An Analysis of Caprolactam Polymerization
M. V. TIRRELL, G. H. PEARSON, R. A. WEISS
AND R. L. LAURENCE
Departments
of
Polymer Science and Engineering
and
C,hemical Engineering
University of Massachusetts
Amherst, Mass.
A versatile model for ecaprolactam polymerization is presented. A deterministic, mathematical basis for obtaining the
most probable distribution of molecular weights in batch polymerization is developed. Continuous polycaproamide production
has been modeled and shown to give other than most probable
distribution in many cases. The effect of adding monofunctional agents has been investigated. Results of some preliminary studies toward determining the optimal reactor configuration are presented.
INTRODUCTION
R e c e n t papers by Reimschussel and coworkers
(1-4) have dealt with various aspects of nylon-6
production by hydrolytic polymerization of c-caprolactam. Focusing on reaction conversion or numberaverage molecular weight as variables of interest,
problems treated have been: condensation equilibrium in the presence of additives ( l ) ,polymerization
in a CSTR and other series configurations ( 2 ) , the
minimum time optimization problem ( 3 ) and the reequilibration of nylon-6 (4).It has been stated or
assumed in all these treatments that a most probable
distribution of polymer chain lengths describes the
reaction product. For a product like nylon-6 that is
typically processed by extrusion and melt spinning,
molecular weight distribution, through its effect on
melt viscosity, is a very important property. The most
probable distribution of molecular weights is found
in most commercial nylon-6 products (18) and does
indeed give a desirable product of uniform quality
CHEMISTRY OF CAPROLACTAM
POLYMERIZATION
There are two basic reaction schemes that are of
commercial interest for the polymerization of €-caprolactam.
The hydrolytic process (5) consists of the reactions illustrated in Table 1. There is an abundance of
kinetic information available in the literature on this
polymerization mechanism (6-10). From a chemical
Table 1. Hydrolytic Polymerization
-
Initiation: ring opening
HN-(CHn)j-C=O
+ HzO
II
NH~(CHZ)~-C--OH
ki’
(MI
(S1)
(w)
Propagation: polycondensation
NH2
(20).
With these factors in mind, we have developed a
treatment of caprolactam polymerization which can
predict not only conversion and number-average
degree of polymerization, but also can describe
mathematically the molecular weight distribution as
a function of reaction time. Our objective has been to
formulate a model adaptable enough to be applied to
a wide variety of reactor conditions and configurations. We are able to show in which reactor designs
a most probable distribution is indeed the expected
product and how it may be drastically different in a
CSTR or in other reactor configurations. In this same
vein, the common practice of adding a monofunctional agent to control molecular weight is examined
in some detail with regard to its effect on molecular
weight and molecular weight distribution.
0
ki
kz
+
+
kz’
- - - -CONH-
+ HOOC----
-
(Sn)
(Srn)
polyaddition
(Sn)
(MI
-- (W)
(Sn+m)
k3
+ HN-(CH&-C=O
----NHz
HzO
s
k3’
- - - -NHCO
(CH2)sNHz
(Sn + 1)
~
Reaction with monofunctional species:
----
NHz
+ HOOC--.---I
k4
K
(Sn)
(Am)
- - -NHCO- - -I
(An + rn)
+ Hz0
(W)
~~
386
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. 15, N o . 5
An Analysis of Caprolactam Polymerization
reaction engineering viewpoint, however, several
features are important. First of all, it has been shown
that this reaction is catalyzed by chain end groups
( B S i ) which change in concentration with reaction
conversion. This imparts an autocatalytic character
to the reaction. Secondly, this polymerization has
both step growth and chain growth character (11).
There are essential differences in the general types
of molecular weight distributions ( MWD ) obtained
in idealized step growth and chain growth polymerizations. The question of MWD will be dealt with in
more detail later in this paper. Thirdly, the reversibility of all the polymerization steps presents a mathematical challenge in modelling the system ( 12).
Caprolactam can also be readily polymerized by a
variety of anionic initiators (5, 13-16). The attractiveness of this type process lies in the speed of the
reaction. The mechanism is entirely chain growth.
This time advantage is largely negated, however, by
the necessity of allowing the narrow distribution
product to re-equilibrate to the apparently desirable
most probable distribution ( 15, 17). This factor, plus
the necessity of maintaining scrupulously anhydrous
conditions, makes this a little used reaction commercially (18). By application of the techniques demonstrated in this paper anionic polymerization of
c-caprolactam can be made a tractable mathematical
modeling problem. However, the remainder of this
paper is devoted to the hydrolytic process, a more
commercially interesting system.
dW
= - k1MW + kl'S1 +
dt
-
k,
n-l
2 2
n=l
m
m
n-1
n=l
n=l
m=l
z
m
dS1
= klMW - kl'SI - k2SI
dt
Sn-msm
m=l
Sn
n=l
z
m
+ k,'W
Sn+@ k3MSl
-+ k;Sp
n=l
m
m
dSn
= ( klMW - k,'S,)
dt
- k2'W(Sn - S,
6 <n
6 <n - 1>
- 1>)
sj 0
21-1
+ 2k,'W
j=n+l
+k3M(Sn-1-sn)
+ k;S1
MODELING TECHNIQUE
-k3'(Sn-Sn+l)
m
2 A,,,
6 <n - 1> - k4Sn
m=l
The hydrolytic process of Table 1 can be represented in the following manner:
kl
B!+WeS,
kl'
Sn
(1)
+ Sm k2 Sn+m+ W
i=
--&Idt
k4A1
sn+k,'W
n=l
2
A n + P (9)
n=l
(2)
k2'
- k4)W(An- Al 6 <n
+ M*Sn+t
k,'
k3
Sn
Sn
+ Am
k4
An+,
k41
- 1> + k:W
2
j=n+l
(3)
Aj 0
j-1
(10)
+W
where the function:
(4)
1This term is not exactly correct since an Si moleaule does not result
from every reaction between water and a polymer chain. The correct
term is
Batch Reactor
The molecular rate equations for the disappearance of the various species can be written as follows:
dM
= - kIMW + kl'S1
dt
. . . kz'W
m
Z Sn/n
*k5a
- 1.
Unfortunately, it cannot be easily
transformed to give the correct term in the moment equations to h e
developed presently. However, it is apparent that the correct term
is some fraction of the term which has been used. Our term represents an upper bound. The small magnitude of the rate constant kz'
(see Table 2) argues for the acceptance of this approximation.
?Solution of this set of equations including this term requires asS2 which has been used previously ( 5 ) and is acsuming that SI
ceptable since both concentrations are quite small. Alternatively, it
could be ignored altogether which has also been done in some theoretical studies of reversible polymerization (12).
=
3This term gives the proper statistical weight to the hydrolysis r e a c
tion of water with the specific amide linkage in a molecule of size i
which will just give a molecule of size n.
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. 15, No. 5
387
M. V.Tirrell, G. H . Pearson, R. A. Weiss and R. L. Laurence
At this point we introduce the discrete transformation of a function (21) :
m
2 s”f(n>
G[fb)lE
(12)
n=l
hence :
m
G(S,) =
C snSn==P(s),
n=l
m
G(A,) =
C PA+,=A(s).
extraction of this information from Eys 5 to 19. First
of all, it should be pointed out that the product distribution obtained from this reaction scheme will be
the sum of the distributions of unterminated and
monofunctionally terminated species. If the set of
Eys 14 to 19 could be solved to give analytical expressions for P ( s ) and A ( s ) , the inverse transformation could be applied to these functions, and analytical expression obtained for MWD as a function
of time. An analytical solution in the present case is
obviously quite out of the question. The alternative
is to obtain the moments of the polymer chain length
distribution (PCLD) defined by:
p k E
n=l
Application of this transformation to Eqs 5-10, yields
the toilowing transformed equations :
dM - - klMW + kiS1-
dt
k3Mpo
+ kip0 - kiS1
(14)
dW
dt
-- - klMW + kl’S1+
k2
m
m
(13)
C nkSn,pk‘= C nkAn
n=l
(20)
n=l
and work with some measure of the character of the
product 14WD defined in terms of its moments,
ppk = Pk
pk’. The simplest function to characterize
the MWD is the polydispersity or dispersion index
Zp, defined by:
+
po2 - k,’WI*,
Differential equations for the moments can be obtained by direct summation according to Ey 20 or by
utilization of the moment generating property of the
discrete transform ( 19,21-23).
Using the latter approach, differential equations for
the moments of the PCLD were obtained:
dP0
k2
= (klMW - kl’S1) - - po2 + k2’W(
2
dt
dP,’
dt
-k3)
- Sl)
=O
(P (s) - -1 P ( s) + Sl) + ks’Sls - k4P( s ) Po)
S
- k4’W(A(s) --A,)
+ k,‘WC [
-1
Aj
j=ntl
j-1
(19)
where Po and pfo are the zeroth moments of the
S and A distributions, respectively.
Our objective being to obtain as much information
as possible about the molecular weight distribution,
in addition to molecular weight and conversion information, there are several ways to approach the
3aa
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. 15, No. 5
An Analysis of Caproluctam Polymerization
boxyl ends will then be the correct term to use in
Eq 29. The results presented in this paper deal only
with the addition of monofunctional acid, the case
represented by E q 29 as written.
The treatment of the terms involving G [
j=n+l
A]
, and the corresponding term for Aj,
i-1
re-
quired a summation approach detailed in Appendix
A. Equutions 14-17 and 23-28 comprise a closed set
of simultaneous, nonlinear, ordinary differential
equations which were solved numerically using a
fourth order Runge-Kutta routine.
As pointed out earlier, this reaction is catalyzed by
chain ends, so we have. .
ki = k,+ k k p p
Continuous Stirred Tank Reactor
For micromixed CSTR‘s, the mass balance equations may be written directly from the batch reactor
Eqs 5-10 by incorporation of the inflow-outflow
terms, ie, ( Min-Mout)/8,etc., where e is the average
residence time. For steady state operation, the time
derivatives vanish. The equations are transformed
and the moment equations developed in a manner
paralleling that for the batch reactor. Non-linear algebraic equations result which are solved by a Newton-Raphson method. The complete set of CSTR
equations is given in Appendix B.
RESULTS AND DISCUSSION
(29)
. . . for each of
the four forward and reverse reactions. The rate constants and equilibrium constants
used in this analysis are tabulated in Table 2. It has
been shown in experimental studies of caprolactam
polymerization (24) that the addition of lactam occurs at the amino end group and is catalyzed by carboxyl groups. This means that only those chains with
uncapped amine ends will be able to add lactam and
only those chains with uncapped acid ends will be
able to affect catalysis. Therefore, if a monofunctional acid reagent is added to the reaction mixture,
some chains will react with it at the amine end and
no longer be able to add lactam. The carboxyl end
group will still be available for catalysis, as will all
the carboxyl groups of the uncapped chains. This is
the case represented by E q 29. If a monofunctional
amine reagent is added, only those chains which do
not react with it remain available for catalysis and
pop in E q 29 must be replaced by po, the zeroth moment of the uncapped chain length distribution. If
one wished to consider the addition of both types of
monofunctional reagents simultaneously (the stability
of the polymer formed would be increased by having
both ends capped) one must include a fifth molecular
rate equation for the reaction with the other type of
monofunctional reagent. The result will be that there
are now three distributions of species in the product
distribution: unterminated, monofunctionally terminated, and difunctionally terminated. The combined zeroth moment of the chains with free car-
Batch Reactor
Figures 1 and 2 compare the model presented in
this paper to the published data of Hermans, et a1
( 7 ) . We find quite a good fit to the experimental
data, especially in the initial portions of the curves.
Along with the conversion curve in Fig. 3, these
curves demonstrate the autocatalytic nature of this
reaction, The overall reaction is relatively slow until
the ring-opening has proceeded to a sufficient extent.
For this reason, some of the ideas of Kilkson (25)
with regard to the concept of slow influx in polycondensation reactions apply to this reaction. The highest rate of conversion roughly corresponds to the
maximum in the I.LO curve.
As pointed out by Hermans, et a1 ( 7 ) ,the changing nature of the reaction medium from predominantly caprolactam to predominantly polymer, makes
determination of rate constants which apply over
the entire conversion range very difficult. The rate
constants we have used, given in Table 2, are a combination of those of Hermans (7) and Wiloth (8,s).
This model may provide a means to determining conI
I
I
I
5
10
I
15
I
M
Table 2. Rate and Equilibrium Constants for Caprolactam
Polymerization at 22O0Ca
kio (kg hr-1
i
mole-1)
1
2
3
4
8.0 x 10-4
1
0.9
1.0
0.9
ki, (kg2 hr-1
mole-’4
0.17
20.0
21.0
20.0
K1
2.2
8.5
1.9
8.5
0
x 10-3
x 102
x
102
Taken from (7-9).
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Yo/. 15, No. 5
Time, h r t .
Fig. I. Monomer concentration 0s time in a batch reactor at
three initial water concentrations:
WJM, = 0.05; ( 0 )
W,/Mo = 0.067; (0)
Wo/M, = 0.1. M is in mole kg-1;
A = 0. Symbols are experimental data of Hemans, et al
(7). Solid line is predicted curve.
(6)
389
M . V. Tirrell, G . H . Paarson, R. A. Weiss and R. L. Laurence
I
0
-
I
C
0
5
10
b
15
a
time, hrr.
Fig. 2. First moment of PCLD us time in a batch reactor at
(0)
three initial water concentrat'ons:
Wo/Mo = 0.05; ( 0 )
W,/M, = 0.067; (0)
Wo/M, = 0.1. p o is in mole kg-I
x 102; A = 0. Symbols are experimental data of Hermans,
et al. (7). Solid line is predicted curve.
stants that provide the statistical best fit to a given
set of data over the entire conversion range. We did
not attempt to optimize the fit to the data in this
manner, however.
Two reactions not considered in the model may
contribute to the discrepancy between the predicted
curves and the data. The first is a cyclization reaction
which, although similar in mechanism to the reverse
reaction 3, produces not monomer but cyclic oligomers. Indeed, commercial products do contain these
species in minute amounts up to a cyclic nonomer
( 18). If kinetic data were available (namely, cyclization constants ) this reaction, or an approximation
to it, could be treated by our model, similar to the
treatment of Mochizuki and Ito (26). The other reaction not considered is the transamidation reaction.
I t is difficult to assess a p r i o ~ ithe effect of this
reaction. Some kinetic data are available ( 5 ) but
inclusion of this reaction in a kinetic model would
be extremely difficult.
Figure 3 gives conversion versus time, as well as
the development of degree of polymerization and
MWD, for a typical initial water concentration with
and without monofunctional acid. Addition of monofunctional acid somewhat decreases the time for attainment of the equilibrium conversion. The molecular weight controlling effect is demonstrated by
the Dp, versus time curve. Considering now the
molecular weight distribution, we note that a value
of 2.0 is characteristic of a most probable distribution. Except for a plateau at about 1.75 in the early
stages of the reaction where polymerization is dominated by the chain growth mechanism polyaddition,
the value of 2.0 is approached as the equilibrium
conversion is neared. It is seen also that addition of
small amounts of monofunctional acid is predicted to
390
I
0
10
20
Ti m e , hrr.
Fig. 3. Predicted curves for
degree of polymerization
in u butch reactor at various
(a) Ao/Mo = 0.00; (b) A,/M,
30
4 0
conuersion x, number average
z,
and,
polydispersity Z, vs time
levels
monofunct'onal acid:
of
= 0.0025; (c) A J M , = 0.01.
broaden the MWD somewhat, especially in the early
stages of polymerization. As with any theoretical result, the validity of the prediction is difficult to argue
in the absence of some of the experimental results it
seeks to predict. This suggests that an experimental
investigation of MWD versus conversion would be
interesting.
The CSTR
Figures 4 and 5 describe the polymerization in a
CSTR. We see that very broad distribution product
may be obtained in a CSTR, consistent with the partial step growth character of the polymerization (19,
23). No data on continuous caprolactam polymerization are available for direct comparison. In Fig. 5
it is seen that control of product molecular weight
and MWD can be obtained by addition of 0.25 to
1.0percent monofunctional acid.
One final point with respect to the CSTR is that
we have assumed perfect mixing. If some degree of
segregation had been introduced, the prediction
would probably be a narrower PCLD in keeping
with the general results for condensation polymerizations (19, 2 3 ) .
Series Configurations
A primary advantage of this deterministic model
is demonstrated in treatment of series reactor configurations. Tubles 3-5 deal with various plug flow
reactor-CSTR schemes for producing nylon-6 continuously. Equal reactor volumes are considered with
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. 15, No. 5
An Analysis of Caprolactam Polymerization
I. r
1. I
,OOt
20
c
4
I
50
100
I50
200
Time, h r t .
Time, hn.
Fig. 4. Predicted curoes for x, so (m/kg
x
102)=, and 2,
os residence time in a CSTR at uarious initial water concentrations: (a) W , / M , = 0.01; (b) W , / M , = 0.05; (c) W,/M,
= 0.1.
the space time, 7,required to give the specified product after each stage r:hown. With no water
removal,
__
using as acceptable product criterion a DP, of 150,
Fig. 5. Predicted curues for x,oPn and 2, us residence time
in a CSTR at various leuels of monofunctional acid: (a) &/M,
= 0.00; (b) A , / M , = 0.0025; (c) A , / M , = 0.01.
it is seen that either series configuration can be used
to obtain a similar product at comparable residence
times. The key point is that it is not recommended to
try to obtain high conversion in the CSTR section.
The largest part of conversion and growth should
Table 3. Two Reactor Schemes, PFR-CSTR in Seriesa
First reactor: PFR
effluent specifications
-
DP,
~~
X
2,
Water removal
after first reactor
TI
~
1
2
3
4
50
100
50
100
* Wo/Mo =
0.05.
0.52
0.84
0.52
0.84
1.56
1.81
1.56
1.81
-
(hrs.)
~
8.0
14.0
8.0
14.0
Second reactor: CSTR
effluent specifications
~
Total space
time (hrs.)
DPn
X
2,
72 (hrs.1
150
150
150
0.88
0.90
0.82
0.89
6.40
55.0
2.68
3.26
2.06
14.0
~~
none
none
complete
complete
150
25.0
8.0
63.0
28.0
33.0
22.0
Table 4. Two Reactor Schemes, CSTR-PFR in Seriesa
First reactor: CSTR
effluent specifications
1
2
3
4
D Pn
X
50
100
50
100
0.41
0.70
0.41
0.70
2,
2.67
4.64
2.67
4.67
Water removal
after first reactor
-
TI (hrs.)
5.0
20.0
5.0
20.0
Second reactor: PFR
effluent specifications
none
none
complete
complete
Total space
time (hrs.)
DPn
X
2,
72 (hrs.1
150
150
150
150
0.90
0.88
0.82
0.88
2.14
3.60
1.50
3.21
15.0
11.0
10.0
6.0
20.0
31.0
15.0
26.0
' Wo/Ma = 0.05.
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. IS, No. 5
391
M . V . TirreU, G . H . Pearson, R. A. Weiss and R . L. Laurence
Table 5. Two Reactor Schemes, PFR-PFR in Seriesa
~
~~~
First reactor: PFR
effluent specifications
DPIl
1
2
Wo/Mo
50
100
X
2,
0.52
1.56
1.81
0.84
TI
-
(hrs.1
8.0
14.0
DPIl
complete
complete
150
150
Total space
time (hrs.)
Z,
X
0.84
0.89
~~(hrs.1
1.54
1.78
9.0
5.0
17.0
19.0
= 0.M.
occur in the PFR with the CSTR used as a starting
or finishing reactor. These same general rules hold
for the cases of complete water removal after the
first stage. Water removal does give a considerable
process time advantage in agreement with results
obtained by other workers (2, 3 ) . This is further
demonstrated in Table 5 where two PFR’s in series
are run with water removal between stages. This
configuration has interest since it is the limiting case
of an infinite series arrangement of CSTR’s. These results show that it is desirable to remove water from
the reaction as early in the process as possible. However, we would not expect it to be desirable to remove the water at a residence time lower than the
point at which the maximum in the p,, curve occurs.
The proper choice of series configuration can provide
a considerable improvement over a single CSTR.
In summary, the deterministic model developed
in this paper has been applied to various single and
series reactor configurations. From this model predictions about MWD as well as molecular weight
and reaction conversion can be made. Distributions
other than most probable result in many cases. Some
tuning of the model is indicated to improve the
agreement between the model and the available
data. A more systematic approach to determining
the optimal reactor configuration through the use of
this model is likely to prove quite fruitful.
NOMENCLATURE
A,
= concentration of monofunctional species of
chain length i (mole kg-’)
DP, = number average degree of polymerization
DP, = weight average degree of polymerization
ki = forward rate constant of ith reaction
k; = reverse rate constant of ith reaction
Ki
= equilibrium constant of ith reaction
M
= concentration of caprolactam (mole kg-’ )
Si
= concentration of difunctional species of
chain length i (mole kg-’)
W = concentration of water (mole kg-’ )
- Z,, = polydispersity = DPJDP,
pk
= kth moment of the distribution of chain
lengths of difunctional species
pk’
= kth moment of the distribution of chain
lengths of monofunctional species
REFERENCES
1. H. K. Reimschuessel and G. J. Dege, J. Polym. Sci., A-I,
9,2343 (1970).
2. K. Nagasubramanian and H. K. Reimschuessel, J. Appl.
392
Second reactor: PFR
effluent specifications
Water removal
after first reactor
Polym. Sci., 16,929 ( 1972).
3. H. K. Reimschuessel and K. Nagasubramanian, Chem.
Eng. Sci., 27, 1119 (1972).
4. H. K. Reimschuessel and K. Nagasubramanian, Polym.
Eng. Sci., 12, 179 (1972).
5. H. K. Reimschuessel in “Ring Opening Polymerizations,”
K. C. Frisch and S. L. Reegan, eds., Ch. 7, Marcel Dekker,
New York, N. Y. (1969).
6. Ch. A. Kruissink, G. M. van der Want and A. J. Staverman, J. Polym. Sci., 30, 67 ( 1958).
7. P. H. Hermans, D. Heikens and P. F. Van Velden, J.
Polym. Sci., 30, 81 ( 1958).
8. F. Wiloth, 2. Physik. Chem., 5, 66 (1955).
9. F. Wiloth, 2.Physik. Chem., 11, 78 (1957).
10. H. K. Reimschuessel, J. Polym. Sci., 41,457 (1959).
11. R. W. Lenz, “Organic Chemistry of Synthetic High Polymers,” Wiley-Interscience, New York, New York ( 1967).
12. C.-R. Huang and H.-H. Wang, J. Polym. Sci., A-I, 10,
791 (1972).
13. J. Sebenda and V. Kouril, Europ. Polym. J., 7, 1637
( 1971).
14. P. Biernacki, S. Chrzczonowicz and M. Wlodarayk, Erirop.
Polym. J., 7, 739 (1971).
15. S. Saunders, J. Polym. Sci., 30,479 (1958).
16. Modern Plastics, p. 70, August 1969.
17. S. Smith, J. Polym. Sci., 30,459 (1958).
18. A. D. Bliss, Foster Grant Co., private communication.
19. R. L. Laurence, unpublished manuscript,
20. S. I. Calouche, Firestone Synthetic Fibers Co., private
communication.
21. W. H. Abraham, Ind. Eng. Chem. Fund., 2,221 (1963).
22. H. Kilkson, h d . Eng. Chem. Fund., 3,281 (1964).
23. Z. Tadmor, Ph.D. dissertation, Stevens. Inst. Tech.,
(1966).
24. D. Heikens, P. H. Hermans and G. M. van der Want,
J. Polum. Sci., 44. 437 ( 1960).
25. H. Kiikson, lnd. hng. Chem. Fund., 7, 354 (1968).
26. S. Mochizuki and N. Ito, Chem. Eng. Sci., 28, 1139
(1973).
APPENDIX A
It can be shown that
-S
s” -dsr}
P(S’)
O
(A-1)
(s’)3
This is not easily converted to a moment equation.
However, since moment equations are the objective
of applying the transformation, they can be obtained
in an equivalent but different form by expanding the
double summation and recombining. We have
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. 15, No. 5
An Analysis of Caprohctam Polymerization
m
C nkSn
pk:=
( A-2 1
n=l
therefore, the following expressions can be evolved
and substituted into equations for the appropriate
order moment
m
m
m
m
01
n2
n-1
n
2
j=n+t
n
c
LJ
.;
1
--=1(2fi-p1-S1)
1-1
6
(A-5)
An identical treatment is used for the corresponding Aj term.
APPENDIX B
Equations for CSTR:
+ k3'b - k3'Sl
Sl,
- SlO
e
(B-1)
= klMW - kl'S1- k2Sp4
POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. IS, No. 5
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