Isocyanate trimerization kinetics and heat transfer in structural reaction injection molding.
The structural reaction injection molding (SRIM) process is used to fabricate composite parts for automotive applications and other uses in which high volume production is necessary. In SRIM, a woven or nonwoven glass fabric preform is placed in a mold and resin is injected. Fast reacting systems are used for low cycle times. The most common type is isocyanurate in which polyol and isocyanate are reacted to form primarily crosslinked polyurethane and cyanurate trimer. In the present analysis, we have considered a proprietary commercial system of this class and addressed issues of creating a kinetics model and then modeling simultaneous reaction and thermal effects during and after mold filling.
The problem addressed is optimization of the balance between the need for a long flow time for large parts and the achievement of relatively complete reaction. We have observed blistering and surface blemishes on large parts that are subjected to high temperature paint bake ovens after molding (1, 3). This has been found to be associated with the presence of residual isocyanate which can react with moisture to produce amine and carbon dioxide, or, it can react with itself to form carbodiimide and carbon dioxide. Increasing the rates and extent of reaction in molding leads to lower residual isocyanate concentration, higher glass transition temperatures and reduced cycle times. Other perspectives on how insufficiently vigorous reactions can lead to premature vitrification and high residual isocyanate levels are discussed in another paper (1).
A mechanistic model of the simultaneous reactions to form urethane and isocyanate trimer has been proposed by Vespoli and Alberino (4). They determined the material constants in a series of reaction equations for a model system. Here we have concentrated on understanding the kinetics associated with a commercial isocyanurate used for molding large parts via SRIM. The kinetics of this system are much slower than those of the model system, but the mechanism of reaction is similar. In order to better understand blistering, the material parameters in the Vespoli/Alberino kinetic model were determined for our resin. The completed kinetic model was incorporated into a numerical model of the SRIM process taking into account the effects of combined heat transfer and reaction.
An in-depth analysis of the numerical simulation predictions has led to the development of some ideas for improving the SRIM process. For instance, the numerical simulations indicate that hot molds should improve isocyanate depletion at the part surface. Further work has shown that three dimensionless groups govern the SRIM process: a Nusselt number and two other groups. Of the three groups, the Nusselt number seems to exhibit the most interesting effect.
In the present case, the Nusselt number relates the ratio of heat transfer between resin and pro-form to that between resin and mold. At high Nusselt numbers, heat transfer between the resin and preform dominates. This results in a longer gel time, a predominance of the urethane reaction, and an increase in the amount of residual isocyanate. Interestingly, the Nusselt number itself depends on the preform geometry (e.g., porosity and filament diameter). The amount of heat transfer between the resin and preform also depends on the initial preform temperature. These results suggest that intelligent choice of preform construction and initial temperature offers another way to control the processability of SRIM resins. Correct selection of preform material could improve isocyanate depletion and reduce blistering and/or increase the gel time on a selective basis.
The work presented here was accomplished in two distinct phases. First the kinetic parameters in the mechanistic model were determined. This required only a simple energy balance consistent with adiabatic temperature rise measurements. The completed kinetic model was then combined with a more complicated thermal analysis of an SRIM mold. This division of work provides for the framework of the paper. We first present the kinetic model, its form, the manner in which the parameters were determined, and the results of the parameter fitting exercises. We then describe the thermal model, the results of nondimensionalization of the energy equations, and the major predictions of the complete kinetic/thermal simulation of the SRIM system.
REACTION KINETIC MODELING
It is a common situation that proprietary resin systems are available to the molder for which the chemical analysis is unknown. In the present case, we are presented with a commercial system consisting of:
1. 4,4[prime] methylene diisocyanate (MDI) with its oligomers and methylene bisphenyl isocyanate, a proprietary catalyst or blocking agent, and;
2. A proprietary polyol to which is added a catalyst (triethylene diamine in a solution of dipropylene glycol).
The equivalent weight of the isocyanate is 153 and of the polyol, 212. Our approach to developing a reaction kinetics model has been to utilize the kinetic scheme presented by Vespoli and Alberino, which assumes that there are two primary simultaneous reactions occurring that produce polyurethane and isocyanate trimer. It is common, however, for commercial systems to contain a blocking agent, such as oximes or phenols, which tie up isocyanate until a temperature of 80 to 150 [degrees] C is achieved in the system, thereby hindering low temperature trimerization of isocyanate. Furthermore, isocyanate can react with itself at high temperatures in an endothermic reaction that produces carbodiimide and carbon dioxide. Thus, to assume that only polyurethane and isocyanate formation occurs is a simplification.
In order to gather data for material parameter determination, adiabatic temperature rise experiments were conducted for various ratios of isocyanate to polyol and different catalyst levels. Parameters for the reaction kinetics were found by an optimization routine that minimizes the error between the actual function of temperature vs. time and that predicted by the model. The reaction model developed using the discovered parameters was then used to develop the thermal model, which is discussed later in this paper.
Adiabatic Kinetic Model
The urethane reaction is straightforward and can be modeled by the following mechanism according to Steinle, et al. (5):
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
The following mechanism is used for the trimerization reaction as proposed by Kresta and Hsieh (6):
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
where we have refrained from using chemical symbols for the sake of clarity. Equations 1 through 7 can be solved simultaneously to yield reaction rate equations for the depletion of polyol, the formation of urethane, the depletion of isocyanate and the formation of trimer (4):
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
Isocyanate participates in both of the reactions above. such that:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
where the following assumptions have been made:
* Pseudo Steady State (i.e. intermediate reactions are at equilibrium).
* Arrhenius temperature dependence of reaction constants.
* [K.sub.1] [is greater than] [K.sub.-1], [K.sub.2] [is much greater than] [K.sub.1], [K.sub.4] [is much greater than] [K.sub.T] and [K.sub.2] = [K.sub.3] = [K.sub.4].
There are seven material parameters in Eqs 8 and 9 that must be determined in order to correlate the behavior of the isocyanurate system: [K.sub.C1], [K.sub.C2], [E.sub.C], [K.sub.A1], [K.sub.A42], [E.sub.A1], and [E.sub.A42]. In addition to the kinetic expressions, a simplified energy balance must also be solved to yield the predicted adiabatic temperature rise. This balance has the following form:
[[Rho].sub.r][Cp.sub.r] dT/dt = ([Delta][H.sub.U][R.sub.U] + [Delta][H.sub.T][R.sub.T]) (13)
where [R.sub.U] and [R.sub.T] are the rates of reaction for polyurethane formation and isocyanate trimerization respectively, and the [Delta][H.sub.i] values denote their respective heats of reaction. The initial conditions for the system of equations above comprise the following:
At t = 0
[C.sub.OH] = [C.sub.OH, o], [C.sub.I] = [C.sub.I, o], [C.sub.U] = [C.sub.T] = 0, T = [T.sub.0]
Equations 8, 9, 10, and 13 constitute what we will refer to as the Adiabatic Kinetic Model. This form of the model was used to determine the kinetic parameters. The more complicated form of the model (used to describe the SRIM process) is presented later.
Adiabatic temperature rise was used to assess the reaction rate of the isocyanurate system. Eighteen different experiments were run corresponding to 14 different combinations of catalyst level and reactant composition. For all cases, the amount of catalyst and polyol added was based on the volume of isocyanate resin used. Table 1 lists all of the pertinent information for each test. The isocyanate index was varied from a low of 0.5 to a maximum approaching 10. This covers a far greater range than the practical use limits recommended by the supplier.
Adiabatic Test Apparatus
Figure 1 shows a schematic of the adiabatic test apparatus used to collect the data. Its major components include:
* An insulated adiabatic chamber.
* A pneumatic powered cartridge gun, which drives resin from two disposable plastic cartridges through a disposable plastic in-line mixer.
* A high speed digitizing temperature measurement system using a thermocouple immersed in the adiabatic chamber.
* Pressure relief and control equipment.
The two reaction components are mixed rapidly in the helical in-line mixer while they are injected by the gun into the chamber. The TS525 Pneumatic Powered Cartridge Gun, from Techcon Systems Inc., is designed to accommodate any combination of 75, 150, and 300 ml Ratio-Pak cartridges. The temperature measuring system consists of a Datascan Universal Analog Input and Output Module (Model #7050) and a Datascan Intelligent System Interface Module (Model #7010), using a J thermocouple. The chamber consists of an 8 mm thick disposable steel cup 100 mm in diameter and 50 mm deep surrounded by polystyrene foam insulation.
The thermocouple is mounted in the test chamber a distance of 15 mm from the bottom. Tape is attached to the cover of the cup near an injection orifice so it can be plugged after resin injection. Two rubber fill plugs are then fitted in the outlet ends of the Ratio-Pak cartridges. The resin components are weighed directly into the cylinders and disposable syringes are used to make minor weight adjustments. The cylinders are then purged with nitrogen and the plungers are inserted in the inlet ends of the cartridges.
The cylinders are inverted, the fill plugs removed, and the cylinders mounted into the cartridge gun. The air supply valve is opened and the bleed valve closed. The test chamber is purged with nitrogen, the gun inverted, and the contents quickly shot in the chamber by increasing pressure to the gun to 30 psi. The tape is placed over the injection hole after the gun is removed. Insulation is then placed over the top of the chamber. After the test, the cartridges, in-line mixer, and test chamber are discarded.
Figures 2 and 3 show typical results generated by the adiabatic testing apparatus. Figure 2 shows plots TABULAR DATA OMITTED of temperature vs. time for isocyanate to polyol volume ratios of 1:1, 1:2, and 1:4. At these volume ratios, the level of polyol is relatively high and so a snap cure is not obvious (except perhaps for the 1:1 system). The temperature profiles shown in Fig. 2 are typical of the urethane reaction; the temperature rises slowly and eventually levels off. All three profiles shown in Fig. 2 are for a catalyst level of approximately 1 wt% (based on initial polyol concentration). Increasing the catalyst level increases the rate of reaction linearly. This increase in reaction rate results in a shorter gel time, which can be deduced from the results in Table 1. Note that the curves in Fig. 2 and 3 are the model predictions and indicate how well the model can correlate experimentally observed behavior.
Figure 3 shows temperature versus time data for higher isocyanate index systems (volume ratios of 2:1 and 4:1). Here the two-step nature of the curing reaction is clearly visible as is the snap-cure phenomenon. We can see from the Figure that the urethane reaction proceeds until a temperature of approximately 80 [degrees] C at which point the trimerization commences. Unlike the urethane reaction, trimerization causes an almost instantaneous temperature rise in the system accompanied by rapid conversion of isocyanate.
Although trimerization proceeds quickly, it may take a long time to get started. For instance, Fig. 3 shows that the 2:1 system cures in about 30 s while the 4:1 system takes more than 100 s to cure. This effect is most likely related to the low volume of polyol in the system resulting in less urethane production. It takes longer for this system to reach the 80 [degrees] C trimerization kick-off temperature. Urethane is also thought to catalyze trimerization, so, until enough urethane is present in the system trimerization occurs slowly (2). These factors combine to explain why the 4:1 system takes longer to cure than the 2:1 system. It also shows that a balance must be maintained between the level of polyol and isocyanate in the initial reactant mixture. Too much polyol and the snap-cure will not occur; too much isocyanate, and the gel time increases.
RESULTS OF KINETIC STUDIES
As stated in the Introduction, our purpose for studying the reaction kinetics of the isocyanurate system was based on a desire to understand the conditions that cause blistering. We chose the Vespoli and Alberino study as a starting point for our work. However, because the exact chemical composition of our commercial isocyanurate system is proprietary, we had no way to know if the model would work for our resin system. Using the Vespoli/Alberino model was a roundabout way of determining the chemical nature of our resin (i.e. if the model worked, then the chemical composition must be quite similar to that used in Vespoli and Alberino's study).
Upon examination of the first results from the adiabatic test apparatus, it was obvious that our resin system is different from the model system. The reaction rate of our system is an order of magnitude slower. While this result did not rule out the possibility that our system is related to the model system it did indicate that new material parameters would have to be determined. Using the adiabatic test apparatus, 18 different sets of data were taken corresponding to 14 different conditions of catalyst concentration and isocyanate index; each data set consisted of 100 time/temperature pairs.
The fitting of the kinetic model to data has involved finding the material parameters that cause the numerical simulation to predict an adiabatic temperature rise equal to that measured experimentally. For each experimental test, the operating parameters were first input to the model (e.g. isocyanate index, initial temperature, duration etc.). The model then predicted the resin temperature as a function of time. At each data point in time, the difference between the actual and predicted temperature was determined and summed to form an error function. An IMSL routine (DUMPOL) was used to minimize this error function. The kinetic model parameters constituted nine degrees of freedom for the optimization (seven parameters and two heats of reaction). This process usually converged after several hundred iterations using the Vespoli/Alberino parameters as an initial guess.
Use of Statistical Methods
The parameters in the kinetic model should, theoretically, be constants. Therefore, in order to correlate all of the data over the whole range of isocyanate indices, one set of constants should be found. This proved to be most difficult. While a set of parameters could be found to correlate the data over a small range (i.e. all the data with the same isocyanate index) no single set of parameters could be found to correlate the entire data range. This is not surprising since Vespoli and Alberino also experienced difficulty describing the behavior of their system at large isocyanate indices (4). Some statistical methods have been used, therefore, in an attempt to produce a model that can correlate a wider range of operating variables.
Individual Data Set Analysis
Rather than try to find a set of material parameters for all or even a subset of the data, we began by optimizing the parameters for each data set. Thus, for 18 data sets of temperature vs. time, 18 sets of material parameters were found for the kinetic model. This data is tabulated in Table 2. With 18 values for each material parameter, some basic statistical functions could be evaluated, such as mean and standard deviation. These values (tabulated in Table 2) could then be used to determine which data sets were not in keeping with the rest of the data.
A specific example may help to clarify the procedure outlined above. Notice in Table 2 that there are 18 values of [K.sub.C1] (one for each data set). The mean of these values is 1217.40 with a standard deviation of 447.8. Notice that for data set 14.10, the optimum value of [K.sub.C1] found is 548.42, almost two standard deviations away from the mean. Likewise, the other parameter values determined for data set 14.10 are consistently off of the mean by at least one standard deviation. Based on these results, the data from set 14.10 were set aside and not used to determine the final optimum set of material parameters. In all, five of the 18 data sets were set aside in this manner.
TABULAR DATA OMITTED
Another benefit of determining multiple values of the material parameters was that plots of these values could be made as a function of chemical composition. Based on the manner in which we conducted our experiments, we plotted the parameters as function of initial isocyanate concentration ([C.sub.I, o). These plots suggested that some of the parameters might best be relaxed from constants to functions of [C.sub.Io]. Observe the dependence of [H.sub.RT] on [C.sub.I, o] shown in Fig. 4. These data indicate a linear dependence of [H.sub.RT] on [C.sub.I, o]. Likewise, Fig. 5 shows that [E.sub.A1] exhibits an almost parabolic dependence on [C.sub.Io].
The argument can be made that the results depicted in Figs. 4 and 5 are artifacts of the manner in which the parameters were determined and bear no further analysis. In order to test this hypothesis, a further statistical analysis was conducted using Statgraphics, a commercially available statistics software package. For each material parameter in the kinetic model, the coefficients in the following function were determined using the 18 values found from the optimization calculations:
[P.sub.i] = [[Lambda].sub.1] + [[Lambda].sub.2][C.sub.C] + [[Lambda].sub.3][C.sub.I, o] (14)
where [P.sub.i] refers to the parameter and [[Lambda].sub.i] are the coefficients. Equation 14 tests for a linear dependence of the material parameters on initial catalyst and initial isocyanate concentration. The null hypotheses [[Lambda].sub.2] = 0 and [[Lambda].sub.3] = 0 were tested for a Type I error (i.e. the error of setting [[Lambda].sub.2] or [[Lambda].sub.3] to non-zero constants when in fact they should be zero). In such a test, the p-value indicates the probability that a Type I error is committed. Thus, a p-value of 0.05 indicates a 5% chance that a coefficient in Eq 14 should be zero or conversely, a 95% confidence that the coefficient is non-zero. Table 3 shows the p-values calculated for the coefficients in Eq 14 for four of the nine material parameters in the kinetic model ([K.sub.C1], [K.sub.A1], [E.sub.A1], [H.sub.RT]). These four parameters were the only ones for which the p-values were less than 0.05. Thus, they are the only parameters for which a dependence on [C.sub.C] or [C.sub.I, o] can be justified based on the experimental data. It is worthwhile to note that the p-values calculated for [E.sub.A1] and [H.sub.RT] are among the smallest found and indicate a 98 to 99% certainty that these parameters are at least a function of [C.sub.I, o].
Table 4 shows the final results of the parameter fitting exercise. The original constants found by Vespoli and Alberino are shown for comparison. Besides the fact that [E.sub.A1] and [H.sub.RT] are now functions of composition, other differences can be seen. The urethane frequency factors and activation energy found are an order of magnitude smaller than those found for the model system. This means that the urethane reaction proceeds much slower for our system. This may be the result of a deliberate modification designed to improve the modulability of the resin for large parts requiring longer flow times and hints at the presence of a blocking agent (9).
For engineers interested in correlating the behavior of a resin system, setting some of the material parameters to be functions of initial composition is an expedient method of achieving the desired results. To those well versed in chemistry and thermodynamics, however, this is troublesome. For instance, if one assumes that the dependence of [H.sub.RT] on [C.sub.I, o] is real and not an artifact, it would suggest that an unaccounted reaction (or combination of reactions) is occurring in the isocyanurate system. The nature of the dependence of [H.sub.RT] on [C.sub.I, o] suggests an endothermic reaction that scales with initial isocyanurate concentration. It is known that carbodimidization is endothermic and can occur with isocyanates (4), but it should only occur at higher temperatures. The dependence of [H.sub.RT] on [C.sub.I, o] suggests a reaction that occurs at all compositions and temperatures.
Table 3. P-Values Calculated by Statgraphics. P-Value P-Value Parameter [[Lambda].sub.2] [[Lambda].sub.3] [K.sub.C1] 0.08 0.0079 [K.sub.A1] 0.036 0.039 [E.sub.A1] *** 0.021 [H.sub.RT] 0.073 0.0018 Table 4. Finalized Kinetic Model Parameters. Vespoli Current Parameter Model(3) Model [K.sub.C1] 6078.00 978.00 [K.sub.C2] 2.97 16.69 [E.sub.c] 5482.00 548.00 [H.sub.RU] 22,595.00 28,095.00 [K.sub.A1] 15,890.00 2883.00 [K.sub.A42] 16.78 1.26 [E.sub.A1] 14,173.00 F([C.sub.I,o]) [E.sub.A42] 6932.00 8156.00 [H.sub.RT] 42,567.00 F([C.sub.I,o])(*) [T.sub.ref] 100.00 100.00 * Specifically: [H.sub.RT] = -5.2055 x [10.sup.6]([C.sub.I,o]) + 6.133267 x [10.sup.4].
We hasten to point out that putting all our eggs into the [H.sub.RT] basket may not be correct. The present situation represents a classic case; that of trying to fit numerous parameters simultaneously. Looking at Eqs 9 and 13 one will notice that an unfortunate combination of parameters arises in the energy balance--[H.sub.RT][K.sub.A1]. Further statistical analysis is required to determine whether it is [H.sub.RT] or [K.sub.A1] that should vary. We have presently not taken the steps to resolve this issue. Either way, the result is the same: the combination of kinetic expressions provided by Hsieh (5) and Steinle (6) is not totally sufficient to describe the polyurethane/isocyanurate system.
A more in-depth analysis is required that will account for other competing reactions and the function of blocking agents. We have empirically accounted for these effects by allowing some of the parameters to vary with initial composition. A more rigorous approach would provide the basis for more understanding. Hopefully, the manner in which we have empiricized our kinetic model provides clues as to the final form of the new kinetic model.
MODELING OF THERMAL EFFECTS IN SRIM
Having found a kinetic model that correlated the data reasonably well, we next turned our attention to modeling the combined heat transfer and reaction effects that take place in SRIM. This process is unique in that it introduces the presence of a fibrous preform within the mold. The preform dramatically changes the thermal balances and, therefore, affects the chemical reaction dynamics. Specifically, the preform affects the material and energy balances in two ways:
1. There is a straightforward volume effect whereby the amount of reacting resin is decreased because of the presence of glass. Thus, on a per volume basis, less heat is generated in the mold. The greater the glass fraction, the greater the influence of heat transfer within the mold and/or the pre-form relative to chemical heat generation.
2. The preform acts as a heat sink and mediates the maximum temperature rise seen in the mold. Thus, the preform can be used to extend the gel time of incoming resin, at least away from the mold wall.
The effects above can be quite large and can affect the gel time of the resin (1). Taken to an extreme, they provide new operating variables that can be used to optimize the SRIM process. In the following analysis, the energy balance is developed for the case in which heat transfer between the resin and mold and between the resin and glass are accounted for. We show that there are design and operating variables associated with the existence of the preform that allow optimization of processing windows.
The present analysis is based on energy balances within the resin and the glass phases which consider heat transfer from the mold wall to the resin and heat transfer between the resin and glass phases. In SRIM, there is a fill step in which resin flows along with simultaneous reaction and a cure step after flow ceases that features continuation of the reactions. In the following development we model reaction in an element of resin at the leading edge of the flowing resin in the fill step to calculate gel times. We can also calculate the behavior of arbitrary elements of resin during the cure step using appropriate initial conditions that correspond to the conditions of the flow element after undergoing the fill step.
Figure 6 gives a physical picture of the conditions existing in an SRIM mold. The injected resin can transfer heat between two media: the mold and the preform. For simplicity, heat transfer between the preform and the mold has been neglected based on the fact that the thermal conductivity of glass is quite small. Heat transfer between the resin and mold wall occurs via conduction while heat transfer between the resin and glass occurs through convection. The following assumptions are also made:
* Negligible heat transfer in the axial (flow) direction, allowing for a one-dimensional analysis through the part thickness.
* Constancy of material parameters such as density and heat capacity.
* Negligible viscous heating.
* Negligible intra-preform conduction (as well as to/from the mold wall).
With these assumptions and Fig. 6, the energy balance can be written as:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
where two equations are needed: one for the preform and one for the resin. The subscripts on temperature and material properties indicate glass or resin. Note that these equations must be solved simultaneously with Eqs 8, 9, and 10 to constitute the SRIM kinetic model (as opposed to the adiabatic thermal model presented earlier).
The quantity [A.sub.v] in Eqs 15 and 16 defines the surface area of glass per unit volume of composite and has the following form for continuous strand random, or chopped random glass preforms:
[A.sub.v] = 2(1-[Phi])/r (17)
where r is the glass filament radius and [Phi] is the porosity of the preform.
The initial and boundary conditions used with Eqs 15 and 16 in this study are as follows and are shown in Fig. 6:
At t = 0 [T.sub.r] = [T.sub.ro], [T.sub.g] = [T.sub.go]
At X = 0 [Delta][T.sub.i]/[Delta]X = 0
At X = H/2 [T.sub.i] = [T.sub.wall] (isothermal case)
[Delta][T.sub.i]/[Delta]X = 0 (adiabatic case)
where the flux at X = 0 is zero because of an axis of symmetry (cf. Fig. 6). At X = H/2, one of two different boundary conditions are satisfied depending on whether an adiabatic or an isothermal analysis is desired. For parameter fitting, the adiabatic analysis was used while for mold flow simulation, the isothermal analysis is more realistic for thin parts.
Besides the different boundary conditions that may exist in an SRIM mold, there are also different types of analyses that are useful: Cure and Fill. The equations derived above are exact for the case where no significant reaction takes place during filling of the mold. This situation arises in small parts where the fill time is substantially shorter than the gel time. In this instance, the equations describe what will take place as the part cures (the Cure case). The initial conditions shown above are still valid, but the initial resin and preform temperature will actually be those existent after the part has been filled.
Equations 15 and 16 can also be used to describe what takes place at the flowing front of the fluid (the Fill case). This is accomplished by imbedding the coordinate system within the fluid and letting it convect with the front. In this case, the partial derivatives, [Delta][T.sub.i]/[Delta]t, in Eqs 15 and 16 must be replaced with substantial derivatives, D[T.sub.i]/Dt, which include a convection term: [V.sub.z]([Delta][T.sub.i]/[Delta]Z). We have assumed, however, that negligible axial conduction exists such that [Delta][T.sub.i]/[Delta]Z is zero. Thus, the convection term is neglected and Eqs 15 and 16 are still valid. The glass temperature will not change since the fluid front will always see glass at its initial temperature. In this sense, Eq 16 is not solved and [Delta][T.sub.g]/[Delta]t is set to zero. The differences between the Cure and Fill cases will become apparent when the predictions of the model are discussed.
Note that our analysis is basically similar to that presented by Gonzales (7). However, we have chosen our single dimension to be part thickness rather than flow length. Thus, Gonzales's analysis preserves the convection term and neglects the conduction term; reaction of resin is also neglected in the fill case because a slow reacting system is selected. These differences become more apparent when the equations are nondimensionalized. The convection term yields a Stanton number whereas our analysis results in a Nusselt number (related to the conduction term) and two other groups (related to the reaction terms) (8).
Nondimensionalization of the Governing Equations
It is instructive to non-dimensionalize Eqs 15 and 16 in order to simplify the analysis of the glass/resin/mold system. We have chosen the following dimensionless variables:
[Mathematical Expression Omitted]
[Tau] = 4[Alpha]t/[H.sup.2]
[Alpha] = k/[Phi][[Rho].sub.r]C[p.sub.r]
[Beta] = [C.sub.OH]/[C.sub.OH, o]
[Mathematical Expression Omitted]
[Xi] = X/H
A = (1 - [Phi])[[Rho].sub.g]C[p.sub.g]/[Phi][[Rho].sub.r]C[p.sub.r]
[Gamma] = 3[C.sub.T]/[C.sub.I, o]
where we have introduced [Beta] and [Gamma], the degrees of conversion for the urethane and trimer. These quantities have been defined based on the initial concentrations of polyol ([C.sub.OH, o]) and isocyanate ([C.sub.I, o]). The quantity "A" is the heat capacity ratio between the resin and preform and will be a constant for a given resin/glass system.
Using the dimensionless variables defined above, Eqs 15 and 16 become:
[Delta][[Theta].sub.r]/[Delta][Tau] = [[Delta].sup.2][[Theta].sub.r]/[Delta][[Xi].sup.2] + [G.sub.U]([Delta][Beta]/[Delta][Tau]) + [G.sub.T]/3([Delta][Gamma]/[Delta][Tau]) - [N.sub.U]([[Theta].sub.r] - [[Theta].sub.g]) (18)
[Delta][[Theta].sub.g]/[Delta][Tau] = [N.sub.U]/A([[Theta].sub.r] - [[Theta].sub.g]) (19)
There are four dimensionless groups in Eqs 18 and 19. One has already been defined (A), the other three groups constitute a Nusselt number and two other groups:
[N.sub.U] = h[A.sub.v][H.sup.2]/4k (20)
[G.sub.U] = [Delta][H.sub.U][C.sub.OH, o]/[Phi][[Rho].sub.r]C[p.sub.r]([T.sub.ro] - [T.sub.go]) (21)
[G.sub.T] = [Delta][H.sub.T][C.sub.I, o]/[Phi][[Rho].sub.r]C[p.sub.r]([T.sub.ro] - [T.sub.go]) (22)
SRIM KINETIC MODEL PREDICTIONS
Basic Model Capabilities
Let us now examine some example calculations and the basic predictions of the SRIM kinetic model. The input variables used for these calculations are shown in Table 5. Note that some of the parameters in Table 5 are constants, such as material parameters, while others are variable, such as operating parameters.
Figures 7 through 10 show the results of the example calculations (Cure case). For Figs 7 and 8, the operating parameters were chosen such that the resultant Nusselt number was equal to 1. This means that heat transfer to the glass will be minimal and that conduction to the mold wall will dominate. Figure 7 shows the resin and glass temperatures as a function of time for the center and near the surface of the part. The initial resin temperature was 70 [degrees] C while the initial glass temperature was only 25 [degrees] C. As time progresses, the Figure shows that the resin temperature in the center of the part goes through a maximum indicative of the reaction exotherm. Near the surface of the part, the resin temperature is mediated by the constant temperature mold surface and remains very close to 70 [degrees] C. The glass temperature profiles are very different from those of the resin. Since the Nusselt number is low for this example, very little heat transfer occurs from the resin to the glass, thus, at both the center and surface, the glass temperature rises little from its initial value.
Table 5. Listing of Input Parameters for Example Calculations. Constants [h.sup.a] = 2.000 x [10.sup.-5] cal/[cm.sup.2] s k = 1.238 x [10.sup.-4] cal/cm-s- [degrees] C [[Rho].sub.r] = 1.000 g/[cm.sup.3] C[p.sub.r] = 4.500 x [10.sup.-1] cal/gr- [degrees] C [[Rho].sub.g] = 2.540 g/[cm.sup.3] C[p.sub.g] = 2.000 x [10.sup.-1] cal/g- [degrees] C Operating Variables [T.sub.go] = 25.000 [degrees] C [T.sub.ro] = 70.000 [degrees] C [T.sub.wall] = 70.000 [degrees] C H = 5.000 x [10.sup.-1] cm [Phi] = 5.500 x [10.sup.-1] [r.sub.[1.sub.b]] = 9.000 x [10.sup.-3] cm [r.sub.2] = 9.000 x [10.sup.-5] cm [C.sub.c] = 8.000 x [10.sub.-3] g/[cm.sup.3] [Mathematical Expression Omitted] [C.sub.OH,o] = 7.500 x [10.sup.-4] mol[/cm.sup.3] a This value is based on Ref. 7. b Note that use of these values for r gives rise to values of [N.sub.u] of 1.0 and 100.0, respectively. c The use of these values gives rise to an isocyanate index of 4.
Figure 8 shows reactant concentration versus time profiles for the center and surface of the part. Looking at both sets of profiles it is apparent that the extent of reaction near the surface is much lower than that at the part center. This is directly related to the temperature profiles shown in Fig. 7. Since the wall temperature is fixed, the reaction rate is slowed considerably. While the residual isocyanate fraction approaches zero in the part center, it barely reaches 0.4 near the surface. Considering the theory that residual isocyanate reacts with water in a foaming reaction, the results in Fig. 8 are highly undesirable. One other facet of Fig. 8 to notice is the low degree of polyol conversion. At both the surface and center of the part, the residual polyol fraction is no less than 0.75. This is due to the relatively small frequency factors calculated for the urethane reaction.
Figures 9 and 10 show the same sets of results as those in Figs. 7 and 8 except that the Nusselt number has been increased from 1 to 100. Now, heat transfer from the resin to the glass preform is dominating. Figure 9 shows the resin and glass temperature profiles vs. time. Notice now that the temperatures of both materials coincide at both the surface and center locations. As a result of increased heat transfer from the resin to the preform, the maximum temperature rise in the center is almost 50 [degrees] C lower than that shown in Fig. 7 (100 [degrees] C vs. 150 [degrees] C). The fact that the resin in the center of the part has been cooled affects the final degrees of conversion. Figure 10 shows the reactant conversions vs. time for the surface and center of the part. There is now a less-pronounced difference between the two part locations. Furthermore, the residual polyol fraction has now decreased to about 0.5. Accompanying this effect, we see that the gel time has increased from about 5 s in the center to more than 25 s; where we have defined the gel time as being the time it takes for 50% of the initial isocyanate to react. This example shows that by increasing heat transfer from the resin to the preform, the gel time increases, the urethane reaction is favored, and a more even degree of cure is achieved throughout the part.
Effect of the Nusselt Number
The example calculations show that changing the Nusselt number can dramatically alter the reaction path. Figures 11 and 12 illustrate this phenomenon in more detail. Here we have plotted the predicted dimensionless gel time in the part center as a function of [N.sub.u] for five different initial preform temperatures; all other operating variables have been held constant. Figure 11 shows results for the Cure case, while Fig. 12 shows the results for the Fill case. Each Figure shows that for [N.sub.u] values less than 1, the preform has little or no effect on the gel time regardless of initial preform temperature. As [N.sub.u] increases, however, the effect of the preform becomes more and more powerful; leveling out past [N.sub.u] values greater than 1000 for the curing case. As one might expect, the gel time increases with decreasing initial preform temperature. For hot preforms, this increase is small while for cold preforms the effect can be quite large; approaching an order of magnitude increase. We notice that for the filling case, the increase in gel time is much more pronounced than for the cure case under the same initial conditions. This occurs because the resin always sees glass at the same temperature and the glass is not allowed to heat. Figure 11 and 12 illustrate, therefore, how intelligent choice of preform construction and initial temperature can be used to control the gel time in the part center. This control is independent of the gel time at the part surface where mold wall temperature is controlling.
Effect of [G.sub.U] and [G.sub.T]
In addition to effect of [N.sub.u], there are also the two other groups to consider: [G.sub.U] and [G.sub.T]. These groups are similar to those which arise in simultaneous heat transfer and nonhomogeneous catalyzed reaction analysis (8). From their makeup it can be deduced that these groups relate the ratio of heat liberated by chemical reaction to the initial temperature difference between glass and mold wall. Because of their complexity, however, the effects of [G.sub.u] and [G.sub.t] are difficult to assess. Specifically, their magnitudes can be changed two different ways:
1. Changing the initial reactant composition.
2. Changing the difference between initial resin and preform temperature.
However, the ratio [G.sub.t]/[G.sub.u] will be more or less equal to the isocyanate index:
[G.sub.t]/[G.sub.u] = [Delta] [H.sub.RT][C.sub.I,o]/[Delta] [H.sub.RU][C.sub.OH,o] [is approximately equal to] [C.sub.I,o]/[C.sub.OH,o] [is equivalent to] Isocyanate Index (23)
Thus, the effects of [G.sub.U] and [G.sub.T] are related in one way to the initial chemical composition of the reactant mass and their ratio will indicate whether the urethane or trimerization reaction will dominate. Taken individually, [G.sub.U] and [G.sub.T] indicate whether heat liberation from reaction or heat transfer from the mold wall will have the greatest effect on the reaction system.
SRIM MOLDING GUIDELINES
At the outset we stated that an in-depth investigation of the predictions of numerical simulations has led to the development of some new ideas for improving the SRIM process. It is not uncommon for molders to use lower mold temperatures to increase gel time. The resin may also be heated prior to injection in order to lower resin viscosity and allow for maximum preform penetration with low injection pressures. Based on the present analysis, these tactics are non-optimum. Using a cold mold means that the trimerization rate at the part surface will be very slow. Because the resin is coming in hot, the reaction rate in the center of the part will be much faster than that at the surface. The result will be a part that is cured in the center but contains an excess of unreacted isocyanate at the part surface. Blistering will always be a concern for parts produced in this manner.
New Control Scheme
Our calculations suggest that there is a different method available to control gel time, namely that of preform construction and the effect of heat transfer from the resin to the preform. Consider this method of SRIM molding: Use a hot mold such that the isothermal gel time of the resin at this temperature is equal to the maximum required fill time. Then, choose the preform and initial resin temperature such that the gel time in the center of the part equals that at the part surface. In this manner, both the center and surface of the part will cure at the same time. Furthermore, residual isocyanate at the surface should be decreased, reducing the blistering problem. Figure 13 and 14 illustrate this concept in more detail.
Figure 13 shows a plot of gel time vs. mold wall temperature for the surface and center of the part (Cure case). The thick line in the Figure corresponds to the gel time at the surface. Note that this gel time will be independent of initial resin or preform temperature for an isothermal mold wall; thus, there is only one curve. The other curves on the graph correspond to the predicted gel time in the center of the mold for different initial resin temperatures (all other variables have been held constant). Two things are noticeable: As the initial resin temperature increases, the gel time in the center of the mold decreases; also, the gel time in the part center is almost independent of the mold wall temperature (this is a 5 mm thick part). Figure 13 illustrates that two different operating variables can be controlled independently to adjust gel time at the center and surface of the part. The surface conversion depends only on mold temperature, while the center conversion depends mostly on initial resin temperature.
Figure 13 shows results calculated for a relatively high Nusselt number ([N.sub.u] = 100). In this case, heat transfer from the resin to the preform will dominate and the preform will slow down the reaction in the part center. Figure 14 illustrates what occurs when the Nusselt number is decreased to 1. Now the effects of the preform are minimal and the gel times in the part center decrease significantly for the same initial resin temperatures. The resin temperature would have to be lowered below 70 [degrees] C in order to achieve a gel time approaching 30 s for a Nusselt number of 1. Thus, depending on preform construction, the resin temperature may have to be reduced considerably in order to increase the gel time in the center long enough to fill the part.
Looking back at Fig. 13, we see that the Figure can be used as an operating guideline. Imagine a part that requires 36 s to fill at a constant volumetric fill rate. In order to gel at the wall, the mold temperature would have to be approximately 90 [degrees] C. If the resin is to gel in the center of the mold in the same amount of time, it will have to possess an initial temperature of approximately 85 [degrees] C given a Nusselt number of 100. For a lower Nusselt number, the resin temperature would have to be reduced accordingly.
The results in Figs. 13 and 14 should not be applied quantitatively; instead, one would like to apply the numerical model to the specific situation at hand. For instance, some of the resin temperatures used are quite high and might be impractical. However, the results illustrate that a different control scheme could be used to optimize the SRIM process. Instead of using cold molds and high initial resin temperatures, hot molds and cooler initial resin temperatures could be used. The use of the preform as a heat sink to control the reaction exotherm in the center of the part forms the basis for this control scheme. The benefit of this approach is that residual isocyanurate at the part surface is minimized, reducing the chance for blistering.
1. A new SRIM molding concept has been put forth in which the preform is used to control gel time in the part center. This concept should be rigorously tested in order to determine its practicality. If successful, the characteristic Nusselt numbers for different preforms should be determined and used to optimize SRIM operations.
2. The presence of a glass preform in the SRIM mold greatly affects the reaction kinetics and heat transfer. For large values of [N.sub.U], heat transfer to the preform dominates and cools the incoming resin.
3. Hot molds should be used in order to improve isocyanate conversion at the part surface. This will prevent difficulties associated with excessive residual isocyanate such as blistering.
4. If a cold mold is used, trimerization occurs too slowly at the part surface and an excess of residual isocyanate results. This excess isocyanate can react with water in a foaming reaction which causes blistering.
5. Preform construction and initial temperature controls gel time in the part center. Mold temperature controls gel time at the part surface. These two independent operating parameters can be used to ensure that the part cures evenly across its thickness while allowing enough time for complete fill.
6. Our results suggest that polyurethane catalyzes trimerization. We have accounted for this effect in an empirical manner. It would be better to rethink the trimerization kinetic mechanism such that this effect is explicitly incorporated. The carbodiimide reaction must also be investigated and incorporated into the total mechanistic model.
7. The kinetics of our system are similar to those of the model system studied by Vespoli and Alberino; this allows for the use of their basic kinetic model. However, modifications must be made to this model in order to correlate the experimentally observed results at higher isocyanate indices.
The authors wish to acknowledge D. R. Stewart for his work regarding the design and implementation of the laboratory adiabatic reaction system.
[Alpha] = Thermal diffusivity.
[Beta] = Urethane conversion.
A = Density ratio.
[A.sub.v] = Specific area of preform, [cm.sup.-1].
[C.sub.C] = Catalyst concentration, g/[cm.sup.3].
[C.sub.I] = Isocyanate concentration, mol/[cm.sup.3].
[C.sub.I,o] = Initial isocyanate concentration, mol/[cm.sup.3].
[C.sub.OH] = Polyol concentration, mol/[cm.sup.3].
[C.sub.OH,o] = Initial polyol concentration, mol/[cm.sup.3].
C[p.sub.i] = Heat capacity, cal/g [degrees] C.
[C.sub.T] = Trimer concentration, mol/[cm.sup.3].
[C.sub.U] = Urethane concentration, mol/[cm.sup.3].
[E.sub.A1] = Urethane activation energy, cal/mol.
[E.sub.A42] = Urethane activation energy, cal/mol.
[E.sub.C] = Urethane activation energy, cal/mol.
[Phi] = Porosity.
[G.sub.i] = Dimensionless group in energy equation.
[Gamma] = Trimer conversion.
h = Heat transfer coefficient, cal/[cm.sup.2] s.
H = Mold cavity thickness, cm.
[Delta] [H.sub.RU] = Heat of urethane reaction, cal/mol.
[Delta] [H.sub.RT] = Heat of trimer reaction, cal/mol.
k = Thermal conductivity, cal/cm-s-[degrees] C.
[K.sub.A1] = Trimer frequency coefficient, [cm.sup.6]/g mol.
[K.sub.A42] = Trimer frequency coefficient, [cm.sup.3]/mol.
[K.sub.C1] = Urethane frequency coefficient, [cm.sup.6]/g mol.
[K.sub.C2] = Urethane frequency coefficient, [cm.sup.3]/mol.
[[Lambda].sub.i] = Coefficient in Statgraphics analysis.
P = Parameter in kinetic model.
r = Glass filament diameter, cm.
R = Universal gas constant, cal/mol K.
[[Rho].sub.i] = Density, g/[cm.sup.3].
t = Time, s.
[Tau] = Dimensionless time.
[T.sub.g] = Glass temperature, [degrees] C.
[T.sub.go] = Initial glass temperature, [degrees] C.
[[Theta].sub.g] = Dimensionless glass temperature.
[T.sub.r] = Resin temperature, [degrees] C.
[T.sub.ro] = Initial resin temperature, [degrees] C.
[[Theta].sub.r] = Dimensionless resin temperature.
[T.sub.wall] = Mold wall temperature, [degrees] C.
[T.sub.ref] = Reference temperature, [degrees] C.
X = Thickness coordinate, cm.
[Xi] = Dimensionless thickness.
1. W. R. Schmeal, G. G. Viola, and J. V. Scrivo, 7th Annual ASM/ESD Adv. Composites Conf. Detroit (1991).
2. J. E. Kresta and K. H. Hsieh, Makromol. Chem. 179, 2779 (1978).
3. W. H. Christiansen, C. D. Shirrell, W. R. Schmeal, D. R. Stewart, D. A. Moore, and D. G. Davis, Internal Shell Report March (1990).
4. N. P. Vespoli and L. M. Alberino, Polym. Proc. Eng. 3, 127 (1985).
5. E. C. Steinle, F. E. Critchfield, J. M. Castro, and C. W. Macosko, J. Appl. Polym. Sci. 25, 2317 (1980).
6. J. E. Kresta and K. H. Hsieh, Amer. Chem. Soc. Div. Polym. Chem. Polym. Prep. 21, 126 (1980).
7. V. M. Gonzalez, PhD thesis, University of Minnesota (1983).
8. R. H. Perry and C. H. Chilton, Chemical Engineer's Handbook, 4th Ed., pp. 4-17, McGraw-Hill, New York (1977).
9. J. H. Sanders and K. C. Frisch, Polyurethane Chemistry and Technology, Part I. Chemistry, pp. 118-22, Inter-science Publishers, New York (1962).
|Printer friendly Cite/link Email Feedback|
|Author:||Viola, G.G.; Schmeal, W.R.|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 15, 1994|
|Previous Article:||Melt crystallization kinetics of syndiotactic polystryrene.|
|Next Article:||Entrance pressure drop studies of corn meal dough during extrusion cooking.|