A Practical Approach To Simulate Polymerizations with Minimal

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Ind. Eng. Chem. Res. 2005, 44, 2634-2648

A Practical Approach To Simulate Polymerizations with Minimal Information L. M. F. Lona*,† and A. Penlidis‡ Departamento Processos Quı´micos, Faculdade de Engenharia Quı´mica, Universidade Estadual de Campinas, UNICAMP, Caixa Postal 6066, CEP 13081-970, Campinas, Sao Paulo, Brazil, and Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

This paper develops an approach based on optimization to allow for the modeling of polymerization reactors even if no information about the specific monomer is available. The algorithm is robust, allowing us to deal with different kinds of polymers, and it is built with practical and easy-to-measure responses in mind. It needs only a minimum of experimental information, thus becoming readily applicable to industrial situations. This approach can be applied successfully if a few or even no parameters (kinetic and diffusional) are available. The first part of the paper will revisit diffusional effects during polymerization, proposing robust expressions for gel, glass, and cage effects, with a minimum number of parameters to be fitted. The second part will estimate kinetic and diffusional parameters at the same time, assuming that no information about the polymer system is available. Results are compared with a great quantity of experimental data. 1. Introduction One of the greatest difficulties when dealing with modeling and simulation of chemical processes is related to the uncertainty and/or lack of information about model parameters. This is even worse in a polymerization reactor scenario because a typical polymerization involves many reaction steps and diffusional effects can be significant as conversion increases, resulting in the need for a large number of parameters. Typical methodologies to estimate parameters usually employ several difficult-to-measure variables in the optimization algorithms and estimate just a few parameters at a time, which are usually valid for a specific polymer system under study. In most of the cases, fairly good initial guesses are needed for these parameters; otherwise, the models may exhibit instability and convergence problems. This work deals with the development of a practical approach in order to obtain kinetic and diffusional parameters related to polymerization reaction models. There are several examples of recent work in the literature dealing with parameter estimation in polymerization systems (e.g., Tefera et al.1,2 and Polic et al.,3 just to mention a few recent efforts). One difference between the estimation of parameters reported in the literature so far and the proposed methodology is that we fit all parameters (kinetic and diffusional) at the same time, using a minimum amount of experimental (laboratory or industrial) data. Besides, the methodology allows us to use easy-to-measure variables as responses in the optimization algorithm, such as conversion and molecular weight averages, at just three or four different points along the polymerization time trajectory. These data can be directly provided from an industrial reactor in operation and/or can also easily come from a minimum of laboratory experiments. * To whom correspondence should be addressed. Fax: 551937883965. E-mail: [email protected]. † Universidade Estadual de Campinas. ‡ University of Waterloo.

These characteristics permit the algorithm to be used in practical industrial situations, in which only limited data are usually available. If a reactor is already in operation but there is no model to simulate it because of the lack of or uncertainty in parameters, this methodology can be easily employed. Simple operational records of the reactor performance over time can provide sufficient data as input in the optimization algorithm. Just five operational conditions would be enough to generate parameters to simulate the reactor not only within the range of operational conditions considered but also to conditions extrapolated from this range. The methodology is developed in order to suggest an algorithm as robust as possible, allowing us to deal with different kinds of polymers. The first part of the paper assumes that diffusional parameters are unknown, but kinetic parameters are available. This situation happens very often, especially when working with copolymerization. In this way, the first part of the paper deals with the development of robust expressions related to diffusional problems, such as the gel, glass, and cage effects, with a minimum number of parameters to be fitted. The second part estimates kinetic and diffusional parameters at the same time, assuming that no prior information about the monomer system is available. The bulk polymerization of styrene (STY) in the batch will be considered as the main case study. After this, the methodology is applied to methyl methacrylate (MMA) polymerization because it is easily extendable to other systems. These two well-understood examples illustrate how practical the approach is for other polymerization systems. The methodology can be applied to any system that is described by the same number of reaction steps. Note that the same set of mathematical equations can be used for MMA and STY because what changes from polymer to polymer is only the values of the parameters. Results are compared with a great quantity of experimental data, indicating that this methodology is practical and can be used with success, especially for the

10.1021/ie049671m CCC: $30.25 © 2005 American Chemical Society Published on Web 12/24/2004

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2635

Figure 1. Number- and weight-average molecular weights versus time during STY polymerization at 50 °C and [AIBN]0 ) 0.1 mol/L (solid lines are from the “standard” model).

modeling of industrial polymerizations, even with no extensive prior information about the polymer system. 2. Phenomenological Model and Optimization Algorithm To illustrate the methodology, we start with a specific system, say, the polymerization of STY, for which we have a detailed mathematical model, like the one presented in Gao and Penlidis.4 Gao and Penlidis4 provide detailed equations and parameter values and have tested the full mathematical model extensively with literature data covering a wide variety of experimental conditions. This model, which employs freevolume theory expressions to describe diffusional limitations (following the work and extension of the work of Marten and Hamielec5 for gel, glass, and cage effects), will be referred to from now on as the “standard” case and will be the basis for comparisons. As mentioned earlier, the reliability of the “standard” model has been tested extensively, not only with literature data4 but also with data from our own laboratories (e.g., see Figure 1 for a case where the “standard” model performs well for fast-increasing weight-average molecular weights). Because all equations are cited in Gao and Penlidis4 in detail, along with database parameter values, they will not be repeated here for the sake of brevity. In the first part of the paper, section 3 uses expressions to represent diffusional limitations with few parameters and as robustly as possible. In the second part of the paper, both diffusional and kinetic parameters will be fitted together in section 5. Even though the free-volume theory is one of the most well accepted to represent diffusional limitations, in this work we will be using correlations based on easy-tomeasure variables, like conversion and molecular weight averages, because we intend to use this methodology in practical industrial polymerization situations. The set of parameters to be obtained from the optimization step is desired to be valid for different levels of temperature (T) and initial initiator concentration ([I]0). Consider, for the sake of illustration, that our example operational conditions cover the ranges of 50 °C e T e 70 °C and 0.01 mol/L e [I]0 e 0.1 mol/L. Now consider a simple factorial design, with five pairs of T and [I]0, as per Table 1.

Figure 2. Conversion for bulk polymerization of STY at T ) 50 °C and [AIBN]0 ) 0.01 mol/L. Table 1. Operational Condtions Chosen as Experimental Points (First Attempt) condition

temp (T), °C

initiator concn ([I]0), mol/L

polymerization time (tt), min

1 2 3 4 5

50 50 70 70 60

0.01 0.1 0.01 0.1 0.05

4800 1800 1000 400 1000

In Table 1, tf stands for the final polymerization time, which is the time that signifies that conversion has reached a certain upper limit. This limit can be governed by the nature of the system; i.e., it could be the time corresponding to the limiting conversion for systems that exhibit it. In other cases, tf can be the time corresponding to a specific conversion level. For instance, it could be the point at which polymerization should be stopped, for cases where there is an operational limit, say, because of gel formation in industrial cases. Similarly, it could be the point at which polymerization is more or less complete for all practical purposes. If experimental data exist (industry or laboratory), it is easy to locate tf. If data are not readily available, one can obtain this information from a model for the process. If neither data nor a model exist, one can get a very good appreciation of tf by running a small number of experiments, for instance, as per the factorial design of Table 1. To illustrate the choice of tf in the cases above, one can look at Figures 2 and 4. From Figure 2, the choice for tf could be in the range of 38004800 min, hence the entry of 4800 min of the first row of Table 1. From Figure 4, the range could be from 1500 to 1800 min, hence the entry of 1800 min of the second row of Table 1. Table 1 represents the four corners of a square (conditions 1-4). In addition, a central point is often used in factorial designs (condition 5). It is well-known that the polymerization rate almost doubles for every 10 deg increase in temperature and increases by about a factor of 2 if the initiator concentration is quadrupled. These simple rules help us realize that the rate of polymerization and, consequently, the polymerization time for conditions 3 and 5 should be almost the same, even before running any experiments. To avoid loss of valuable information, condition 5 can thus be modified. Our choice for the new condition is T ) 50 °C and [I]0 ) 0.02 mol/L, but of course another condition could be

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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 Table 2. Some Examples of Correlations Proposed To Describe the Gel Effecta gt ) (P1/Mw)P0 exp(-P4 - P2x - P3x2) gt ) (P1/Mw)P0 exp(-P2x - P3x2) gt ) P1 exp(-P2x - P3x2) gt ) exp(-P2x - P3x2)

(a) (b) (c) (d)

a Consider k ) k g , where x ) conversion, M t t0 t w ) weightaverage molecular weight, P0, P1, P2, P3, and P4 ) parameters to be fitted, and gt ) gel effect correlation.

Figure 3. Profiles of gt, gp, and gt/gp at T ) 50 °C and [AIBN]0 ) 0.01 mol/L using the “standard” model.

Figure 4. STY conversion (T ) 50 °C and [AIBN]0 ) 0.1 mol/L) using different ways to represent gt and gp.

chosen as well, as long as the polymerization rate does not match exactly any of the other four conditions. For T ) 50 °C and [I]0 ) 0.02 mol/L, the polymerization time is around 3300 min. Three variables were considered as responses: conversion, number-average molecular weight, and weightaverage molecular weight. These were taken at three different times along the polymerization trajectory: at the end of polymerization (tf); at a midpoint in time (0.5tf); and in the beginning of the reaction (0.1tf). Time tf varies widely depending on the T and [I]0 used. This is essential because the set of parameters to be obtained after the optimization step is desired to be valid to describe polymerizations over a broad polymerization time range (note that in Table 1 the higher tf is more than 10 times the lower one), as well as out of the T and [I]0 range used during the optimization step. 3. Fitting Diffusional Parameters In what follows, the optimization algorithm used to fit the parameters is the Levenberg-Marquardt one, which interfaces with a Runge-Kutta method to integrate differential (model) equations that describe the system. The steps followed to obtain practical correlations for gel, glass, and cage effects are shown, keeping robustness needs in mind. If one wants to simulate a

new polymer system with unknown and/or uncertain diffusional parameters, one needs only to use the correlations proposed in this section directly and fit the parameters for the specific process. The correlations were chosen in order to have as few parameters to be fitted as possible, in combination with a small amount of easy-to-measure experimental data (conversion and molecular weight averages). In this way, we can cover a large number of polymerization systems with minimal effort, following exactly the same steps as the ones used herein to validate our approach with STY and MMA. 3.1. Gel Effect. Initially, a gel effect correlation was obtained. We considered that all of the other kinetic parameters as well as expressions to represent glass and cage effects were known, using values from the database of Penlidis.6 To fit the gel effect parameters for both systems (STY and MMA), experimental data and results from the “standard” model were used. Several possibilities were tested, based on observations of profiles of kt versus conversion obtained from correlations based on free-volume theory. The best results were obtained using the group of correlations shown in Table 2, in which gt is an expression that represents the gel effect and modifies the termination rate constant kt0 [which is the chemically controlled termination rate constant at zero (initial) to low conversion levels, i.e., in the absence of diffusional limitations]. The criterion adopted to choose the best correlation was to consider the one that gives the best predictions with respect to conversion and number- and weight-average molecular weights but at the same time using the smallest possible number of parameters. On the basis of this criterion, expression (b) was considered the most appropriate. There is no improvement in the results if eq a is applied, despite one more parameter used. Equations c and d generate a much poorer profile, as can be seen in Figure 2. A similar behavior is obtained for the other operating conditions (plots for different operating conditions will be shown selectively throughout the paper for the sake of brevity). It is well-known that kt and, consequently, gt curves have a similar behavior over time for all bulk polymerizations, decreasing abruptly at the onset of the gel effect (see Figure 3). Because eq b was chosen in order to mimic the gt curve behavior from Figure 3, it is expected that eq b represents well the gel effect for all polymerization reactions that have kt behavior similar to the one in Figure 3. What will change from polymer to polymer will be the time at which the reaction becomes diffusionally controlled (point of decrease of gt in Figure 3), but this will, of course, be captured by the different parameter values fitted for each case. In this work, it is shown that this correlation works very well for MMA and STY under both chemical and thermal initiation. Although in this first attempt we need four parameters to describe gt, it will be seen from the analysis

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2637 Table 3. Expressions Proposed for the Cage Effecta attempt

correlation

observation

1 2 3 4

cage ) 1 cage ) exp(-C1x - C2x2) cage ) exp(-C1x - C2x2) step function: cage ) 1, for gt > gt,crit; cage ) C1, for gt e gt,crit step function: cage ) 1, for gt > gt,crit; cage ) exp(-C1x), for gt e gt,crit step function: cage ) 1, for gt > gt,crit; cage ) exp(-C1xnorm), for gt e gt,crit

at all times (using all options for gp) at all times at all times, with a restriction: cage ) 1 if cage > 1 C1 and gt,crit are parameters to be fitted

5 6 a

C1 and gt,crit are parameters to be fitted C1 and gt,crit are parameters to be fitted

Consider f ) f0 × cage.

below that overall only three parameters will be enough to represent both gel and glass effects, which is a considerable reduction in the number of parameters compared to the number of parameters used in such models in the past. Gao and Penlidis4 have made extensive use of freevolume theory models to take into account diffusional limitations in polymerizations. These models work very well provided that one has an equally extensive set of experimental data for parameter estimation. If the polymerization system is fairly unknown without prior information, the effort needed to estimate parameters in mechanistic models can be rather extensive. Here we will try to demonstrate a very practical approach with a minimum number of parameters and an equally minimal effort in experimentation and analysis. Back to eq b of Table 2, extensive testing was done in order to verify the robustness of the correlation with respect to adjusting the parameters and checking the influence of the initial guesses. It was observed that parameter P0 in eq b usually varies from 0.3 to 0.4 for all initial guesses tried, so the final expression used in this work to handle the gel effect was

gt ) (P1/Mw)0.35 exp(-P2x - P3x2)

(1)

Simulation results showed that there was no loss in accuracy when eq 1 was used, and, therefore, one less parameter is employed. 3.2. Glass Effect. The second step was to find a reliable expression to represent the glass effect and fit its parameters. In this case eq 1 was also used, so parameters for gel and glass correlations have to be fitted together. All other kinetic parameters were considered known, including the expression that represented the cage effect. In the same way, we considered here kp ) kp0gp, in which kp0 is the chemically controlled propagation rate constant (below a critical conversion). Equation 2 appeared to be a good representation of the glass effect:

gp ) exp(-B1x - B2x2)

(2)

The behavior of kt and kp (as well as that of gt and gp) along the polymerization time was analyzed using the “standard” model for verification purposes. It was observed that gt and gt/gp profiles over time exhibited similar behavior, as can be seen in Figure 3. This is because when gp starts being different from unity, gt values are very low (zero or close to zero in all cases tested), so then the ratio gt/gp remains equal or very close to (an already low) gt. In other words, when propagation becomes diffusion-controlled and even the movement of small molecules starts being restricted, macroradicals are restricted anyway.

It was also observed that gt and gp profiles over time obtained from eqs 1 and 2 do not exhibit the same behavior as that obtained from the “standard” model; however, gt/gp profiles are sufficiently similar. On the basis of these two facts, an expression to represent the ratio gt/gp was proposed, as can be seen in eq 3.

( )

gt P1 ) gp Mw

0.35

exp(-P2x - P3x2)

(3)

Results obtained from several approaches are shown in Figure 4. The same trends are observed for other operating conditions. The approach using eq 3 has the advantage of using fewer parameters and still showing a good agreement. In another attempt shown in Figure 4 and based on the observations from Figure 3, gp is used only when a critical value of gt is reached; otherwise, gp stays equal to unity. This critical value (gt,crit) is also a parameter to be fitted:

if gt e gt,crit, gp ) exp(-B1x)

(4)

Results obtained from this approach are equally good, but two more parameters have to be adjusted, so finally eq 3 was adopted to represent diffusional effects for both the propagation and termination steps, with a total of only three parameters. 3.3. Cage Effect. The initiator efficiency (f) can vary during polymerization, especially at higher conversion levels, and we refer to this here as the “cage” factor. An expression to represent the “cage” factor is proposed, based on the behavior of typical profiles of the initiator efficiency versus conversion obtained from the “standard” model. All parameters related to gel, glass, and cage effects must be adjusted together, while the rest of the kinetic parameters are assumed known. Table 3 shows all of the attempts proposed. The symbol f0 represents the initiator efficiency under chemical control (i.e., no diffusional limitations). We observed an improvement going from attempt 1 to 6. Attempt 6 improved the overall picture, including both conversion and molecular weights. Because conversion after the break point (see attempts 4-6) is high, the exponential expression proposed in attempt 5 has to have a fairly accurate value of C1 to be sensitive to even small changes in conversion (in the high conversion regime). To overcome this problem, attempt 6 uses a “normalized” conversion (xnorm) in the exponential part, calculated as follows:

xnorm )

x - xcrit 1 - xcrit

(5)

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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005

Figure 6. (a) Conversion of STY at 70 °C and [AIBN]0 ) 0.1 mol/ L. (b) Molecular weights of STY at 70 °C and [AIBN]0 ) 0.1 mol/ L.

Figure 5. (a) Conversion of STY at five different operating conditions used to fit the parameters. (b) Number-average molecular weight at five different operating conditions used to fit the parameters. (c) Weight-average molecular weight at five different operating conditions used to fit the parameters.

where xcrit can be the conversion corresponding to gt,crit, or it can be a fixed value larger than 0.5. This approach makes xnorm vary over a broader range, making the exponential part less sensitive to C1. Even though “cage” profiles are visually similar for attempts 4-6 (not shown), an improvement in the weight-average molecular weight description is achieved when attempt 6 is used. To summarize the results obtained so far, Figure 5 shows conversion and molecular weight profiles for all five conditions used to fit the parameters (dashed lines; see Table 1, considering condition 5 as T ) 50 °C and [I]0 ) 0.02 mol/L). Figure 5 also shows curves obtained

when the “standard” model is used (full lines). Equation 3 and attempt 6 (Table 3) were used to represent the ratio between the gel and glass effects and the cage effect, respectively, and five parameters were fitted in total. This is a small number of parameters compared to other correlations representing diffusional limitations reported in the literature (e.g., 5 versus 14 parameters; see, for example, the cases described by Gao and Penlidis4 and Marten and Hamielec5). It can be seen (Figure 5) that the same set of parameters is able to provide good descriptions for a very broad range of conditions and polymerization times. The polymerization time for the slowest reaction (tf ) 4800 min) is 12 times the polymerization time for the fastest reaction (tf ) 400 min). Besides, the five parameters fitted are able to describe a very broad range of molecular weights as well. 4. Analysis and Validation of the Methodology 4.1. Validation Remarks. To verify the reliability of the proposed approach to estimate diffusional parameters, a validation step was performed. At this point, eq 3 and attempt 6 from Table 3, with the five parameters estimated from the optimization algorithm, were used to describe diffusional limitations (gel, glass, and cage effects). Results obtained in terms of conversion and number- and weight-average molecular weight profiles are compared with experimental data generated at the University of Waterloo (Figure 6) and with experimental data from the literature4 (Figure 7).


Figure 7. (a) Conversion of STY at 60 °C and two different initial initiator concentrations. (b) Molecular weights of STY thermal polymerization in bulk at 120.4 °C.

Many tests were performed, but only selective ones are shown herein. A similar degree of agreement between experimental and simulated profiles was also obtained in all other operating conditions tested. Figure 6 shows conversion and molecular weight profiles when T ) 70 °C and [I]0 ) 0.1 mol/L. To check the extrapolative capacity of this approach, the same set of estimated parameters was used to evaluate the performance of thermal polymerization of STY, in which no initiator was added. This situation is completely different from conditions in which the parameters were fitted. We can see a reasonable agreement with experimental data4 in Figure 7b, which is a good confirmation of the validity of the methodology. The parameters were found to be equally valid for cases with chain-transfer agent added to the system. 4.2. Robustness and Initial Guesses. Two very important aspects when working with parameter estimation are the robustness of the algorithm and the quality of the initial guesses. Many reports in the literature that deal with parameter estimation try to improve the numerical values of the parameters that are already reasonably known. When this is the case, a good idea about the initial guess is possible. The higher the uncertainty about the initial guess is, the more robust the algorithm must be to absorb bad initial starts.

In the present work, no previous knowledge about the parameters to be fitted was assumed, so an analysis was performed in order to bound their ranges. In addition, procedures to increase the robustness of the algorithm were developed and will be discussed further. Taking a look at eq 3, we observe that we have to find an order of magnitude for the initial guesses of three parameters (P1, P2, and P3). When eq 3 is analyzed, it is reasonable to say that P1 might have the order of magnitude of molecular weight. Also, as in a power series, the term P3 might be larger than P2 because conversion squared is a smaller number than conversion on its own. Taking into account that gt/gp curves over time (or conversion) for a wide variety of polymer systems under bulk polymerization might exhibit similar behavior, an analysis of the behavior of profiles generated from eq 3 was conducted, trying to yield good ranges for P2 and P3. Keeping in mind also that gt/gp must vary between 0 and 1 (0 e gt/gp e 1), we obtained the following order of magnitude for the initial guesses of the parameters in eq 3: 60 000 e P1 e 120 000, -4 e P2 e -2, and 10 e P3 e 40. The ranges shown represent just a suggestion for an initial guess. They do not mean that an initial guess out of this interval will lead to a nonconvergence problem. They just mean that it is not necessary to try values outside this range because the ones given appeared to be the most efficient during optimization. Attempt 6 of Table 3 involves two parameters that have to be estimated. Parameter gt,crit might have a very low value, with an order of magnitude similar to gt when diffusional limitations dominate the process. From this point on, a steep decrease in “cage” over time (or conversion) is observed. By analysis of the behavior of “cage” vs “time” profiles at different values of C1, it is possible to obtain the following ranges for the initial guesses of gt,crit and C1: 0.000 05 e gt,crit e 0.0003 and 10 e C1 e 50. Once again, observations regarding the ranges made previously are also valid here. These ranges do not pose any limits on the parameter estimation. One can have as the initial guess C1 ) 40, for example, and obtain finally C1 ) 55.6, which is out of the range of the initial guesses. 4.3. Extrapolation to Other Polymer Systems. As mentioned before, profiles of gt/gp and “cage” might behave in a similar way for other kinds of polymers for bulk polymerization over the full conversion range. Therefore, the proposed equations that describe diffusional limitations should work for other polymers. For all polymers, gt/gp will reach zero at the end of polymerization because termination will be diffusioncontrolled before propagation. At time zero, gt/gp will be equal to unity and will show a dramatic reduction as diffusional limitations become important. What changes from polymer to polymer is when this drastic reduction will start and how long it will take until gt/gp reaches zero. For different polymer systems, this characteristic is related to conversion. Equations used so far to describe diffusional limitations are functions of conversion and should be valid for all kinds of polymers; however, the five parameters should assume different (numerical) values depending on the polymer system. The range of initial guesses for these parameters should be valid for all kinds of

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Table 4. Estimated Parameters Using Responses Collected at Different Points over Time parameter

initial guess

condition 1

condition 2

condition 3

condition 4

P1 P2 P3 gt,crit C1

80 000 -3.0 15.0 1.00 × 10-4 15.0

71 773 -2.4947 15.039 1.135 × 10-4 37.798

170 778 -2.3445 15.587 0.671 × 10-4 52.463

69 527 -2.708 15.273 0.606 × 10-4 38.262

65 556 -2.700 15.356 0.975 × 10-4 41.741

polymers or most of them because a very broad spectrum was considered. To verify the robustness and applicability of the proposed approach in describing diffusional limitations for other polymer systems, a test was performed using MMA, which presents much more severe diffusional limitations compared with STY. An analysis was performed using responses collected at different points over time, as can be seen in Table 4, emphasizing more the autoacceleration region (with the polymerization time greater than 0.75tf). Columns 1 and 2 of Table 4 show the parameters that have to be fitted and their initial guesses, respectively. Columns 3-6 show fitted parameters obtained when responses are collected at different points over time, as described below. (i) Condition 1: 0.1tf, 0.5tf, 0.75tf, and tf. (ii) Condition 2: 0.1tf, 0.5tf, and tf. (iii) Condition 3: 0.1tf, 0.5tf, 0.7tf, 0.75tf, and tf. (iv) Condition 4: 0.5tf, 0.7tf, 0.75tf, and tf. MMA polymerization presents stronger diffusional limitations compared to STY polymerization, so condition 2 is not representative of the system, presenting always a delay in describing the autoacceleration region. More pieces of information are needed in this region, so conditions 1, 3, and 4 [which bring information at high conversion (0.75tf)] seem to be more effective for all polymerization scenarios analyzed under different temperatures and concentrations. In fact, profiles with parameters generated at conditions 1, 3, and 4 are similar in quality because all of them generate similar fitted parameters, as opposed to condition 2 (Table 4), which is not able to predict the autoacceleration region properly. Because the aim of this work is to develop a procedure that uses fewer experimental points to estimate the parameters, one can conclude that condition 1 is the best one to be adopted. Conditions 3 and 4 bring information also at conversion 0.7, very close to x ) 0.75, both in the autoacceleration zone. This is a repetitive kind of information, thus not offering any advantage for the parameter estimation procedure. In the same way, there is no advantage in considering responses at the beginning of the region, when the system is chemically controlled (compare conditions 3 and 4). Figure 8 shows conversion and molecular weights for MMA polymerization obtained from the approach proposed in this work, at conditions of temperature and initial initiator concentration different from the ones used to fit the parameters. Experimental data from Gao and Penlidis4 are also presented. A good agreement can be observed even at conditions beyond the range of T and [I]0 considered during the optimization step, which means that extrapolation is reasonable. A very important characteristic related to the choice of the initial guesses was observed. Despite MMA presenting stronger diffusional limitations than STY,

Figure 8. (a) Conversion in MMA polymerization at 70 °C and [AIBN]0 ) 0.0258 mol/L. (b) Molecular weights in MMA polymerization at 70 °C and [AIBN]0 ) 0.0258 mol/L. (c) Conversion in MMA polymerization at 90 °C and [AIBN]0 ) 0.0258 mol/L.

it was possible to use the same set of initial guesses in both cases, and probably the initial guesses used here would be a good set to be used for other kinds of polymers. Table 5 shows the fitted parameters in both cases.

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2641 Table 5. Comparison between Fitted Values for STY and MMA Polymerization fitted parameter

initial guess

STY

MMA

P1 P2 P3 gt,crit C1

80 000 -3.0 15.0 1.00 × 10-4 15.0

92 121 -3.195 14.489 1.37 × 10-4 38.32

71 773 -2.4947 15.039 1.135 × 10-4 37.798

The convergence criterion used in this work stops the search procedure when the error reaches a minimum acceptable value. The error is the difference, in module, between the real and calculated values of conversion and number- and weight-average molecular weights, as follows:

error ) min

∑∑| 2j)1i)1 1m

n

xreal - xcalc xreal

|

2

Mnreal - Mncalc

|

Mnreal

+ Mwreal - Mwcalc

| | 2

+

Mwreal

|

2

where m ) number of different times over the polymerization trajectory [m ) 3 for STY (0.1tf, 0.5tf, and tf) and 4 for MMA (0.1tf, 0.5tf, 0.75tf, and tf)] and n ) number of operating conditions chosen (n ) 5 in our case). The ratio, used in the error equation above, is considered in order to guarantee similar order of magnitude for each one of the three terms of the equation to be minimized. 5. Kinetic Part and Polymerization Steps Three different monomers were considered to obtain a better overview about how reaction rates for each polymerization step behave and how they are affected by temperature. The three monomers chosen were STY, MMA, and vinyl acetate (VAc). Taking into account the kinetic parameters available in the literature and in the database of a Homopoly simulator,6 typical rate constants for chemically controlled propagation, termination, and transfer to monomer as a function of the temperature were plotted, as can be seen in Figure 9. All units are in L/(mol s). We notice that termination rate constants are almost linear in relation to the temperature because of their low activation energy, even over a large range of temperatures (from 0 to 120 °C) (see Figure 9b). From Figure 9a,c, we can see a strong nonlinearity of kp0 and kfM as a function of the temperature; however, if the ratio between kfM and kp0 (kfM/kp0) is plotted against temperature, a much more linear behavior is observed, even for the wide range from 0 to 120 °C (see Figure 10). Besides, if ratios between rate constants are used, these ratios will change in a much narrower range if compared with isolated rate constants, thus reducing problems related to the initial guesses. Polymerizations are usually conducted over a relatively narrow range of temperatures. If this same analysis is performed considering temperatures from 50 to 70 °C, a range often employed, a linear variation with temperature can be observed for all monomers consid-

Figure 9. (a) Propagation rate constant versus T for three different monomers. (b) Termination rate constant versus T for three different monomers. (c) Transfer to monomer rate constant versus T for three different monomers.

ered, as shown in Figure 11. In fact, this linear relationship is still valid for a much broader range. The number of parameters to be fitted either in an Arrhenius expression or from these linear relationships is the same (two parameters); however, in the latter case, a much more robust algorithm is possible, without loss of information. 6. Fitting Kinetic Parameters According to the shape of all curves in Figure 11 and taking into account the order of magnitude of these variables, one can propose the following expressions to

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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 Table 6. Parameters Related to Termination trial

initial guess

fitted parameter

1

P6 ) 300.0 P7 ) 3.0 P6 ) 200.0 P7 ) 2.0 P6 ) 100.0 P7 ) 1.0 P6 ) 400.0 P7 ) 4.0

P6 ) 296.1 P7 ) 4.85 P6 ) 276.6 P7 ) 2.8 P6 ) 120.0 P7 ) -0.27 P6 ) 390.6 P7 ) 5.5

2 3 4

Figure 10. Ratio kfM/kp0 versus T for three different monomers.

Figure 11. (a) kt0/kp0 versus T for three different monomers, from 50 to 70 °C. (b) kfM/kp0 versus T for three different monomers, from 50 to 70 °C.

be used in the optimization step:

kt0/kp0 ) 103[P6 - P7(T - Tinf)]

(6)

kfM/kp0 ) 10-6[P8 - P9(T - Tinf)]

(7)

and

where P6, P7, P8, and P9 ) parameters to be estimated, T ) temperature (°C), and Tinf ) lower limit of temper-

ature in the range considered during optimization. In this specific example, Tinf ) 50 °C. Observe that both factors (103 and 10-6) that appear multiplying the expression inside the brackets in eqs 6 and 7 account for the order of magnitude for these two ratios for all three monomers that are being considered. According to the plots, one can also see the range in which parameters must remain for many classes of monomers: 30 < P6 < 300, 30 < P8 < 300, P7 is around 100 times smaller than P6, and P9 is around 100 times smaller than P8. It is important to notice that the relationship between P7 and P9 and between P6 and P8, respectively, simplifies the task of guessing too many initial values. Ranges presented above are broad in order to accommodate as many different kinds of monomers as possible. We can see, for example, that ranges of kfM/kp0 obtained from Figure 11b and considered in this work might cover many monomers. Transfer to monomer sets an upper limit to the molecular weight of the polymer that can be produced, and kfM/kp0 values change according to monomer type. For VAc, it is well-known that its kfM/kp0 value is high. On the other hand, the kfM/kp0 values for STY and MMA are quite low. Monomers considered in this work might represent the upper (VAc) and lower (STY and MMA) limits of kfM/kp0 if compared to other kinds of monomers, which might have their kfM/ kp0 values between these two limits. 6.1. Termination Step. We will now verify the robustness and efficiency of this approach by fitting parameters related to the termination step. In this case, seven parameters have to be estimated together: five related to diffusional limitations (section 3) and two related to the termination step (kinetics). Taking a look at Figure 11a for STY, one can see that P6 and P7 are equal to 300.3 and 4.84, respectively. To mimic a situation in which no information is available regarding the monomer, many pairs of P6 and P7 were used, within the range of previously established initial guesses. Pairs out of this range were also considered. The set of trials and their respective parameters can be seen in Table 6. When a good initial guess is used (trial 1), fitted parameters reach values close to the “real” ones. Observe that P7 goes from 3.0 to 4.85 (the “real” value is 4.84). When the optimization algorithm is initiated with bad guesses (trials 2-4), although fitted parameters differ from trial to trial, all pairs of parameters can generate reasonable predictions in terms of conversion and molecular weight. Table 6 shows that, even when bad initial guesses are used, parameters tend to go to the correct region (observe that P6 tends to increase if the initial guess is smaller than 300 and vice versa), but local minima that cause small errors are found along the path. These differences in kinetic parameters are compensated for

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2643 Table 7. Effect of Initial Guesses Related to Termination on Diffusional Parameters fitted parameters initial guess

trial 1

trial 2

trial 3

trial 4

P1 ) 80,000 P2 ) -3.0 P3 ) 15.0 gt,crit ) 1.0 × 10-4 C1 ) 15.0

81 536 -3.065 13.17 1.0 × 10-4 22.96

113 710 -2.512 13.02 0.7 × 10-4 16.06

95 143 -4.737 12.25 1.132 × 10-4 39.76

72 900 -3.104 14.56 0.7 × 10-4 17.24

Table 8. Initial Guesses and Estimated Values for P8 and P9 trial

initial guess

fitted parameter

1

P8 ) 100.0 P9 ) 1.0 P8 ) 200.0 P9 ) 2.0 P8 ) 300.0 P9 ) 3.0

P8 ) 116.581 P9 ) 2.952 P8 ) 174.5 P9 ) 3.157 P8 ) 229.38 P9 ) 1.88

2 3

Table 9. Initial Guesses and Estimated Values for P7 and P9 initial guess

fitted parameter

“real” value

P7 ) 2.0 P9 ) 1.0

P7 ) 4.77 P9 ) 2.952

P7 ) 4.84 P9 ) 2.49

by different values that diffusional limitation parameters assume, depending on the direction of search taken by the optimization algorithm, thus guaranteeing small deviations only. Table 7 shows examples of how diffusional limitation parameters can be affected depending on the quality of the initial guesses related to termination. This proposed approach proved to be flexible and robust, allowing good reactor predictions for a large range of initial guesses. This would not be possible if kinetic equations based on regular Arrhenius expressions were used (convergence, in such a case, was possible only for a very narrow range of initial guesses). This proposed approach gave us also an indication as to when the initial guesses for P6 and P7 were badly chosen (much lower than the “real” values). kp0 increases more than kt0 with temperature, so profiles of the ratio kt0/kp0 are decreasing with temperature for all polymers. When very low values of P6 and P7 were used as initial guesses (trial 3 of Table 6), the algorithm tried to increase the termination rate not only by moving P6 to higher values of the y intercept but also by inverting the slope of the line, making it ascending. We can see in Table 6 that trial 3 is unique in that P7 went to negative values. In this way, when one is running the optimization software and observes that P7 is becoming negative, it is possible to stop the program and make another (higher) initial guess. 6.2. Transfer to Monomer Step. The next step will be the estimation of parameters related to transfer to monomer. In this case, the five parameters related to diffusional limitations, the two related to termination, and the two related to transfer to monomer must be fitted together, resulting in a total of nine parameters. Equation 7 and ranges previously established for P8 and P9 initial guesses were considered. According to Figure 11b, P8 and P9 will be 74.16 and 2.49, respectively, for STY. Pretend we do not have any information about the polymer, so we do not have a good starting value for these two parameters. To verify the robustness and capacity of generalization of this approach, initial guesses over a broad range established previously were

Figure 12. (a) STY bulk conversion at T ) 60 °C for four different initial initiator concentrations. (b) STY bulk conversion at T ) 70 °C and initial initiator concentration [AIBN]0 ) 0.1 mol/L.

tested. Table 8 shows three different pairs of initial guesses and their respective estimated values. The program converged for all initial guesses tested. Once again, despite generating different values for P8 and P9 during optimization, the sets of parameters obtained from the three example trials above are able to describe the behavior of the polymerization reactor well for all operating conditions tested. The difference among pairs of fitted parameters related to transfer to monomer obtained from trial to trial (see Table 8) is mainly due to the very broad range of initial guesses that this approach allows us to make. When initial guesses for parameters related to termination and transfer to monomer are taken not very far from the “real” values, we can see that fitted parameters become closer to the slope and y intercept displayed in Figure 11 (see Table 9). Table 9 shows that parameters related to slopes (P7 and P9) move close to the “real” values even when the initial guess is about 2.5 times smaller than the “real” value. Figure 12 shows predictions obtained when diffusional and kinetic parameters fitted by this proposed approach are used. Experimental data obtained from this research (Figure 12b) and from Gao and Penlidis4 (Figure 12a) are also shown. Many different initiator concentrations and temperatures were considered, including conditions out of the range used during the

2644


optimization step. A satisfactory agreement was observed in all cases. It is important to notice that, at this point, no parameters from the literature related to either kinetics or correlations for gel, glass, and cage effects are required to simulate the polymerization reactor. Potentially, this approach could describe how a polymer system behaves even with little prior information about polymerization variables. The only information required for the program is that related to the initiator. Estimating parameters related to the initiator will be treated further in section 8, but before starting to deal with initiator parameters, the proposed approach to fit kinetic and diffusional parameters simultaneously will be tested using another kind of monomer.

Table 10. Initial Guesses and Estimated Values for P6, P7, P8, P9, and P10 trial

initial guess

fitted parameter

1

P6 ) 30 P7 ) 0.3 P8 ) 30 P9 ) 0.3 P10 ) 0.5 P6 ) 60 P7 ) 0.6 P8 ) 30 P9 ) 0.3 P10 ) 0.5 P6 ) 100 P7 ) 1.0 P8 ) 100.0 P9 ) 1.0 P10 ) 0.5

P6 ) 40.29 P7 ) -0.0065 P8 ) 43.035 P9 ) 0.368 P10 ) 0.95 P6 ) 60.22 P7 ) 0.34 P8 ) 37.24 P9 ) 0.522 P10 )0.642 P6 ) 91.83 P7 ) 1.42 P8 ) 74.17 P9 ) 3.085 P10 )0.484

2

3

7. Generalization to Other Kinds of Monomers This section will further verify the applicability of this methodology to MMA. Termination by both combination and disproportionation takes place in general, but the proportion between them will depend on the kind of polymer. Styrenics exhibit termination via mainly combination; on the other hand, (meth)acrylates will exhibit both; therefore, more effort will be needed during estimation of the parameters when dealing with MMA. In principle, the total termination rate constant is equal to the termination rate constant by combination plus the one by disproportionation; therefore, disproportionation can be represented as a fraction (between 0 and 1) of the total termination rate constant. Mathematically, we can write

kt0 ) ktc0 + ktd0 ktd0 ) kt0 × fraction

(8)

Via an analysis similar to that of Figure 11, it can be verified that “fraction” has a linear dependence on the temperature over a large range of T (from 30 to 180 °C). One can also notice that its value does not vary considerably with temperature (for instance, from 50 to 70 °C, “fraction” changes by less than 10%). These two characteristics do not change very much for other kinds of polymers and are therefore reasonable assumptions to make. We verified two possibilities for describing the term “fraction”. (a) In the first one, we assumed that “fraction” has a linear dependence on the temperature; therefore, two more parameters have to be fitted:

fraction ) P10 + P11T in which P10 and P11 are parameters. (b) The second considers “fraction” as a constant; hence, one more parameter is to be adjusted:

fraction ) P10 in which P10 is a parameter. Results showed that there is no advantage in using option a because the same quality in predictions is possible from option b, at the same time saving one parameter. In relation to the initial guess for parameter

P10, we know this term can vary from 0 to 1, so an initial guess of 0.5 seemed to be a reasonable one. Many tests were performed using different initial guesses for parameters related to transfer to monomer and termination (P6, P7, P8, and P9) within the range previously defined, starting from the lower limit (P6 ) P8 ) 30) and moving toward the upper limit (P6 ) P8 ) 300). Better results were obtained when the initial guesses remained between the lower limit and a point in the middle of the range. It is a good policy to try initial guesses first around the middle of the range. Table 10 shows, as examples, initial guesses and their respective fitted parameters related to termination and transfer to monomer using three different cases: (a) trial 1, initial guesses are at the lower limit; (b) trial 2, initial guesses are close to “real” values; (c) trial 3, initial guesses are close to the middle of the range. We can see that in trial 2 fitted parameters stay close to the initial guesses. It can also be verified that when a very low value for parameters related to termination is guessed, the slope (P7) changes direction, exactly what occurred for the STY situation (discussed earlier). Figures 13 and 14 show profiles of conversion and molecular weights for poly(MMA) obtained when fitted kinetic and diffusional parameters were used (10 parameters). Experimental data are also shown.4 Different conditions of temperature and initial concentration of the initiator were used. These figures also show profiles obtained when only diffusional parameters are fitted (section 3). A reasonable agreement can be observed in both cases when compared with experimental data. 8. Estimation of the Kinetic Parameters Related to Initiation Many initiators are well-known and frequently used. Therefore, often, estimation of parameters related to the initiation step may not be necessary. By using the methodology described so far and obtaining parameters related to the initiation step from the literature or an industrial source, one can describe the behavior of a polymerization reactor. If the initiator parameters are not reliable or in the case of dealing with a new initiator, the approach presented in the previous sections must be extended to take into account initiation parameter estimation. The rate of initiation is strongly dependent on the temperature, characterized by large values of activation energy. A nonlinear dependence with the temperature is observed even for narrow intervals of T.


Figure 13. (a) MMA polymerization at T ) 90 °C and [AIBN]0 ) 0.01548 mol/L when just diffusional parameters are fitted and when they are fitted together with kinetic parameters. (b) Molecular weights for MMA polymerization at T ) 90 °C and [AIBN]0 ) 0.01548 mol/L when just diffusional parameters are fitted and when they are fitted together with kinetic parameters.

In general,

rI ) 2fkd[I]

(9)

where rI ) initiation rate, f ) efficiency factor (0 e f e 1), [I] ) initiator concentration, and kd ) initiator decomposition rate constant. The rate constant follows an Arrhenius expression: kd ) kd0 exp(-EA/RT). In the beginning of polymerization, f assumes a constant value (f0) and decreases later at high conversions. This can be represented by f ) f0 × cage, in which “cage” has already been described in section 3.3. So, the initiation rate can eventually be described by

rI ) 2f0kd0 × cage exp(-EA/RT)[I]

(10)

In the description of a polymer system, both kd0 and f0 appear together, so in this work, the product of these two terms will be considered together. The same initiator can behave differently, depending on the kind of monomer used. To have an idea about how different pairs of initiator/monomer behave as a function of the temperature, a set of three monomers (STY, MMA, and VAc) and four initiators (AIBN, AIBME, di-sec-BPODC, and BPO) will be analyzed. Data related to the activation energy and preexponential and efficiency factors were taken from the Homopoly

Figure 14. (a) MMA polymerization at T ) 70 °C and [AIBN]0 ) 0.0258 mol/L when just diffusional parameters are fitted and when they are fitted together with kinetic parameters. (b) Molecular weights for MMA polymerization at T ) 70 °C and [AIBN]0 ) 0.0258 mol/L when just diffusional parameters are fitted and when they are fitted together with kinetic parameters.

database,6 but they can also be found in other sources (e.g., Sanchez and Myers,7 Dhib et al.,8 or in industrial catalogues of initiators). Figure 15 shows the efficiency factor, initiator rate constant, and their product as a function of the temperature. There are 12 curves in each plot, representing all possible combinations between monomers and initiators. In many cases, there is complete superposition of curves. We can see that f0kd values (Figure 15c) are spread over a range twice as small as that for kd, which is good from an optimization point of view. The most different curve is related to polymerization of STY using di-sec-BPODC, with all other curves remaining within a much smaller range (0-0.003). Despite the variation in f0kd values and the diversified set of initiators/monomers, it was observed that the activation energy varies over a rather narrow range, even if the pair di-sec-BPODC/STY is considered. The following range was observed: 2.72527 × 104 < EA < 3.1991 × 104 (cal/mol). This is again a very good characteristic from an optimization point of view because the exponential dependence between the activation energy and temperature requires a very good initial guess for EA, to avoid typical nonconvergence problems during the parameter estimation step. A value of around 3 × 104 seems to be a very good initial guess for the activation energy.

2646


pair is not considered, the range becomes 3.118 × 1015 < f0kd0 < 44.895 × 1015. In summary, the approach to fit data related to initiation takes into account the following expressions and initial guesses:

f0kd ) P10 × 1015 exp(-P11 × 104/RT)

(11)

in which the initial guesses are P10 ) 100 or less and P11 ) 3.0. Considering the approach related to the initiation step presented above, all parameters were adjusted together: diffusional, kinetic, and those related to the initiator. Figure 16 shows experimental data and conversion profiles for different operating conditions obtained in three different situations: Case a: just diffusional parameters are fitted (5 parameters), as shown in section 3. Case b: diffusional and kinetic parameters are fitted together (9 parameters), as shown in section 6. Case c: diffusional, kinetic, and parameters related to initiation are fitted together (11 parameters), as shown in this section. A good agreement can be seen in the results for all cases tested. Curves related to cases b and c are very close to each other in all plots. Table 11 shows initial guesses and fitted parameters used to construct Figure 16. Although the fitted value for P10 shown in Table 11 is very close to the initial guess, convergence is obtained even when different orders of magnitude are used (P10 ) 10 as the initial guess, for example). 9. Concluding Remarks

Figure 15. (a) f0 as a function of the temperature. (b) kd as a function of the temperature. (c) f0kd as a function of the temperature.

Analyzing how the preexponential factor f0kd0 behaves (observe that here we are talking about kd0 and not kd), we can notice the range 3.118 × 1015 < f0kd0 < 290.714 × 1015. This is a much larger range compared with the EA range, but this parameter allows for a more imprecise initial guess because there is no exponential dependence on temperature, making nonconvergence problems rarer. A reasonable initial guess for f0kd0 is around 100 × 1015 or smaller. This is because the pair AIBME/STY has a high value (290.714 × 1015). If this

This paper presents a practical approach, based on optimization techniques, to allow the modeling and simulation of a polymerization reactor, even if no prior information about the system is available. Because the program is supposed to deal with several different kinds of polymers, it has to work with a large range of initial guesses, and so the software was built with the robustness of the algorithm in mind. The algorithm was developed in order to use easy-tomeasure responses and as few experimental/industrial data as possible, mimicking real industrial situations. These data can be provided from an industrial reactor in operation, without making it necessary to run additional experiments in the laboratory. The methodology was applied to STY and MMA polymerization as examples. It was observed that, even though the polymerization time for these two monomers is very different and poly(MMA) exhibits much more severe diffusional limitations, parameters can be estimated from the same set of initial guesses. Results were shown at three levels: (a) parameters related to gel, glass, and cage effects are fitted, assuming that kinetics are known (sections 3 and 4); (b) parameters related to kinetic and diffusional limitations are fitted, assuming that the initiator parameters are known (section 6); (c) information about kinetic and diffusional limitations and the initiator is not available (section 8). Comparison with experimental data showed a good agreement for all three levels studied, for both polymers. This methodology can be extended to other kinds of

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2647 Table 11. Initial Guesses and Fitted Parameters Used for Figure 16 P1 guess fitted

80 000 94 839

P2 -3 -3.24

P3

gt,crit

C1

P6

P7

P8

P9

P10

P11

15 12.01

10-4

15 23.2

200 222.6

2 3.25

100 111.8

1 -2.37

100 100.03

3 3.13

1× 0.93 × 10-4

is expected when dealing with a CSTR at steady state because no variation over time is observed. Acknowledgment The authors thank Mike J. Leamen for help with the experiments and the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Canada Research Chair (CRC) program, and the Sa˜o Paulo State Research Aid Foundation (FAPESP) of Brazil for financial support. Nomenclature C1, C2 ) parameters to be fitted in Table 3 [CTA] ) chain-transfer agent concentration [mol/L] f ) initiator efficiency gt ) gel effect correlation value gt,crit ) parameter to be fitted when attempts 4-6 of Table 3 are used [I] ) initiator concentration [mol/L] kd ) rate constant of initiator decomposition [L/min] kfM ) rate constant of transfer to monomer [L/(mol min)] kp ) rate constant of propagation [L/(mol min)] kt ) overall rate constant for termination [L/(mol min)] ktc ) rate constant for termination by combination [L/(mol min)] ktd ) rate constant for termination by disproportionation [L/(mol min)] [M] ) monomer concentration [mol/L] Mn ) number-average molecular weight Mw ) weight-average molecular weight PS ) polystyrene rI ) rate of initiation T ) temperature t ) time x ) conversion Indices 0 ) initial condition or zero conversion crit ) critical f ) final p ) polymer m ) monomer

Literature Cited

Figure 16. (a) Conversion of STY at 60 °C and two different concentrations of initiator [AIBN]0. (b) Conversion of STY at 45 °C and four different concentrations of initiator [AIBN]0. (c) Conversion of STY at 60 °C and two different concentrations of initiator [AIBN]0.

reactor/polymerization, such as copolymerization, solution polymerization, and continuous stirred tank reactor (CSTR) operation. Less effort is expected when dealing with solution polymerization because diffusional limitations are not as pronounced. In the same way, less effort

(1) Tefera, N.; Weickert, G.; Westerterp, K. R. Modeling of Free Radical Polymerization up to High Conversion. I. A Method for the Selection of Models by Simultaneous Paremeter Estimation. J. Appl. Polym. Sci. 1997, 63, 1649. (2) Tefera, N.; Weickert, G.; Westerterp, K. R. Modeling of Free Radical Polymerization up to High Conversion. II. Development of a Mathematical Model. J. Appl. Polym. Sci. 1997, 63, 1663. (3) Polic, A. L.; Lona, L. M. F.; Duever, T.; Penlidis, A. A Protocol for the Estimation of Parameters in Process Models: Case Studies with Polymerization Scenarios. Macromol. Theory Simul. 2004, 13 (2), 115. (4) Gao, J.; Penlidis, A. A Comprehensive Simulator/Database Package for Reviewing Free-Radical Homopolymerization. J. Macromol. Sci., Rev. Macromol. Chem. Phys. 1996, C36 (2), 199. (5) Marten, F. L.; Hamielec, A. E. High-Conversion DiffusionControlled Polymerization of Styrene. I. J. Appl. Polym. Sci. 1982, 27, 489.

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(6) Penlidis, A. Homopoly Simulator; Institute for Polymer Research (IPR), University of Waterloo: Waterloo, Canada, 1996. (7) Sanchez, J.; Myers, T. Peroxides and Peroxide Compounds: Organic Peroxides. Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed.; Wiley: New York, 1996; Vol. 18. Sanchez, J.; Myers, T. Initiators: Free Radical Peroxides. Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed.; Wiley: New York, 1996; Vol. 13.

(8) Dhib, R.; Gao, J.; Penlidis, A. Simulation of free radical bulk/ solution homopolymerization using mono and bifunctional initiators. Polym. React. Eng. J. 2000, 8 (4), 299.

Received for review April 22, 2004 Revised manuscript received September 21, 2004 Accepted September 29, 2004 IE049671M

A Practical Approach To Simulate Polymerizations with Minimal

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