a study of chain addition polymerizations with temperature variations-ii thermal runaway and...

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J. A. Biesenberger, R . C a p i n pi n , a n d ] . C. Yang

Table 1. Kinetic Steps and Rate Functions for Free Radical Polymerizations~~

Reaction Nature Rate Function~~

{Decomposition of initiator, or initiation of kinetic chains)dm Po

P + m A I

P + m A

Px + P

{Initiation of polymer chains, or propagation of kinetic chains} R i = ki[m,l

R, = kJm1 P Iropagation of polymer and kinetic chains

Rt = kt[PI2

where kt = kt, + k x+r

*mx + mr

Termination of kinetic and polymer chains by combination

Termination of kinetic and polymer chains by disproportionation

rate k d , i.e., k i = f k,. The use off is customary and

accounts for imperfect catalytic efficiency.

Partly dimensionless material and energy balances for

free-radical polymerization in a well-mixed batch re-

actor with heat removal, as they appeared in the first

paper l), ppear again in Table 2, but without the

gel-effect function (G = 1)and with typographical er-

rors corrected. Corresponding characteristic times ap-

pear in Table 3. It should belnoted that characteristic

time for initiator depletion, &, has been defined moreappropriately in the present study

- -

than in previous ones, hi 1, 4). In this way initiator

efficiency is properly included, and the resulting mod-

ified dead-end parameter

a; h,/A; = ~ k / f (2)

becomes analogous, as it should, to dimensionless group

$ used in an early paper 5 ) which correctly charac-

terized the dead-end phenomenon. In fact a; = 2 .Completely dimensionless balances are shown in Tu-

bles 4 and 5. New dimensionless variables and parame-

ters have been introduced in order to conform, where

possible, to existing literature on thermal ignition in com-

bustion processes (6-8), which has come to our atten-

tion subsequent to completion of our early studies on

ignition in polymerization processes 1,3). It must be

emphasized that, while thermal ignition during

polymerization does not generally result in flames ordetonation, the possibility of runaway temperatu re with

thermal instability is very real, as it is with all exother-

mic chemical reactions, and its formal description is

identical to that of thermal explosions (9). Only the

specific kinetics involved and the possible influence ofrunaway temperatures on polymer properties are as-

pects of ignition phenomena that are unique to

polymerization.

It should be noted that a new dimensionless tempera-

ture 8 and a new characteristic time had have been

introduced. The former conforms to the literature (6-9)

and automatically gives rise to the latter (see E q 5-2 in

Table 5 ) , which is identical to Frank-Kamenetskii'sadiabatic induction period , and which turns ou t to be

more useful in formulating an ignition criterion, (a < I ,

than A was in our previous paper (I), yT< I , as we

shall presently show.

Table 2. Partly Dimensionless Balance Equations for BatchPolymerizations

Initiator balance

dmo-t l m..exp[E;(

41Monomer balance

dm - m, e x ~ [ ;( z ]dt x& T' + 1

+ (3 + 2r) mmt exp[ E&( &)]Am

+ (2A,aifr ) x ~ mzexp[ 2 - E;) ( T ' T l ) ]

Energy balance

Assumptions: constant p and QSSADefinitions of dimensionless variables

(2-5)

I02 POLYMER ENGINEERING A ND SCIENCE, FEBRUARY, 1976 Vol. 16 No. 2



A Study of Chain Addi t ion Polymerizat ions wi th Temperature Variat ions: I I .

Table 3. Some Kinetic and Thermal Characteristic Times

Process

Table 4. Completely Dimensionless Balance Equatio ns forBatch Polymerizations

Energy Balance

6 = B6- 0 - 0,)

Monomer balance-1st moment

Initiator ba la nc eqt h moment

Second moment

(3 + 2r) (1 - @) (1 - 14

aB exp (A)( 2 + r) (YN),(~- @)'

aBd

d7where . =- nd 7 = t/AR

Instantaneous DP

Dimensional analysis has indicated that eight dimen-

sionless parameters are necessary to describe our sys-

tem. They are listed in Table 6 and their significance in

terms of characteristic times is shown in Table 7.

Table 5. Some Approximate Equations

Dimensionless monomer balance

Partly dimensionless energy,balancF

Completely dimensionless,energy,balance

(5-3)

Instantaneous DP . -

Assumptions: LCA, in addi tion to those made in Table 2.

Table 6. Dimensionless Parameters

Definition Classification

eR= E ( T ~ TJ/R,T: ThermalE = RTE Thermala = URgT,2/(-AH)I Aa,E,,[m,[mJd Thermal

B = -AH)E,,[~~,/PC,R,T,~ Monomer(reactant)

b = (-AH)fA,,E,,[ml,[m,l~ exp[-(E,, - El)/ InitiatorRgToI/AipCuRgT,2 (reactant)

Initiator(reactant)

(vN), Aap[mlo exp[-(Eap - E~/RgT,IIAi[m~l~ Polymer(product)

r A,IAt Polymer(product)

exp(- ERTd

EE[= EJEap

Table 7. Significance of Parameters

RESULTS

In the p resent analysis all equations (in T a bl e 4 ) were

solved numerically on a high-speed digital computer.

Two approaches were used. In the first, specific val-

ues were assigned to all reaction and reactor parameters

listed in Table 8, and the balance equations were sub-

sequently solved. Thermodynamic and transport prop-

ert ies were chosen with reference to a particular system,

mostly styrene, initiator properties with reference to

three specific initiators, azobisisobutyronitrile (AIBN),

benzoyl peroxide (BP), and di-t-butylperoxide (DTBP),

and heat transport parameters with reference to our own

ignition-point apparatus (to be described in the third

paper of this series). Feed conditions were systemati-

cally varied. The occurrence of runaway vs non-runaway

POLYMER ENG INEERING AN D SCIENCE , FEBRUARY, 1976, Vol. 16, No. 2 103



1.A . B ie se n be rg e r, R . C a p i n p i n , a n d ] . C. Yung

Table 8. Reaction and Reactor Parameters forBatch Polymerizations

Monomer-polymer systemThermodynamic -AH, PC,Kinetic r, A,, E,, At, t

Feed conditions L~I,,,m,],; TI)Heat transport TR, UlI where I = V/A,

Initiator-type Ai = f , Ei = E d

Thermodynamic and Kinetic Properties

Monomer Styrene MMA AN

Density, glcc 0.906 0.950C cal/g C 0.4 0.3BulkConc., moleslliter 8.7 9.5(-AH),Kcallmole 16.7 13.5A,, liter/mol-sec 4.5 x 106 9.0x l o 5

A,, iterlmol-sec 6.0 10' 1 1 x 10'E,, Kcallmole 6.4 4.7E,,Kcal/mole 1.9 1.2

Initiator Kinetic Constants

0.8060.415.218.4

3.0 x 1073.3x 10'24.15.4

Benzoyl Azobis-iso- di-tert-Butyl

peroxide butyronitrile peroxideInitiator (BPI (AIBN) (DTBP)

Ad, sec-' 1.0 1019 1.0 x 10'5 4.3 x 10'5Ei, calimole 30.0 30.5 37.0f 1 .o 0.6 1 .o

was thus observed as a function of initiator type, feed

conditions, and heat transport parameters.

The second approach was similar to that employed byBarkelew (10). Dimensionless parameters, listed in

Table 6, were varied, one at a time, in order to trace the

ignition boundary in dimensionless space. Upper and

lower bounds were placed upon the dimensionless pa-

rameters by using the extreme values computed from allpossible combinations of properties (Table 8) for the

three-by-three system comprising styrene, acrylonitrile

and methyl methacrylate monomers and AIBN, BP and

DTBP initiators within the following range of feed con-

ditions: 50°C T , 200°C; 0.005 < [rn,], 0.20.Additional constraints imposed upon these computa-

tions were first that the finalDP, XN) , , in all cases was not

permitted to fall below 500 and second that all polymeri-

zations were of the bulk-type. Typical values of the

dimensionless parameters for this three-by-three sys-

tem are listed in Table 9. Parameter 'a' is not shown

since it contains the heat transport property Ull which is

peculiar to a given reactor.

More than sixty-two computer runs were made using

the first approach. Param-eters and variables for some of

these are listed in Table 1 0-15;where case numbers arelisted. Those cases omitted have been tabulated

elsewhere (2, 3 ) . From Table 11 it can be seen that

initiator type (Ai ,Ei),eed conditions ([m,],,[rn],, T o ) ,

coolant temperature (TR) nd heat transport characteris-

tics (Ull)were varied. It is clear from Table 13 that all

cases shown lie on the boundary between runaway and

uon-runaway. Some results have also been plotted in

Figs. 18 .

Temperature profiles for cases 1-3 and 4-6 were pre-

viously plotted in Figs. 15 and 16, respectively, of refer-

ence 1 . They show sensitivity of runaway to feed tem-

perature (cases 1-3) and initiator concentration (cases4-6).DP drift profiles for these cases have been plotted

in Figs. 1 and 2 of this paper and the dispersion profile

for cases 1-3 appears in Fig. 3 .

Temperature profiles showing sensitivity to coolant

temperature ( T R) nd heat transfer coefficient (Ull) are

illustrated in Figs. 4 and 5 , respectively. It should be

pointed out that in all cases cited thus far, a$ has had a

value less than unit.

The graphs in Figs. 6-8,however, correspond to cases

42-49 for which a has a value slightly in excess of unity.

We note that sensitivity of runaway with respect to the

parameters varied has disappeared. Furthermore, in

contrast to cases 1-3 (Fig. 3 ) or which runaway contrib-uted to broadening the product dispersion, examination

of Fig. 8 reveals that high-temperature runaways pro-

duce less dispersed products than their low-

temperature counterparts. The explanation is evident

from the graph. Broadening is due to a sudden thermal

quench which is possible in cases 46 and 47, but which is

prevented by early dead-ending in cases 48 and 49.

Figures 9-22 illustrate some results using our second

computational approach. These graphs are entirely in

dimensionless form and thus do not correspond to a

Table 9. Some Values of Dimensionless Parameters

PAN PAN PAN PMMA PMMA PMMA PS PS PSBP AlBN DTBP BP AlBN DTBP BP AlBN DTBP

T,, = 80°C

B 57.46 58.34 69.72 34.71 35.1 7 41.02 33.1 1 33.52 38.78b 25000 1256 21 5000 1182 59.29 14010 704.3 55.77 13070E 0.04277 0.0421 0.03525 0.03672 0.03625 0.031 4 0.03430 0.03388 0.02929'EE: 1.829 1.832 1.859 1.571 1.576 1.637 1.467 1.473 1.545( Y N ) ~ , 66140 27270 468600* 6468 2667 32410 9252 241 3 29320f 1 .o 0.6 1 .o 1 .o 0.6 1 .o 1 .o 0.6 1 .or 1 .o 1 .o 1 .o 0 0 0.0 0.0 1 .o 1 .o 1 o

To = 120°C

B 46.36 47.06 56.25 28.01 28.37 33.14 26.72 27.04 31.29

b 2803 135.8 14500 196.0 9.476 1379 142.1 10.85 1587E 0.04762 0.04698 0.03924 0.04088 0.04036 0.03455 0.0381 0.03772 0.0326EE: 1.829 1.832 1.859 1.571 1.576 1.637 1.467 1.473 1.545'y ) 9190 3655 391 90 1330 528.8 4010 231 4 581.9 441 8

1 .o 0.6 1 .o 1 .o 0.6 1 .o 1 .o 0.6 1 .or 1 .o 1 .o 1 .o 0.0 0.0 0.0 1 .o 1 .o 1 .o. nusual ly la rge va lue of v,) 1s d u e to the sma l l va l u e of (K,),, for DTBP

104 POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1976, Vol. 16 No. 2



A Study of Chain Addi t ion Polymerizations wi th Temperature V ariat ions:11.

particular system. Parametric sensitivity of runaway is

shown in Figs. 9-11 and the absence of sensitivity in Fig.

12.

Table 10. Fixed Parameters

Figure 13 illustrates how we typically distinguished

between runaway and non-runaway polymerizations byexamining their temperature profiles for the onset of

Figures 14pward concavity (increase in slope, _).eUI

and 15 illustrate how we typically determined whether

r f sec mol sec mol cc C mot moPK or not a runaway was sensitive to Darameter variation. In

A,, Ep, At, Ei, PC,, -AH, -AS ,I lmol Kcall I /mol Kcall call Kcall call

these cases a qualitative judgment was made as to how

sharply the tempera ture maximum 8, rises in response

to a variation in a reaction Darameter, i .e .. how steeD is

1 [ .5 x lo6 6.4 6 x lo7 1.9 [~:~~6.7 32

+Cases 1-14. 40, 41, 46-49

;Cases 15-39. 42-45, 50-62

Used to compute T,'Cases 1-14

Cases 15-62

the 6 , vs parameter curve. W e observe in Figs. 14 and

15 that the disappearance of runaway sensitivity, (igni-

tion behavior) is gradual, i.e., is not itself sensitive to

Table 11. Adjustable Parameters

Case sec-I Kcallmol molK moll1 "C C callcc sec C

Ad = Ailf, Ed = 6, [ m J [ml TO TR, UII x 103,

123456789101 1

1213142122

23242526272829303132333435383940

41424344454647484950515253545556

575859606162

1015

1015

1015

1015

10151015

1015

1015

1015

1015

1015

10151015101510'5

10'510'5

1015

10'510'5

1015

1 0l5

10lS

10l5

10'51 or310l5

4.3 x 10154.3 x 10154.3 x 1015

10151015

101510151015

1015

10131013

1013

1013

1013

1013

10131013

1013

1013

4.3 x 10'51 OI5

10'51 0l5

10'5

10'51OrJ

1OL5

30.530.530.530.530.530.530.530.530.530.530.530.530.530.530.530.5

30.530.530.530.530.530.530.530.030.030.030.037.037.037.037.030.5

30.530.530.530.530.530.530.530.530.530303030303030

303030303030

0.020.020.02

0.020.020.020.020.020.020.020.020.1560.158

0.1 00.0320.0340.020.020.00400.00420.160.180.00860.00880.0400.0440.00200.00220.086

0.0870.0010.001 20.0010.00200.00870.00870.00870.00870.200.200.1 00.100.010.010.01

0.010.010.010.010.010.01

8.7 588.7 598.7 608.7 678.7 678.7 678.7 678.7 678.7 678.7 678.7 676.7 676.8 678.7 678.7 708.7 70

8.7 708.7 578.7 578.7 598.7 608.7 708.7 708.7 708.7 708.7 908.7 908.7 1008.7 1008.7 1208.7 1208.7 70

8.7 708.7 808.7 808.7 808.7 808.7 1008.7 1008.7 1008.7 1008.7 88.58.7 89.08.7 93.68.7 93.78.7 111.48.7 111.78.7 1 1 5.5

8.7 115.88.7 116.08.7 120.08.7 121.08.7 121.78.7 122.0

47474747474739.5404147474747477070

705757596070707070909010010012012070

708080808010010010010088.589.093.693.7111.4111.71 15.5

1 1 5.81 1 6.0120.0121 o121.7122.0

0.450.450.450.450.450.450.450.450.450.59980.59940.450.450.45

1212

121.81.81.81.81.81.81.81.81.81.81.81.81.81.86.72

6.721.81.81.81.8

16.514.612.811.98.08.08.08.08.08.010.29

10.2910.2914.414.414.414.4

POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1976, Vol . 16 No. 2 105



I A . Biesenberger, R . Cap in p in , an d J. C . Yang

Table 12. Dimensionless Parameters

Case € a eyT B = H$c O R €€: b = B/& i

1234

56789

1011121314212223242526

27202930313233343538394041424344

45464748495051525354555657585960

6162

0.031 70.031 80.031 90.0327

0.03270.03270.03270.03270.03270.03270.03270.03270.03270.03270.03290.03290.03290.031 60.031 60.0318

0.03190.03290.03290.03330.03330.03520.03520.0310

0.03100.03260.03260.03290.03290.03390.03390.0339

0.03390.03580.03580.03580.03580.03510.03520.03560.03560.03740.03740.03780.03780.03780.03820.0383

0.03840.0384

0.8270.7570.6920.507

0.5040.5000.3790.3790.3790.5070.5070.4940.4870.3791.991.981.972.011.952.13

1.961.871.822.071.951.911.892.041.941.941.851.881.871.681.531.41

1.191.491.331.160.9321.971.901.931.911.831.791.811.771.751.911.80

1.721.69

30.830.630.429.1

29.129.129.129.129.129.129.122.422.729.135.935.935.938.638.638.4

37.935.935.935.435.431.531.534.834.831.631.628.628.633.633.633.6

33.624.224.224.224.231.831.730.930.928.128.127.527.527.526.926.8

26.726.6

- .04-1.13-1.22-1.81

-1.81-1 3 1-2.48-2.42-2.36-1 .81-1.81-1 .81-1.81-1.81

0000

00

0000

0

0

0

0

00

0

00000

000

00000

00000

000

00

0.1 840.1920.2000.364

0.3620.3580.2720.2720.2720.2720.2720.2720.2720.2720.07840.07790.07740.09830.09540.136

0.1 420.4900.4780.01 120.01 050.1040.1 030.01270.01 210.1390.1330.1 0560.1 0501.471.341.24

1.041.061.061.061.060.02050.02090.03480.03500.2020.2040.2310.2330.2340.2660.274

0.2800.283

1.471.471.471.47

1.471.471.471.471.471.471.471.471.471.471.471.471.471.471.471.47

1.471.471.471.471.471.461.461.541.541.541.541.471.471.471.471.47

1.471.481.481.481.481.471.471.471.471.471.471.471.471.471.471.47

1.471.47

16816015280.0

80.581.5

1071071071071078283

107456459462393405280

26773.375.1

31 593350

302306

27482882

226237271273232527

3222.822.822.822.8

15501520

88783313913811911811710197.6

95.294.1

1.151.171.181 oo

1.111.111.061.081.121.101.101.131.141.461.321.331.341.301.341.23

1.341.411.441.271.341.371.381.301.361.351.421.401.411.571.721.86

2.221.761.982.262.811.331.381.361.371.431.461.451.471.491.371.45

1.521.54

parameter variation. Thus, we conclude that parametric

sensitivity is not parametrically sensitive. However, a

change in the slope at the inflection point was detec ted

(Fig. 16)which aided us in making the necessary qualita-

tive judgment .Thermal ignition boundaries for those runaways

which were parametrically sensitive have been plotted

in Figs . 17-22. They represent the product of many

numerical solutions of our dimensionless system equa-

tions and they cover a wide range of values for eachdimensionless parameter. They should thus apply to a

wide class of free-radical chain-addition polymerizations

providing, of course, that such systems exhibit “ideal-

like” behavior up to the point of onset of thermal runa-

way. We note that the ignition boundaries are limited to

values of B and b in excess of 20 (Fig. 17) and 100 (Fig.

18), espectively, below which runaway was adjudged to

be insensitive to these parameters by the procedure

described above.

In F ig . 18 it is noted that when b 2000, the curves

become relatively independent of 6 . Figure 2 0 shows

the effect ofparameter EE ; It should be pointed out thatfor a given monomer-initiator system, the value of&Is

constant.In Fig. 22, a comparison between computer ignition

boundaries and the theoretical ignition boundary,

Z e,a,O,) = 1, is shown. It is seen that for very large

values of B and b , the computer boundaries approach

106 POLYMER ENGINEERING A ND SCIENCE, FEBRUARY, 1976, Vol. 16 No. 2



A Study of Chain Addi t ion Polymerizat ions wi t h Tem perature Variat ions: 11.

Table 13. igni t ion Var iables

Ca s e T,, C T,, C Tf, C IG $ 1 ~ ' G T;G

12

34

567

8

9101112131421222324

25

26

272829303132

3334

3538

394041

424344

45464748495051525354

55

5657585960

6162

585960

67

67676767676767676767

70707057

57

59

607070707090

90100

100120120

7070

8080

80

8010010010010088.589.093.693.7

111.7111.4115.4115.8116.0120.0121 o121.7122.0

62 47 No142 142 Yes171 171 Yes

74 47 No

150 150 Yes160 160 Yes68.8 39.5 No

171 171177 17781 47

152 15274 47

151 151

183 18393 70

211 211221 221

76 57200 200

70.6 59

187 18795.6 70

152 15185 70

302 302118.5 90

223 233118 100

299 299140.7 120246 246

97 70167.5 163

100 80124 80139 131

159 159130 100150 100164 155168 165100.0 89.2307.8 307.8122.4 93.9267.2 267.2143.3 111.6216.0 210.5147.4 115.7198.6 116.0229.5 225.3139.2 120.2148.9 121.2

213.1 213.1231.7 231.7

YesYesNoYesNo

YesYesNoYesYesNoYesNo

YesNoYesNoYesNo

YesNoYesNoYesNoYes

-

-

--

-

-

NoYesNoYesNo

YesNo

NoYesNo

No

YesYes

-

0.188 0.47 6.5- -

- -

0.253 0.40 4.0

- -

0.132 0.30 2.4

-

0.231 0.45 3.5

0.180 0.37 2.7

- - -

- -

0.047 0.28 7.5

-

0.042 0.22 6.5

0.065 0.23 6.6

0.181 0.20 5.2

0.005 0.25 6.5

0.058 0.27 6.3

0.006 0.22 6.1

0.075 0.24 5.4

0.066 0.29 6.1

- -

- - -

- -

-

- -

- -

- - -

-

- -

0.0111 0.25 5.9

0.022 0.29 6.5

0.110 0.26 5.2

- - -

- -

- - -

- - -

0.117 0.25 4.8-

- -

0.138 0.25 4.60.133 0.23 4.2

. . . . . . . . . . . . . . . . . . . . . . . . ~ ~~

* These values correspond t o = 2 at whic h temperature IS arbitrarily defined as

having ignited

the theoretical boundary. From numerous computer

solutions, the results show that when B 500 and b2000, the computer ignition boundary approaches the

theoretical boundary to within 6 percen t. However, for

free-radical polymerizations, the value of B rarely ex-

ceeds 100 and therefore, the computer boundary will

deviate from the theoretical one. Thus the ignition

criterionZ(E,a,O,J> 1.35(for0, = 0)was recommended(3)rather than Z e,a,0,) > 1. It was found that when 0,

decreases below 0 (for fixed values of the o ther parame-

ters), the value of I ( ~ , a , 0 ~ )t ignition decreases and

approaches 1. This result is confirmed by the corre-

Table 14. Polymer izat ion Var iables

Ca se T,, C Tf, C 7; clrf Of D-E

1234

5678

91011121314

21

22232425

26

2728

2930

313233

343538394041424344

4546474849

50515253

54555657

585960

6162

62 47142 142171 171

74 47

150 150160 160

171 171177 17781 47

152 15274 47

151 151183 18393 70

211 211221 221

76 57200 200

70.6 59

187 18795.6 70

152 15185 70

302 302118.5 90233 233118 100

299 299140.7 120246 246

97 70167.5 163

100 80124 80139 131

159 159130 100150 100164 155168 165

110.0 89.2307.8 307.8122.4 93.9267.2 411.3

143.3 111.6216.0 210.5147.4 115.7198.6 116.0

229.5 225.3139.2 120.2148.9 121.2

213.1 213.1231.7 231.7

68.8 39.5

2.557.443.74

4.363.25

2.54

1.68

5.09

3.050.903

374

564

344

257

17510.18.83

7.69

8.21

6.94

7.87

8.37

7.25

6.92

8.99

203

234

438

176

153

168

173

167

205119

3.85

2.3716965.4

2.852.03

7.44

9.1 4

7.67

50.1 1

6.78

945.3

131.1

157.3

162.2

176.5170.5

6.875.95

0.3950.999

0.9990.564

0.9990.999

0.3060.9990.9990.551

0.9990.4650.9990.9990.278

0.9990.999

0.331

0.9990.405

0.9990.9990.9990.0380.9990.378

0.9990.0470.9990.4500.9990.4440.9990.9990.9990.999

0.9990.9990.9990.9990.9990.07970.9990.1390.9990.6450.9990.7020.9880.9990.7050.761

0.9910.999

0.970 No0.795 Yes0.595 Yes0.970 No

0.654 Yes0.576 Yes

0.970 No0.593 Yes0.520 Yes0.970 No0.732 Yes0.970 No0.699 Yes0.445 Yes0.950 No0.789 Yes0.734 Yes

0.950 No0.624 Yes0.950 No

0.622 Yes0.936 Yes0.510 Yes0.950 No0.883 Yes0.950 No

0.744 Yes0.950 No0.772 Yes0.950 No0.633 Yes0.970 No0.859 Yes0.629 Yes0.460 Yes0.329 Yes

0.301 Yes0.704 Yes0.489 Yes

0.385 Yes0.369 Yes

0.950 No0.876 Yes0.950 No

0.911 Yes0.950 No0.752 Yes0.950 No0.800 Yes

0.705 Yes0.590 No

0.532 No

0.467 Yes0.408 Yes

sponding values of I , which was close to 1, for cases

1-14 of Table 12.

Ultimate dispersion (DN)for a number of computer

runs as a function of parameter a has been plotted in

Fig. 23. In essence this graph demonstrates how tem-

perature variations affect polymer dispersion.

CONCLUSIONS

The following conclusions are evident from the resultsin the tables and figures.

Chain-addition polymerizations are susceptible to

thermal runaway and, under appropriate conditions,

thermal ignition. In the thermal ignition region the

POLYMER ENGIN EERING AND SCIENCE , FEBRUARY, 1976 Vol. 16 No. 2 107



J. A. Biesenberger, R . Capinpin, and J. C. Yang

Table 15. Polymer Properties

1234

56789

1011121314212223242526

27282930313233343538394041424344

45464748495051525354555657585960

6162

62142171

74

150160

171177

81152

74151183

93211221

76200

70.6

18795.6

15285

302+1 18.5233118299+140.7246

97167.5100124139

159130150164168110.0307.8122.4267.2143.3216.0147.4198.6229.5139.2148.9

213.1231.7

68.8

321268293316

28729632629330030127830327330531027027831 5290320

29030930132025030927231 527231 528930225932031 4314

31 431431031 131 1314.5250.0308.5281.5308.5281.5308.5281.5281.5320.0314.5

281.8286.0

47142171

47

150160

171177

47152

47151183

70211221

57200

59

18770

15170

302++90

233100299++120246

701638080

131

159100100155165

89.2307.8

93.9267.2111.621 0.5115.7116.0225.3120.2121.2

213.1231.7

39.5

208268293208

287296208293300208278201273305222270278222290222

29023030022225022227222227222228920825528930431 3

31 5281302309310235.6250.0235.6243.0235.6275.0235.6267.0281.5235.6235.6

281.8286.0

9462904786528485

8448837563796379637963796379491 349866379142414151406553453696398

611888838669973291 76

1939019170341903260062410

5950019171906

1182010790

9991

83591896189618961896425041 73499649788607852475447472742565476347

621261 54

4270139010365279

1996173055191033

905306512742799

951775382

8780

1559320

2041

541407621152700

86508915318888

30618350

501 1442172

1096066774092

26211409

979771738

1123

1189159

25641309235314121228234521 72

12301148

76.26

2.073.64 .4.431.89

2.813.082.1 53.793.841.843.181.863.1 43.591.926.1 56.621.926.421.99

4.641.642.1 52.08

1.914.852.0

30.21.874.512.044.212.424.991.83

1.932.554.691.841.832.01

1.99

1.822.801.792.662.681.821.76

2.332.48

31.3

17.1

10.7

,LCA used for ( d < , b u t ot for ih) , ,

+Tm (TJ,

(T,),,, refers to the ceiling temperature corresponding to the monomer conversion at T

Ti > Td,

system is sensitive to parameters such as To, T B , [mO],,

[m],nd U/Z. The latter conclusion is apparent from the

results in the tables and graphs corresponding to cases

1-41 and 50-62.

Sensitivity (IG)does not always accompany runaway,

and runaway does not generally occur early with respectto monomer and initiator conversion, i.e., at low con-

version. We conclude that in such cases the early runa-

way assumption (ERA) employed by Semenov (3, 9) is

not always valid for polymerizations.

Thermal runaway transforms an otherwise conven-

tional polymerization (isoth CONV) into a dead-end

polymerization (thermal D-E), even under conditions

such that a; < 1.Cases 1-41 and 50-62 are examples of

this.

Runaway always reduces polymerization time, Tf andultimate DP, FN)f, often dramatically; but, interest-

ingly, it does not always increase ultimate dispersion,

(DN)f,of the degree-of-polymerization distribution,

D P D . In support of this observation are cases 42-45,

108 POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1976, Vol . 16, No. 2



A Study o f Chain Addi t ion Polymerizations wi th Tempe rature Varia t ions: 11.

46-49 (compare (DN),values in Table 13 ,compare (DN),

values in Table 17 of reference 1.

Initiators are susceptible to runaway in th e following

increasing order: DTBP, BP, AIBN; that is, lower values

I 1 I I I I

0 2 4 6 8 1 0 1 2 1 4T

Fig. 1 . DP dri f t prof i les f o r cases 1-3 i n Tab les 11 15.

CASES 4 and 5

-CUM. Yn

--- NST. TN

/-\I \\

\

\\

I \

\\\

I \

0 0 1 0.2 0.3 + 0.4 0.5 0.6 0.70

Fig . 2. DP drif t proji les fo r cases 4 und 5 i n Tables 11-15.

of T , are requ ired to produce runaway. Compare cases

15-17 with 18-20 and 32-33 with 36-37.

It appears possible to meet design objectives of high

conversion in short time rfwith an acceytable targetdegree of polymerization xN),by utilizing high values of

T o with concomitantly large temperature increases due

to incomplete removal of reaction exotherm, as oftenoccurs in industry. In fact, rising temperature profiles

lie in the direction of the optimum with respect to

minimum polymerization time (11)and, under appro-

priate D-E conditions, also with respect to minimum

drift dispersion in DPD (1, 12).

5 - CASES 1-3

/-0=6O0C

I c0 0.2 0.6 0.8 1.0

0.4 (P

Fig. 3. Cum ulat ive dispersion index dr qt prof iles o r cases 1-3 nTables 11 -15 .

0.3

0.2

T'

0 1

0

-0.1

-0.2' I I I I

Fig. 4 . Temperature dri f t prof iles o r cases 7 9 in Tables 11-15.

4

POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1976, Vol . 16, No . 2 109



J. A, Biesenberger, R . Cap in p in , an d J. C . Y a n g

0.20

A good parameter for predicting isothermal

CONV/D-E polymerization is a; = B/b , as previously

concluded 1, 4, 5). Large values of b indicate early

runaway with respect to initiator decay. We note that

cases 31 and 35, for example, which have the smallest

values of a; and the largest values of b, are extremely

conventional by nature and exhibit early runaway withrespect to $. Furthermore, their ultimate DPDs are

broad as expected because the tendency of their DPs o

-

-

0.5

0.4

0.3

0 2

T '

0.1

0

-0.1

-0.2

CASES 10 and II

U/I = 5.998 x

cal /cc-sec-oc

I I I

0 0 2 0.6 0.8 1.0

Fig. 5 . Tem perature dr i f t profi les f o r cases 10 and 1 1 in Tab le s11-15.

0.4 4

0.25

[m.].=o.0020

i r .0014

/ ///- 0.0010

0 I 2 3 4 5

zFig . 6 . Temperature drift profiles f o r cases 42-45 in Tables11-15.

110

- - - - I

4 1L I ' I ' I l l 1 I I ' I I '.4 05 0.6 07

0.3 (90 ai m

Fig. 7 . LIP dr if t profi les fo r cases 46-49 in Tables 11-15,

16.5 X I O - '

01 1 1 1 1 1

1 1 1 1 1 1 1

0.2 03 O A 0.5 0.6 070.1

Fig. 8. Temperature and cu mulat ive dispers ion index dr if t pro-

f i les fo r cases 46-49 in Tables 1 1 -1 5.

POLYMER ENGINEERING A ND SCIENCE, FEBRUARY, 1976, Vol. 16, No. 2



A Study of Chain Addition Polymerizations wit h Temperature Variations: I I .

10

Q

drift downward, owing to the conventional nature

(under isothermal conditions) of these polymerizations,

is reinforced by runaway temperatures. Even such

polymerizations dead-end under runaway conditions,

however. We note that in cases 31 and 35,T , > (TJ,..

This indicates that the results obtained for these cases

using the irreversible model are subject to doubt and toobtain more accurate theoretical predictions a kinetic

model with depolymerization should be used.

By contrast, cases 42-45 and 46-49 are strongly D-E

and borderline D-E, respectively, by nature (under;

isothermal conditions) according to their values of ak

-

8 -

I

6.32

b* 56

*0.04

?q=O.O

F i g . 9. Dimensionless temperature drift profiles showingparametric sensitivity with respect to parameter “a”.

1

o -2.00

b=256

E =0.04

€ E ; = 1.467

3000

A,. 0.0

r =1.0

I I I I6 8 10 12 14

t

Fig. 10. Dimensionless temperature drift profiles showingparametric sensitivity with respect to B .

POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1976, Vol. 16 No. 2

and b < 100. In fact, the former possess the largest

values of a;. It is interesting to note that they no longer

exhibit parametric sensitivity. We might even term

their runaway as “stable”. Furthermore, their ultimate

values of(DN),ndicate that dispersion has actually been

reduced by runaway. The same phenomenon was ob-

served in cases 7 9 (Table 17) of reference 1, and thereason for it is believed to be the compensating effect

(12) of rising temperatures, which tend to decrease

a = 1.89

8 932

b.256

€ =0.04

EE;=I.467

IL/ =3000

0 4 12 16 208 z

F i g . 1 1 . Dimensionless temperature drift profiles showingparametric sensitivity wit h respect to 0,.

8.17

‘b=256

€ =o.wE E : = 1.467

Wn&= 0008 =o.or = 1 0

0 2 4 6 8 10z

F ig . 12. Dimensionless temperature drift profiles showing non-sensitivity wi th respect to parameter “a”for small value of B .

1 1 1



J. A . Biesenberger, R . C a p i n p in , a n d ] . C. Yang

2.0

1.6-

1.2

dudr

0.8

( x ~ ) ~ ~ ~ ~ ,n the natural tendency of (;;N)inst to drift upward

under isothermal conditions (1).

In general, runaway is accompanied by small values of

‘a’ E , and large values of B , b, OR. Of course, a

necessary condition for runaway is a small value of E to

insure the requisite sharp upward curvature of the heat

generation curve (3), T)g O). It should be pointed out

that the small value of E is due mainly to the large value

of Ei or free-radical initiators, since E , and E t or most

chain addition monomers are small. Parametric sensitiv-

ity (IG)occurs for large values ofB and b , and disappears

when B <20 andb < 100.This is evident in Figs. 13-15 .

-

-

-

-

-

-

4

12

10

B = 32

8 80.04

4€:=1.467

(u, 3000

eR o.o

r = 1.0-

-

B =50

b.256

E = 0 0 4

E,’=1.467

(~,b=3000

c), = o0

r.1.o

I

0.0-

-

0 2

Fig. 13 . d8ld r us r fo r two values of parameter “a”

I

em

Fig. 14 . 8 us “a” fo r various values of b .

I4

b * 256

6 . 0 0 4

eE * 14672-3000

e , = o or - 1 0

10-

em

6 -

4 -

0 I 1 I I I I I I12 13 I4 I5 16 I7 I8 19 2 0

Fig. 1 5 . Om us “a ” fo r various values of B .

F i g . 16.Plot of the maximum slope of the curves shown in Fig. 14

us h.

195-

G E =1467

, + o= 3000

e , = o oRUNAWAY r.10

I I I I I

20 30 40 50 60 70 80 90 0

F ig . 17. Computer ignition boundaries plotted as “a” vs Bcurves with b as parameter.

112 POLYMER ENGINEERING A ND SCIENCE, FEBRUARY, 1976, Vol. 16 No. 2



A Study of Chain Addition Polymerizations with Temperature Variations: I .

b = 256

E . 0 0 4

€ E , = 1.467

Y”L=MOO

r = 0UNAWAY

I I I I I 1 I1

Its disappearance is coincident with the negation of the

ERA.

We have also noted that while D-E is exacerbated bylarge values of a; = B / b , due to either large T oor small

[m,] ,or both, susceptibility to runaway is actually miti-

gated. In fact, large values of a; sometimes will cause

sensitivity (IG) to disappear altogether. This may beobserved, e.g., in graphs of ‘a’ vs B(Figure 17) as bincreases (& decreases) at fixed values ofB. The expla-

€ = n o 4

SL = , 4 6 7

Ha.O 0

1 M ’ Y ) O OR M A W b Y

t a r , , I , ~ I , I I , I I I , r = I o , , ,

17

,0302 104

b

Fig. 18. Com pute r igni t ion boundaries plot ted as “a” vs bc urve s w i th B as parameter.

1.61 I I I I I I I

20 30 40 50 60 70 en 90 100

Fig. 19. Com puter igni t ion boundaries plot ted as “a” vs Bc urve s w i th E as parameter.

I65 I I I 1 I I I

20 30 40 50 60 TO 80 9011

Fig. 20. Com pute r igni t ion boundaries plot ted as “a” vs Bcurves wi th , as parameter.

nation is that D-E causesf(.) to drop sufficiently o offset

the rise in g ( 8 ) due to climbing temperatures.

. Criteria for the ERA are (3):

and

They insure that monomer and initiator do not drift

ap reciably from their initial values, i.e., [m] = [m] ,

way. These inequalities can also be derived by stating

that the characteristic times for monomer and initiator

depletions, h, and A t respectively, are much greater

than A during the period prior to the onset of ignition.

an8 [m,] =[mO],,espectively, prior to onset of runa-

---Theoretical IG Bound.-Computer Runs

1 - - - - - -- - ------I.5

0’5

tJ ’ ’ I 1 ’ 1 I I I

0.02 0.06 a10 014 0.18 022 0.26

€

Fig. 22. Com parison be tween theore t ical igni t ion bounda ry andcomp uter igni t ion boundaries.

POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1976, Vol. 16, No. 2 113



J. A . Biesenberger, R . Capinpin, an d] . C. Yang

t II / I

MWEAK-E

v I l l l l l l l l l l8 10

0 2 4 a 6a4 - R.A Q I

F i g . 23. Ultimate cumulative dispersion index vs “a” f o r con-

ventional, weak D-E and strong D-E polymerizations.

The use of the characteristic time approach is reasonably

since the temperature rise during the induction period

is relatively small.

For chain addition polymerizations B virtually always

exceeds 20, but not by much. Therefore, runaway is not

expected to be early with respect tom. This is confirmedby our observation that runaway was generally found to

occur at values of @ between 0.25 and 0.35.

As suggested in our previous paper (I), yT is an ac-

ceptable IG parameter, but the IG criterion is not as

simple as expected yT < 1. We have since found that

parameter a = cyT is more convenient, but the IG

criterion for the special case OR = O is not as simple for

polymerizations either as expected ( a ) from

Semenov’s theoretical analysis. Even our modified

theoretical criterion (2, 3),I(a,c,OR)> I, is not entirely

adequate for the more general case where On is not

necessarily zero andE

is not neglected with respect tounity. A primary reason for the latter is that the ERA,

which is inherent to the theory, is not generally valid for

polymerizations. I t should be pointed out that the mod-

ified theoretical criterion for OR = 0 approaches

Semenov’s criterion (to within 99.9 percent) when E <0.0006. However, E for most chain addition polymeriza-

tion is greater than 0.025. Consequently, our modified

theoretical boundary departs from Semenov’s bound-

ary.We have found in the present study on the basis of

numerous computations that the value of ‘a’ for chain-

addition polymerizations with OR = 0 at the IG point lies

within the range

1 5 a I G 5 2 (5)

(6)

where the upper limit confirms the criterion,

l a , , 6,) > 1.35

which was proposed elsewhere (3) for chain addition

polymerizations in the special case On = 0 and which we

thus recommend as the criterion to use even with mod-

era te deviations from the ERA. We note that when E =

0.04, for example, criter ion alG 2 corresponds to yT 5

1 as we suggested in an earlier paper (1).Cases 11,22,

25,27,3 1,33,3 5,37,39 ,51 and 53correspond to 0, = 0and B > 20, b >> 100 (i.e., ERA valid). For all these

cases I I G = 1.35 as expected.

We have found that in general the upper limit ofalG n

range (5) corresponds to large values of the pair B , b and

that the lower limit corresponds to small values. In fact,

when aIG akes on values at or below the lower limit, B

and b are such that IG sensitivity is gone.

SOME GENERALIZATIONS

A summary of general conclusions concerning the role

of various dimensionless parameters in characterizing

reaction behavior is presented in Table 16. The sig-

nificance of ‘a’ in determining thermal runaway and the

significance of B and b in characterizing its parametric

sensitivity, as well as the validity of the ERA, have been

established. However, the effect of temperature profile

on polymer properties is a more complex matter and

requires further elaboration. I ts elucidation will estab-

lish the quotient of reaction parameters Blb = ak as an

important property parameter of the polymer product.

varies with parameter ‘a’ for three classes of kinetic

behavior: conventional, borderline (weak) D-E and

strong D-E. All classifications are based upon parameter

a; and thus refer to how the polymerizations wouldbehave under isothermal conditions. It is instructive to

observe the wide variety of polymer dispersion charac-

teristics possible when thermal parameter ‘a’ is varied

from adiabatic conditions (a = 0) through the ignition

region (1 a 5 2) towards the quasi-isothermal regime

a >> 1). It is also interesting to note that runaway

temperatures generally appear to inflect the most dam-

age upon conventional-type polymerizations.

In summary: our findings show that conventional

polymerizations produce narrower polymers under

quasi-isothermal (Q-I) conditions than under runaway

(R-A) conditions but the re is a peculiar minimum at the

ignition boundary; borderline D-E polymerizations alsoproduce narrower polymers under Q-1 onditions than

under runaway conditions, which are also narrower than

their corresponding conventional counterpar ts, but

there is now a maximum at the ignition point; and,

strong D-E polymerizations actually produce narrower

Figure 23 illustrates how ultimate dispersion

Table 16. Summary of Effects ofDimensionless Parameters Under

Non-Runaway Conditions

ObjectivelParameter a B b E EE , OR Y~),,

Avoid R-A - - - s -

Avoid IG - s s - - - -s I - - - -vo id D-Ehort T~ s s -

Large (xd, - s l l s s - I

? = small, = large value of paramete rson ly when tE : > 1

114 POLYMER ENGINEERING AND SCIENCE, FEBRUARY, 1976, Vol. 16 No. 2



A Study of Chain Addition Polymerizations with Temperature Variations: 11.

polymers under runaway conditions than under Q-I

conditions with a similar maximum dispersion at the

ignition boundary.

These interesting results can be explained on the basis

of a coupling effect between concentration (monomer

and initiator) drift and temperature drift in terms of E 9

4 5 in Table 4 . As we have previously shown, (1, 5 ) ,ultimate DP and DPD are direct products of the drift

characteristics of (;;N)inst. It is thus convenient when

attempting to explain the above results to consider the

thermal effect to be superimposed upon a “natural

isothermal behavior (12). This is the reason we have

chosen to classify reactions according to the isothermal

behavior of (;;N)inst. Three classes of isothermal behavior

asdetermined by ai are illustrated in Fig. 24 and three

classes of thermal behavior as determined by a (or I) in

Fig. 25. Thus, nine combinations of drift behavior are

possible, which account for the wide variety of results

observed when temperature variation acts upon ulti-

mate polymer dispersion.

The fact that thermal runaway exacerbates dispersion

in conventional polymerizations (curve I in Fig. 23) is

easily explained by the fact that drift dispersion due to

both CONV (Fig.24)and quasi-adiabatic (Q-A) (Fig.25)

behavior tend to drive &,Jinst in the same direction, viz.,

downward, and thus they reinforce one another. The

polymerization dead-ends efore the thermally-induced

(D-E) upward sweep of ( x ~ ) ~ ~ ~ ~an effect drift recovery.

The minimum at the R-A boundary is due to just such

recovery which is additionally reinforced by the

/STRONG D-E

05

(P

Fig. 24. Instantaneous DP drift profiles f o r conventional, weakD-E and Strong D-E systems.

POLYMER ENG INEERING AN D SC IENCE, FEBRUARY, 1976, Vol. 16 No. 2

I

0

a = 2

0.5 0

F i g . 2.5.Dimensionless temperature drift profiles o r three typesof thermal behavior: quasi-isothermal, runaway and quasi-adiabatic.

thermal-quench (declining) portion of the R-A curve in

Fig. 25.

Drift dispersion under runaway conditions is greater

than under Q-I conditions for weak D-E polymeriza-

tions (curve 2 in Fig. 23) simply because runaway trans-

forms weak D-E into strong D-E, thereby producing a

sharp upward drift FN)inst.he maximum at the R-A

boundary is due to continual reinforcement in the drift

of FN) i n s t ,ownward, followed by upward, and then

again downward, caused by the alignment of the “R-A”

curve ofFig. 25 with the “weak-D-E” curve ofFig. 2 4 . It

is apparent that cases 46-49 (Fig. 8 ) ie to the left of thismaximum.

Finally, in perhaps the most unexpected result, viz.,

that of strong D-E (curve 3 in Fig. 23), we find that

runaway drift produces t he least dispersion. This is due

to he cancellation of the sharp upward D-E drift of

(x,Jinst (Fig. 24) by the steep rise in temperature which

acts to force xN) ins tn the opposite direction. Both effects

are present in E 9 2 4 Table2 ) ;one is a drop inm, due to

D-E and the other is a rise in T ’ convolutedby a negative

temperature coefficient E l p E l < 0).The maximum

in (DJf is due to the thermal quench in the R-A profile,

which reinforces rather than cancels the upward drift in

ACKNOWLEDGMENT

( X N ) i n s t .

This work was sponsored in part by a grant from the

National Science Foundation (GK-34079).

115



I . A . Bicscnbcrgcr, R . C a p i n p i n , a n d J .

NOMENCLATURE

A = rate constant pre-exponential

A, = wetted surface area

CP = heat capacity

D N = dispersion index

E = activation energy

l

E/R,T,

f ( ~ )

= dimensionless activation energy defined as

= defined as (1 @) (1 +)*/a

)8

1 + E 0defined as exp (

heat effect number defined in reference 1.

heat of polymerization

ignition function or (with brackets) initiator

concentration

rate constant

characteristic length defined as VIA ,

fictitious initiator concentration

monomer concentration

free-radical intermediate concentration

rate of reactioninitial rate of reaction defined as (k& [ m ] ,

initial rate of initiation defined as k i [ m, ] ,

universal gas constant

ceiling temperature

maximum temperatu re

temperature

dimensionless temperature defined as

T To)/Totime

overall heat transfer coefficientvolume

degree of polymerization

instantaneous degree of polymerization

[mold

Greek Symbols

dimensionless temperature

characteristic time

k th moment of the degree of polymerization

distribution

C . Yang

initial kinetic chain length defined in Table 6

density

dimensionless time

monomer conversion

initiator conversion

Subscripts

apparent (lumped)

decomposition

final

generation

initiation

monomer consumption or maximum value

(as in 0,)

initial value

propagation

heat removal or reservoir (as in T R )

termination

termination by combination

termination by disproportionation

x,y-mer

REFERENCES

1. J. A. Biesenberger, and R. Capinpin, Polym. E ng. Sci., 14,

2. J. A. Biesenberger, R. Capinpin, and D. Sebastian, A m .

3. J . A. Biesenberger, R. Capinpin, and D. Sehastian, to be

737 (1974).

C h e m . Soc., Diu. o fP o lym . Prep . , 16, 365 (1975).

Dublished in A.C.S . Adu. in Chem. series.

4. . A. Biesenberger and R. Capinpin,J.App l . Polym. Sci . , 16,

695 (1972).

5. Z. Tadmor and J. A . Biesenberger,]. Polym. Sci., B3, 753

6. P. H . Thomas, Proc. Roy. Soc., A262, 192 (1961).

7. W . Squire, Combust ion Flame, 7, 1 (1963).

8. J. Adler and J. W . Enig, Combust ion Flame, 8, 97 (1967).

9. D. A. Frank-Kamenetskii, “Diffusion and Heat Exchange in

Chemical Kinetics,” Princeton University Prcss (1955).

10. C. R. Barkelew, Chem. E ng. Prog. Sym p. Ser. No . 25,55,37

11. M . E. Sacks, S-I. Lee, and J . A . Biesenberger, C h e m . Eng.

12. M . E. Sacks, S-I. Lee, and J . A . Biesenberger, C h e m . Eng.

(1965).

(1959).

Sci., 27, 2281 (1972).

Sci., 28, 241 (1973).

a study of chain addition polymerizations with temperature variations-ii thermal runaway and...

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