Batch solution polymerization of methyl methacrylate: Parameter estimation

The Chemical Engineering

Journal, 39 (1988)

Batch Solution Polymerization

175

175 - 183

of Methyl Methacrylate:

Parameter

Estimation

S. R. PONNUSWAMY and A. PENLIDIS* Institute for Polymer Research Ontario N2L 3GI (Canada)

(ZPR), Department

of Chemical Engineering,

University

of Waterloo,

Waterloo,

C. KIPARISSIDES Department

of Chemical Engineering,

(Received March 25,1988;

Queen’s University,

Ontario

K7L 3N6 (Canada)

in final form July 18,1988)

ABSTRACT

Conversion, number- and weight-average molecular weight data obtained over different operating conditions from a batch solution methyl methacrylate polymerization were used to estimate two kinetic parameters of a batch reactor model: the initiator decomposition rate constant and the propagation rate constant. Property-time histories were adequately fitted by the model using both reported data from the literature and our own experimental observations. This model should find use in the design, simulation, optimization and control of polymer reactor systems for the production of poly(methy1 methacrylate).

1. INTRODUCTION

Parameters in bulk and solution polymerization reactor models are usually obtained from very careful measurements of conversion and molecular weights. Usually, the monomers and initiators are specially purified so that the amount of impurities present in the reaction mixture is as low as possible. However, it is not always economical to maintain these laboratory conditions in polymerization reactions carried out in pilot plant or commercial scale reactors. As a result, conversion and molecular weight predictions that are obtained by using polymerization reactor models with parameter values available in the literature do not match data measured in the field. The discrepancy between the predicted values and the actual measurements is *To whom correspondence 0300-9467/88/$3.50

Kingston,

should be addressed.

generally attributed to variations in the values of the kinetic rate constants and other model parameters. If one has a mathematical model of the process, it sometimes becomes necessary to re-estimate at least some of the parameters of the model so that results measured in the field and model predictions are in agreement. Once the model parameters are established under or close to actual production conditions, a certain model could subsequently be used wtih a higher degree of confidence in developing optimization or control strategies, or in predicting polymer properties under different reactor operating conditions. A mechanistic model for a polymerization reactor is usually obtained in terms of a number of simultaneous coupled non-linear ordinary differential equations. The measurable outputs from a polymerization reactor are usually conversion, and less frequently, number-average and weight-average molecular weights. When actual measurements of conversion, number-average and weight-average molecular weights are available for different experimental conditions, the problem becomes one of estimating the unknown parameters appearing in the differential equations based on multiresponse data. A number of publications have dealt with the problem of estimating such unknown parameters by minimizing the sum of squares of the differences between the measured and computed values of the output. Rosenbrock and Storey [l] described methods for minimizing such an objective function by integrating the differential equations and calculating the outputs at those points in time where measurements are available. They recommended a standard hill climbing 0 Elsevier Sequoia/Printed in The Netherlands

176

procedure to solve the optimization problem and estimate the parameters. They also suggested a numerical procedure to compute confidence intervals for the parameters by linearizing a non-linear model around estimated parameter values. Donnely and Quon [ 21 used a quasilinearization method to estimate the parameters in a set of ordinary differential equations describing several chemical engineering systems. Nieman and Fisher [3] incorporated linear programming ideas into the previously mentioned quasilinearization methods and solved a constrained parameter estimation problem, whereby the search was restricted in a small parameter space around the initial guess. Wang and Luss [4] proposed a technique to substantially enlarge the region of convergence, thereby also avoiding the problem of convergence to non-global optima. Hwang and Seinfeld [ 51 tried to overcome the problem of multiple optima using varying weights along the data vector length. It was concluded that when weighting of the data did not affect the final parameter values, the global optimum had been reached. Bergman et al. [6] used a combination of a quasilinearization and a Gauss-Newton method to estimate unknown parameters in a biomedical model. Hosten [7] carried out a comparative study of short-cut procedures for parameter estimation in a catalytic reaction system. He reported that a maximum likelihood treatment proved to be a laborious task requiring up to 100 times more computer time than the short-cut methods. Watanabe and Himmelblau [8] examined a number of aspects concerning parameter estimation in kinetic models. They reported on a class of methods in which derivatives are approximated by proper finite difference schemes to yield a set of algebraic equations that can be solved to give values for the unknown coefficients. Kalogerakis and Luss [9] presented an alternative development of the quasilinearization method for parameter estimation and a computationally more efficient implementation of the algorithm. Recently, Jang et al. [lo] compared two approaches to the on-line identification of parameters and states in systems described by non-linear ordinary differential equations. The first approach, based on an extended

Kalman filter, was found to be sensitive to several factors: the initial guess of the states, the statistics of input such as variance and level of input noise, and the measurement noise. The second approach, which is based on the application of non-linear optimization methods to minimize a suitable objective function of the error, while computationally more intensive, proved to be superior in terms of tracking and robustness in the presence of modelling errors. Most of the above studies dealt with the development of efficient parameter estimation algorithms. Testing of the algorithms was done with simulated data where the error structure was assumed to be well known. However, model development and parameter estimation, when actual plant or pilot scale data are used, is not a trivial exercise. An example of the difficulties that may be encountered is given by Biegler et al. [ 111; these authors considered a parameter estimation problem formulated by the Dow Chemical Company for comparing existing parameter estimation methods based on industrial data. Based on the results of the comparison they suggested guidelines for parameter estimation when dealing with real plant data. Parameter estimation with polymerization models was used in a series of recent publications concerned with commercially important polymer systems; Lord [12] and GarciaRubio et al. [ 131, investigating the kinetics and modelling of bulk styrene-acrylonitrile copolymerization, employed the minimization of a weighted non-linear least squares criterion involving molar conversion and residual monomer mole fraction. Bhattacharya and Hamielec [14] looked at the bulk thermal copolymerization of styrene-paramethyl styrene. A sixth-order Runge-Kutta differential equation solver within a GaussMarquardt optimization scheme was used for parameter estimation. Similar schemes were also employed by Jones et al. [ 151 (methyl methacrylate-paramethyl styrene), and by Yaraskavitch et al. [ 161 (paramethyl styreneacrylonitrile). In this paper we report on the parameter estimation of a batch polymerization reactor model using data from the literature as well as our own experimental results obtained from a batch solution polymerization of methyl

177

methacrylate (MMA) in a 5-1 pilot plant reactor.

directly measured. Variables I, M, po, p2, x, M, and M, are the dependent variables in this

2. MATHEMATICAL

set. The independent variables (degrees of freedom) are the initial concentrations of initiator and monomer, temperature, and, obviously, time. The analysis assumes that the independent variables are not correlated.

MODEL

DEVELOPMENT

A state-space model for the free radical solution polymerization of MMA has been developed by Ponnuswamy [ 171. The state and measurement equations are as follows: dI dt

3. EXPERIMENTAL

=-kdI

W-4

dM = -k,M& dt dEro = (k,S

dt

dll2 -

dt

=

(lb) + k,M)Ao

(k,S + k,M)X2

+

(1- G!)kJo2

+ k,X,hz

+ &X1’

UC) (14

A, = (2fk, 1/k,)“2

Xl =

(24

2fk, I + (kp M + k,M

X,=h,+

(k,M

+ k&ho

+ k,S + k,Xo)

t2b)

2k*Mb (k,M

(2c)

+ k,S + k,Xo)

x = (MO - M)/M,

(34

Mn = W&Mo)ho

t3b)

M,=

(3c)

Wm~2lWo)

In the above equations I, i%f,p. and ,u2 denote initiator concentration, monomer concentration, and the zeroth and second moment of polymer chains respectively. X0, hi, and X2 represent the zeroth, first and second moment of polymer radicals respectively. Equations (3a) - (3~) represent measurement equations for monomer conversion x, number average molecular weight M,, and weight average molecular weight M,. All the other symbols are defined in the Nomenclature (Appendix A). The parameters that appear in the state eqns. (1) and (2) are kd, k,, k,, y, k,, km, and f. Variables I, M, p, and c(~ represent the

states of the model; not all of them are necessarily measured directly. Variables x, M, and M, are the output variables, which are

SECTION

A series of experiments at different levels of temperature (T), volume fraction of solvent in the reaction mixture ($,), and initial initiator concentration (IO) have been conducted, and important process output variables such as conversion, number average and weight average molecular weights have been measured at different sampling times during the course of the polymerization. The measurement techniques employed and accuracy of measurements have been reported in Ponnuswamy et al. [ 181. Raw data collected from these experiments are reported in Appendix B. Kinetic parameters for MMA polymerization have been reported in the literature (Brandrup and Immergut [ 191, Schmidt and Ray [20], and Louie et al. [ 211). First, we used these literature parameter values and obtained model predictions for conversion, M, and M,. Figures 1 and 2 show the comparison between model predictions and experimental results for 4, = 0.60. The model trends are quite satisfactory and in the right range numerically. A higher polymerization temperature, for the same 4, and I, levels, gives higher conversion (Figs. l(a) and 2(a)), and lower molecular weights (Figs. l(b, c) and 2(b, c)). For the same temperature and & levels, a higher initial initiator concentration gives higher conversion levels (Figs. 2(a) and l(a)), and lower molecular weights (Figs. 2(b), l(b) and 2(c), l(c)). However, one can also see from Figs. 1 and 2 that there is a certain discrepancy between model and experiments, particularly in the runs conducted at 65 “C. This discrepancy shows the necessity to re-estimate at least some of the model parameters so that the model might become more finely tuned. This is described in the following section.

lr

0.6 0.4

0

00 0

oo 0

0

0

0

0.2 00

001

800

I

I

I

800

00

160

24.0

I

I

160

240

0

0.01u I 0.0

I

I

I

80.0

a) I

I

16.0

I

24.0

bl I

00

0.0

I

I

80.0

I

I

160

I

240

0 70 0.6 a IO

4.0-

40 C)

00

I 0.0

I

I

800 TIME

I

I

1600

I

2400

(min)

C) I

00 0.0

I

I

80.0

TIME

I

16.0

I

!

24.0

(min)

Fig. 1. Comparison of experimental and predicted data - model parameters from the literature, case I.

Fig. 2. Comparison of experimental and predicted data - model parameters from the literature, case II.

4. MODEL

by combination accounts for about 40% of the total termination in MMA polymerization. When V, the ratio of k,, to k,, is set equal to 0.5, theoretical polydispersity predictions vary from 1.8 upwards, consistent with experimentally observed polydispersity values. However, it should be noted that the contribution of disproportionation to the overall termination increases with an increase in the reaction temperature [23, 151. This latter effect of temperature on the modes of termination was not taken into account in this study, and for the temperature range used a value of v = 0.5 was deemed satisfactory. The rate constants for chain transfer to monomer and solvent are very low compared to the rate constant for propagation. The numerical values of these two chain transfer constants were taken from Brandrup and Immergut [ 191 and are cited in Appendix C. The termination rate constant k, was taken from Schmidt and Ray [20] (see Appendix C),

PARAMETER

VALUES

It is well known that not all initiator radicals produced during the initiator dissociation reaction are able to initiate a polymer chain. The initiator efficiency f, which is defined as the fraction of radicals formed in the primary step of initiator dissociation that are successful in initiating polymer chains, lies for all practical purposes in the range between 0.3 and 0.7 (Hamielec [22]). In this study, a value of 0.5 was chosen for the initiator efficiency, which is a reasonable estimate used by other workers as well [ 20). It has generally been accepted that in MMA polymerizations termination was predominantly by disproportionation. However, polydispersity values (M&V,) in the range between 1.7 and 2.0 obtained experimentally suggest that both modes of termination should be taken into account. Odian [23] and Jones et al. 1151 report that at 60 “C, termination

179

and so two parameters, namely kd and k,, remain to be re-estimated from experimental data. Each kinetic rate constant can be expressed in terms of two additional parameters through an Arrhenius relationship (4)

kj = A, exp(-Ej/RT) where kj represents the rate constants kd or

k,,Aj is the Arrhenius frequency factor Ad or A,, and Ej is the activation energy Ed or E,. According to eqn. (4), a total of four parameters, namely two pre-exponential Arrhenius frequency factors and two activation energies, are to be estimated. Hence, the problem reduces to one of estimating numerical values for the parameters appearing in a set of non-linear ordinary differential equations. It is well known that A, and Ej ineqn. (4) are highly correlated, and hence it is necessary to reparameterize the equation around the mean temperature of the experimental runs in order to reduce the amount of correlation. According to this scheme (see e.g. Agarwal and Brisk [24]) (5) and Aj'

= Aj exp(-Ej/RT,)

(6)

To in the present case is equal to 70 “C. 5. PARAMETER

ESTIMATION,

RESULTS

AND

DISCUSSION

A finite difference Levenberg-Marquardt (LM) method (Beck and Arnold [25], Bard [26]) available in both the IMSL Library (routine ZXSSQ) and in routine UWHAUS was employed for parameter estimation in combination with another IMSL routine (DGEAR, Gear [27]), which was used to integrate the set of ordinary differential eqns. (1) and (2). The Levenberg-Marquardt method combines the Gauss and steepest-descent methods. The steepest-descent method works well during the initial iterations, but the approach to the minimum becomes progressively slower. On the other hand, the Gauss method

works well and converges fast in the neighbourhood of the minimum, but does not perform in a satisfactory way (or it may even diverge) when the initial guesses are away from the minimum. The LM method starts as a steepest-descent method and progressively switches to the Gauss method when one is close to the minimum of the objective function, thus ensuring convergence as well as a reasonable speed for the algorithm. Numerical integration of the differential equations representing the batch reactor started at t = 0 with an initial guess for the unknown parameters as discussed in the previous section. At times ti at which experimental data were available, the computed values of the output ypi were compared with the measured values of the output ymi. An error (deviation) vector ei at time ti was then formed as ei = (y,i - ypi). If a is the vector of the four parameters, then a scalar function J(a) can be formed by summing the weighted squares of the errors, i.e. J(a) = f: eirWei

(7)

i=l

It should be noted here that yPi and ymi are vectors consisting of the predicted and measured values of conversion, M, and M,, respectively, and that W in eqn. (7) is a weighting matrix. The elements of the diagonal matrix W are proportional to the reciprocal of the error variance of the corresponding measurement. In eqn. (7) n is the total number of measurements that were taken. Note that at each sampling time ti three measurements, namely conversion, number and weight average molecular weights, were made. A total of six different experimental runs is reported in Figs. 1 through 4. Since the orders of magnitude of the different measurements are not the same, it is necessary to weigh these measurements such that each measurement contributes approximately the same to the objective function, which is the weighted sum of the squares of the errors in eqn. (7). For example, conversion lies between 0.0 and 1.0, whereas the order of magnitude of M, and M, is 50 000 -100 000.Hence, it is necessary to assign a smaller weight to the molecular weight part of the objective function. It was assumed that there was no correlation among

180

the three measurements, so the weighting matrix W contains only diagonal elements. After forming the objective function according to eqn. (7) by a proper choice of the weighting matrix W, the next step is to minimize the objective function by the proper choice of the parameter vector a. One of the main difficulties associated with least quares estimation is how to carry out successfully the optimization of J(a) for a highly nonlinear model. Because of the complex nature of the state trajectories, local optima may be encountered, and hence the choice of the initial guesses for the parameters is important. The kinetic parameters reported by Schmidt and Ray 1201 were found more reliable and so they were used as the initial parameter values. A total of 159 data points obtained from six different experimental conditions were used for estimating the parameters. I .o

It is noted here that it may be possible to estimate all four parameters by using conversion data only. This is equivalent to assigning zero weights to the molecular weight data through matrix W in the objective function. When the parameter estimation program was run with W = diag.(l.O, 0.0, O.O), conversion predictions agreed very well with the experimental conversion data, but molecular weight predictions differed significantly from the measurements. This suggested that molecular weight data should also be considered. By trial and error simulation runs a weighting of W = diag.(l.O, 1 X 10e8, 1 X 10e8) was eventually chosen. Appendix C gives the final parameter values obtained. Figs. 3 and 4 show model predictions (solid lines in the plots) along with the experimental data. From these figures it can easily be seen that model predictions for conversion are in good agreement with the experimental data for all temperature levels. It can also be

‘.OY

0)

0.0 0.6 0.4 Cl A -

0.6 0.6

16.0

80.0

0.0

70 75

MODEL

0.2 _r/

O.oJ 0.05

RESULT3

0.0.

24.0

al 80.0

0.0

16.0

24.0

I

10.01

6.0 -0 4.0-A-a 0.2 b)

0.01

1

I

I

I

16.0

80.0

0.0

I

I

I

24.0

0.0 / 0.0

I

1 80.0

I

I 16.0

I

I 24.0

b)

--0

0 A

65 70 75

0.6 0.6 0.6

0.10 010 0.10

4.0-

c) I

0.0 0.0

I

I

80.0

TIME

I

I

16.0

(min

I

24.0

1

Fig. 3. Experimental and predicted data - re-estimated model parameter values, case I.

I

0.0 0.0

c) I

I

8OQ

TIME

II

16.0

I

24 .O

(min)

Fig. 4. Experimental and predicted data - re-estimated model parameter values, case II.

181

seen that there is an improvement in predicting molecular weight data compared to the predictions obtained before (see Figs. 1 and 2). It should be kept in mind that the molecular weight data which were obtained from size exclusion chromatography (SEC) are subject to measurement errors, and so the reliability of these measurements is not as good as the one of conversion measurements. It may be possible to further improve the parameter estimates if more accurate and reliable SEC measurements are available. A sound engineering judgement is always necessary in order to interpret and appreciate the results of a parameter estimation stage, results which can further be used for reactor optimization or control studies.

ACKNOWLEDGMENT

5 M. Hwang and J. H. Seinfeld, AZChE J., 18 (1972) 90. 6 R. N. Bergman, R. E. KaIaba and K. Springran, J. Optimiz. Theor. Appl., 20 (1976) 47. 7 L. H. Hosten, Comput. Chem. Eng., 3 (1979) 117. 8 K. Watantabe and D. M. Himmelblau, AZChE J., 29 (1983) 789. 9 N. Kalogerakis and R. Luss, AZChE J., 29 (1983) 858. 10 S. Jang, B. Joseph and H. Mukai, I and EC, Proc. Des. Deu., 25 (1986) 809. 11 L. T. Biegler, J. J. Damiano and G. E. Blau, AZChE J., 32 (1986) 29. 12 M. G. Lord, M. Eng. Thesis, Dept. Chem. Eng., McMaster Univ., 1984. 13 L. H. Garcia-Rubio, M. G. Lord, J. F. MacGregor and A. E. Hamielec, Polymer, 26 (1985) 2001. 14 D. Bhattacharya and A. E. Hamielec, Polymer, 27 (1986) 611. 15 K. M. Jones, D. Bhattacharya, J. L. Brash and A. E. Hamielec, Polymer, 27 (1986) 602. 16 I. M. Yaraskavitcb, J. L. Brash and A. E. Hamielec, Polymer, 28 (1987) 489. Ph.D. Thesis, Dept. Chem. 17 S. R. Ponnuswamy, Eng., Univ. of Alberta, 1984. 18 S. R. Ponnuswamy, S. L. Shah and C. Kiparissides, J. Appl. Polym. Sci., 32 (1986) 3239. 19 J. Brandrup and E. H. Immergut, Polymer Handbook, Interscience, New York, 1975. 20 A. D. Schmidt and W. H. Ray, Chem. Eng. Sci., 36 (1981) 1401. 21 B. M. Louie, G. M. Carratt and D. S. Soong, J. Appl. Polym. Sci., 30 (1985) 3985. Reac22 A. E. Hamielec, Notes on Polymerization tion Engineering, McMaster Univ., 1986. Wiley, 23 G. Odian, Principles of Polymerization, New York, 1981. 24 A. K. Agarwal and M. L. Brisk, Znd. Eng. Chem., Proc. Des. Dev., 24 (1985) 203. 25 J. V. Beck and K. J. Arnold, Parameter Estimation in Engineering and Science, Wiley, New York, 1977. Estimation, 26 Y. Bard, Nonlinear Pammeter Academic Press, New York, 1974. 27 G. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-HaU, New Jersey, 1971.

The authors are grateful to NSERC, Canada, for financial support.

APPENDIX

6. CONCLUSIONS

In this work we have reported on the estimation of parameters of a non-linear model for a batch polymerization reactor, when measurements of conversion, numberand weight-average molecular weights are available at different experimental conditions. It was found that proper selection of the elements of a weighting matrix in the objective function is important. It was further noted that after the parameter estimation stage, the model was able to predict measured conversion and molecular weight results reasonably well. Some slight discrepancies observed in the molecular weight predictions are attributed to difficulties associated with the molecular weight measurements using SEC.

REFERENCES H. H. Rosenbrock and C. Storey, Computational Techniques for Chemical Engineering, Pergamon, New York, 1966. J. K. Donnely and D. Quon, Can. J. Chem. Eng., 48 (1970) 114. R. E. Nieman and D. G. Fisher, Znd. and Eng. Fundam., 9 (1970) 28. B. C. Wang and R. Luss, Znt. J. Control, 31 (1980) 947.

A: NOMENCLATURE

Ad, A,, arrhenius pre-exponential frequency factors for initiator dissociation, At

e

Ed,Em Et

propagation and termination respectively. A prime denotes reparameterized rate constants error vector (deviations between measured and predicted outputs) activation energy for initiator dissociation, propagation and termination respectively (Cal mol-‘)

182

f

Ymi

initiator efficiency initiator concentration (mol 1-i) initial initiator concentration (mol l-1) objective function for optimization, see eqn. (7) dissociation rate constant (s-l) rate constant for transfer to monomer (1 mall’ s-l) propagation rate constant (1 mol-’ s-i) rate constant for transfer to solvent (1 mol-’ s-l) rate constant for termination by combination (1 mol-’ s-l) rate constant for termination by disproportionation (1 mol-’ s-l) = 4, + ktd monomer concentration (mol 1-i) initial monomer concentration (mol 1-i) number average molecular weight (g mol-‘) weight average molecular weight (g mol-‘) universal gas constant (1.987 cal mol-’ K-’ ) solvent concentration (mol 1-l) time time at which measurements are available temperature of reaction (K or “C) weighting matrix in the objective function, see eqn. (7) monomer molecular weight (g mol-‘) conversion measured output vector at time

Ypi

predicted

I IO

J(a) k, kn kJ k, k tc k td kt

M MO

M,

WV

R S t ti

T W Will X

ti

output

vector

at time

ti

Greek symbols x0, x19 zeroth, first and second moments of live radicals respectively (mol x2 1-1) zeroth moment of dead polymer PO (mol 1-l) I42

second

moment

of dead polymer

(mol 1-l)

i.

= kt,lkt volume fraction reaction mixture

of solvent

in the

APPENDIXB:EXPERIhcfENTALDATARECORDS

Experiment number 1 Temperature Initiator concentration Volume fraction of solvent Time (min)

Conversion

30 60 90 120 150 180 210 240 270

0.0600 0.1370 0.2210 0.2870 0.3660 0.4370 0.4940 0.5510 0.6050

Experiment number 2 Temperature Initiator concentration Volume fraction of solvent

65 “C 0.050 mol 1-i 0.600 MW 85400 83100 82100 76900 74200 77700 74500 70700 60700

148000 145000 139000 134000 130000 132000 127000 122000 111000

70 “c 0.050 mol 1-i 0.600

Time (min)

Conversion

M,.,

MW

20 40 60 80 105 120 140 160 180

0.1080 0.1970 0.2850 0.3590 0.4450 0.4800 0.5340 0.5780 0.6170

76400 73400 64832 63861 66900 65200 62100 59400 52800

126000 123000 111980 112284 111000 110000 107000 102000 95000

Experiment number 3 Temperature Initiator concentration Volume fraction of solvent

75 “c 0.050 mol 1-l 0.600

Time (min)

Conversion

Mn

WV

20 40 60 80 100 120 140 160 180

0.1980 0.3130 0.4280 0.5220 0.5940 0.6530 0.7530 0.7650 0.8230

65500 54000 48800 46500 45700 43800 34400 36000 32100

105000 93800 86700 82700 81400 78400 64800 68900 63000

183

Experiment number 4 Temperature Initiator concentration Volume fraction of solvent

A P P E N D I X C: N U M E R I C A L V A L U E S

65°C 0.100 mol 1-1 0.600

Time (min)

Conversion

Mn

Mw

30 60 90 120 150 180 210 240 270

0.0900 0.1890 0.2860 0.3690 0.4500 0.5170 0.5770 0.6450 0.6860

76900 58900 51900 54800 45700 49200 44800 41900 40600

125000 106000 97200 97900 88900 88800 86200 83900 83300

Parameters Parameter

L i t e r a t u r e value

Ad Ed Ap Ep

1.69 X 1014 30000 4.92 X l0 s 4353

Ad t

70 °C 0.100 mol 1-1 0.600

Time (min)

Conversion

Mn

Mw

20 40 60 80 100 120 140 160 180

0.1340 0.2490 0.3740 0.4520 0.5360 0.5950 0.6340 0.6750 0.7140

57900 46000 43500 42300 39000 38300 34800 34100 33800

95550 78100 75700 71700 69200 67900 64000 61500 62800

2 . 6 8 6 × 1017 35010 8.73 × 10 a 9540

Correlation matrix Ad f

Experiment number 5 Temperature Initiator concentration Volume fraction of solvent

E s t i m a t e d value

Ed Apr Ep

Ed

Ap t

Ep

1.0 --0.05 0.7968

1.0 0.0471

1.0

1.0

0.014 0.7905 --0.0496

Confidence limits for each parameter

Experiment number 6 Temperature Initiator concentration Volume fraction o f sol/Jent

75 °C 0.100 mol 1-1 0.600

T i m e (rain)

Conversion

Mn

Mw

15 30 45 60 75 90 105 120

0.2000 0.3250 0.4430 0.5250 0.6090 0.6750 0.7480 0.7800

44500 41000 40700 35100 35800 32600 31900 30500

72200 71400 69300 63000 64200 58700 58400 57500

Ad t

Ed

Ap t

Ep

0 . 1 1 8 8 × 10 - s 0 . 1 1 1 3 × 10 - 4

36870 33160

714 682

10850 8230

Rate constants For chain transfer T(°C)

C s X 104

C m X 104

60 70 80

0.190 0.567 0.910

0.00515 -0.0240

C s = k s / k p ; C m ffi k m / k p.

For the termination rate constant At = 9.8 × 107 E t = 701.0