ii- Fe when f reaches minimum, the second partial derivative term is

(391

User specifled relative and absolute convergence cri- teria are used to check for convergence. When the change of the norm value is smaller than the relative convergence criterion, the relative convergence is reached. When the norm value is smaller than the absolute convergence criterion, the absolute convergence is reached. If either convergence is reached, the estimation is stopped, and the parameter values are acceptable. If neither convergence is reached after a certain number of simulations are completed, the estimation process will be stopped and the user is prompted to restart with a different set of initial values. The current simulation run limit to stop the estimation process is set at 1000.

Because SRIM/RTM is a n extremely non-linear problem, the current estimation results show a slaw convergence: anywhere from 40 to 200 simulations with different parameter values are needed to find the best parameter values. Experimental temperature data are the only independent variables used for the regression and kinetics parameters. An exotherm is important for the parameter estimation. When the heat generated by the chemical reaction is too small or too slow to increase the temperature at the sensor location, there will be no exotherm peak. When there

is no exotherm, the resin temperature change is caused by heat transfer from the mold. The resin tern-. perature is not influenced by the kinetics values. In this "no exotherm" scenario, regardless of kinetics values, the simulated temperature curves will be the same. Material properties and molding conditions have the same heat transfer effect. Therefore, the normf(5) will not change. There will be no difference between the norms. There will be no searching direction for parameter estimation and the estimation will fail.

5 RESULTS

For verification purposes, the parameter estimation technique was applied to both constant volumetric flow rate and constant pressure injections conditions. Specifically, temperature data published by Chen (1 3) and Gonzales-Romero (3) were used to demonstrate parameter estimation for injection into a radial mold under constant pressure and constant volume conditions, respectively.

Chen (13) injected a urethane resin (consisting of 100 parts by weight of 1,l-bis (4-cyanatophenol) ethane monomer, two parts by weight of nonylphenol and 0.15 parts by weight of 8% zinc naphthenate). Because this is a thermally activated material, the stepwise kinetics model (Eq 1) was selected for parameter estimation. The resin was injected into a radial mold with a 10.5 cm radius. The mold cavity height and mold temperature were varied. Porosity of the fiber preform ranged from 0.82 to 0.84. The thermai- couple was located near the midplane of the cavity at a radial position 5 cm away from the inlet port. Mold injection conditions are summarized in Table 1. Fig- ures 4 through 7 compare temperatures predicted b y the parameter estimation model with Chen's expern- mental data. Six parameters were estimated for each run: A,, 4, El, &, n, and "2. In each case, the estimated temperatures accurately predict the resin exotherm. For the run at 132°C mold wall temperature, there was no exotherm (-. 4). The model predictions of temperature agree well even in this case. Table 2 summarizes the kinetics parameters values for these runs. From the Table, it can be seen that each run produces a unique set of parameters. These results indicate that there is more than one set of parameters which can represent the cure kinetics. To ensure that each set of parameters represents the op- timum solution for a particular set of data, the parameter estimation was repeated for several initial parameter values. Regardless of the initial trial vector, the optimization converged to within 5% of the values reported in the Table.

Parameter estimation was also performed for constant volumetric flow rate conditions. Gonzalez- Romero (3) studied a styrene-dimethac-rylate (SDM) copolymer that has a rapid chemical reaction. Be- cause this resin is considered a mix-activated resin, a chain-wise kinetics model was selected for parameter



Table 1. Mold Injection Conditions for Chen’s Data.

132 140 150 157

10.5 10.5 10.5 10.5

ParameterlMold Temperature (“C)

cavity radius R, (cm) cavity height H (cm) porosity @ injection pressure P, (MPa)

296 304 299 300 inlet resin temperature To(K) mold temperature T,(K) 405 41 3 423 430 r(cm)’ 5.0 5.0 5.0 5.0

Filling time 52 108 22 42

0.399 0.742 0.432 0.450 0.81 9 0.805 0.836 0.842 0.20 0.07 0.35 0.08

5** 0.4 0.1 0.1 0.1

*r is sensor location from center gate “5 is sensor location from cavity center plane

Flg. 4. Temperature us. time for Chen’s 121 experimental conditions JlOa (Table 1) . mold temperature is 132°C. initial temperature is 23°C.

Fig. 5. Temperature us. time for Chen’s 12) experimental conditions M29 (Table I ) , mold temperature is 140°C. initial temperature is 31 “C.

300

250

200 h c

8.

2 a 2 150 Y

E + 100

50

0

300

250

h

2 200 2 a Y

2-D simulation - experimental data 0

0 100 200 300 400 500 600

Time (sec)

I I I

0 100 200 300 400 500 600

Time (sec)


Robert J. Duh, Susan Mantell, Jefley H. Vogel, and Robert S. Maier

Fig. 6. Temperature us. time for Chen’s (2) experimental conditions 512 (Table 11, mold temperature is 150°C. initial temperature is 26°C.

300

250

200 h

u, 2 2 150 %

1 Y

5 r- 160

50

0 0

2-D simulation - experimental data 0

estimation. Gonzklez-Romero conducted six experiments in a radial mold (10 cm radius] at flow rates ranging from 22 to 28 cm3/sec. The mold cavity depth ranged from 0.52 cm to 0.63 cm and fiber preform porosity was 0.764. The temperature sensor was located 3.2 cm from the mold center near the cavity center line. Mold injection conditions are summarized in Table 3. Kinetics parameter estimates for each of the six experiments are shown in Table 4. Figure 8

Flg. 7. Temperature us. time for Chen’s (2) experimental conditions J11 (Table I), mold temperature is 157°C. initial temperature is 27°C.

100 200 300 400 500 6001

Time (sec)

shows a typical comparison between predicted temperatures and data for these injection conditions. Gonzalez-Romero also obtained kinetics parameter values by curve fitting the chain-wise kinetics model to DSC data for this same resin. To demonstrate the significance of the proposed parameter estimation technique, temperatures for Gonzales-Romero’s ex- periment at 24 cm3/sec were predicted using the average of the values reported in Table 4 and also using

250

200 h u, 2 1

2-D simulation - H 150 E

experimental data 0

100 G

50

0 1 I I I I 1

0 100 200 300 400 500 600

Time (sec)



Table 2. Kinetic Parameter Estimation for Chen's Data: A,, A,, E,, 5, n,, and n,.

A1 A2 €1 E2 n1 n,

140 577095 143928 8245.8 881 6.7 3.02 1.11 150 4921 53 232272 8319.1 6572.9 0.94 1.38

Mold Temperature ("C)

132 592643 108790 8235.3 5000.4 2.59 1.17

157 784390 191221 8037.2 5874.3 3.99 1.79 Initial 283587 92055 8220.0 1625.0 3.00 3.00

Table 3. Experimental Conditions for Gonzalez-Romero Data. ~~ ~ ~ ~~~~~

Parameter Exp. 1 Exp. 2 Exp. 3 Exp. 4 Exp. 5 Exp. 6

cavity height H (cm) 0.59 0.58 0.63 0.59 0.52 0.53 porosity @ 0.764 0.764 0.764 0.764 0.764 0.764 flow rate Q(cm3/sec) 24 22 25 24 27 28 inlet resin temperature To (K) 303 297 294 303 294 300 mold temperature T,,, (K) 395 41 1 399 395 407 41 5 r (crn)' 3.2 3.2 3.2 3.2 3.2 3.2

cavity radius R,,, (crn) 10 10 10 10 10 10

5" 0.1 0.1 0.1 0.1 0.1 0.1

* r IS smnsor lmation fmm center gate "c is sensor location from cavity center plane

Table 4. Kinetic Parameter Estimation for Gonzalez-Romero Data: A, Ep B,, B,, and k/k, .

Exp. Ax Ex 8, ( x 1 O-zo) 0 2 k/kP

#1 945760.55 -8463.2 1 8.6937 191 05.07 27.12 #2 587310.07 -8346.00 6.2727 19207.1 0 7.23 #3 9431 16.1 8 -8123.96 9.0163 19030.03 17.13 #4 105621 8.63 -8413.79 9.1866 19166.94 23.89 #5 920659.00 -8466.69 8.7324 19185.81 17.11 #6 1051 509.41 -8571.69 9.0598 1931 1.1 0 11.29 Initial 1012000 -8453.00 9.2278 19093.00 25

Fig. 8. Temperature u s . time using estimated parameter ual- ues OfA, Ex, B,. B2. and kJk, at Gonz&z-Romero's (3) ewperimen- tal conditions # I (Table 3).

I I $ I 0 50 100 150 200 250 300

Time(sec)


Robert J. Duh, Susan Mantell, Jefley H. Vogel, and Robert S. Maier

Table 5. DSC Obtained Kinetic Parameters From Gonzalez-Romero.

Parameter Value

1.15 X lo6 2.05 X -8.63 x 103

1.91 x 104 25

the DSC obtained parameters (Table 5). Figure 9 shows the two predicted temperatures compared with the data. The DSC obtained parameters predict the exotherm peak and shape accurately but the timing of the peak is too soon. On the other hand, the parameters obtained from molding conditions accurately pro- duce the exotherm peak, shape, and timing. This result demonstrates the significant improvement in predicting cure kinetics by the proposed approach.

6 CONCLUSION

A new approach to estimating kinetics parameters for FTM and SRIM applications has been presented. Parameters are obtained by curve fitting a kinetics model to temperature data recorded during actual molding. Models for the thermal behavior during molding under constant pressure and constant flow rate were developed and linked to an optimization algorithm. The user provides an initial estimate for the parameters and the model updates the parameters until the temperature data are predicted.

Rg. 9. Comparison of temperature us. time with DSC parameter vd- ues and estimated parameter vd- ues of A,, E, B,, B,, and kJkp at Gonzcilez-Romem’s (3) ewperimen- tal conditions # 1 (Table 3).

This parameter estimation method has been vali- dated for radial and rectangular geometries under constant flow rate filling and constant pressure filling (20). Different types of resin, both mixing activated and thermally activated systems, are included in these validation studies. A case study is presented in which temperatures during molding were predicted using kinetics parameters obtained by a DSC and those obtained by the proposed optimization algorithm. In this case study the parameters found by optimization more accurately predict the resin temperature during molding.

The optimized technique is a powerful approach that can be applied to not only predicting kinetics parameters during molding, but also other molding constants required to simulate processing (including: thermal conductivity between the fiber and resin, and fiber preform permeability). These properties are often difficult to measure directly through experiments. By comparing model predictions of temperature or flow front position with experimental data, these parameters could be varied until the difference between the model and the data is a minimum.

7 ACKNOWLEDGMENTS

The authors appreciate the Army Research Labora- tory’s funding of this study. Additional thanks go to Professor Christopher Macosko at the Chemical Engi- neering and Material Science department, University of Minnesota, who provided the chemistry information and the experimental results.

250

50 simulation (estimated average) __ simulation (DSC) - - ~ ~ - ~

experimental 0

I I I I

0 50 100 150 200 250 300

Time (sec)



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