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Reversible polycondensation in a semi-batch reactor
Chemical Engineering Science, 1969, Vol. 24, pp. 125-l 39.
Pergamon Press.
Printed in Great Britain.
Reversible polycondensatlon in a send-batch reactor DUNCAN A. MELLICHAMP Department of Chemical and Nuclear Engineering, University of California, Santa Barbara (First received 5 February 1968; in revisedform 20 May 1968) Ah&r& -The material balance equations describing reversible polycondensation, both polymerization and rewmngement, are derived and applied to the special case of semi-batch reaction. Solutions of the N + 1 equations for values of N up to 40 are presented A collapsing of the equations to a set of three non-linear ordinary daferential equations is directed and shown to be *orous for the calculation of munber average properties. Finally, the collapsed equations are coupled with a normal distribution function to calculate distribution of the species and weight average properties, these results comparing very favorably with solutions obtained from the individual material balance equations. INTRODUCTION IN THE past
several years considerable attention has focused on the theoretical analysis of polymerization reactions. Recent kinetic work has evolved from the predominantly statistical approach employed by earlier workers[l]. In the continuing attempt to obtain better estimates of the distribution of reacting species, use has been made of generating functions[2,3], Ztransforms [4,5], the continuous variable tech-
volatility or insolubility of the A molecules to withdraw them from the reacting mass. Polyesterification processes, for example, utilize the relatively high vapor pressure of the glycol constituent of monomer vis-h-vis the polymer chains themselves to force the polymerization reactions to completion. As an illustration, poly(ethylene terephthalate) is produced by a polycondensation of the diester of ethylene glycol and terephthalic acid:
nique[6,7], and the faster computers which are now available to solve the system equations [8,9]. One important class of polymerization reactions, polycondensation, is accurately characterized only through the use of reversible kinetic expressions. For example, the condensation of ABA type monomer is described by a sequence of reactions of the form.
HOCHzCHzO
0
+HOCHCHO 0 2
HFt 0
ABA+ ABA # ABABA+ ABABA+
A
ABA # ABABABA+ 0 0 0
A(BA), + A&A).
0
HOCHzCHzO A
# A(BA),+, + A
where the A molecule which is split off in the polymerization reaction is not inert but can react reversibly to break down developed polymer chains. Commercial polymerization processes of this nature typically make use of the relative
+ HOCH2CHeOH. Pressures down to several millimeters Hg absolute are required to drive the reaction to the higher molecular weight polymer suitable for commercial films or fibers. Most published treatments of polycondensation systems start with irreversible mechanisms, either a priori, by neglecting the A type molecule split off in the reaction, or by assuming the A 125
D. A. MELLICHAMP
molecule is some conveniently innocuous species which desires not to participate further in the reaction scheme. In an attempt to introduce a more realistic model, Abraham[4] utilized an extension of the Z-transform approach to the reversible case; however, he was unable to obtain an analytic solution to the transformed partial differential equation. This paper discusses alternate methods of analyzing a reacting polymer system. The rate expressions for all polymeric species are written. This analysis differentiates between polymerization reactions and chain rearrangement, i.e., reversible reactions of the form A(BA), + A(BA), @ A(BA),+
forming reactors required to obtain high molecular weights in continuous flow systems. The results of this work indicate strongly that the assumption of the Flory-Schultz distribution to estimate weight average molecular weight, second and higher moments, or polydispersity functions introduces much less error than the assumption of an irreversible reaction mechanism. In particular, there appears to be little error introduced for batch reactions (1) where physical constraints (reactant carryover at high vapor flow rates) limit the rate at which polymerization can be carried out, or (2) in high molecular weight reactions which are mass transfer limited to the extent that instantaneous point chemical equilibrium of the reactants can be assumed.
A(BA),+,_,
which under certain conditions can be the major mechanism of species redistribution. The entire set of N equations is truncated to some manageable level and solved (together with necessary vapor-liquid equilibrium expressions) by brute force techniques using the digital computer. The particular case of semi-batch reaction is investigated at some length. The set of N reaction equations is “collapsed” by means of standard summation techniques to a set of three nonlinear differential equations. Although these equations are not amenable to analytical solution and do not contain explicit information about the concentration of the individual species they do furnish an easy way to solve for the system number average molecular weight as a function of reaction time. From an engineering point of view this is a primary objective since the most important polymer in particular intrinsic viscosity, properties, are primarily functions of the number average molecular weight. However, one result of this study has been to show that the Flory-Schultz distribution can be coupled with the “collapsed’ equations to correlate accurately results obtained by solution of the individual species reactions. Hence the collapsed equations can be used with a very real saving in computation time to describe approximately a given reactor system: batch, series of CSTRs, or the film
DESCRIPTION
OF REACTION
SYSTEM
A. Polymerization Smith .and Sather[8] and Abraham[4] both considered condensation of homologs of the bifunctional molecule AB (AB), + (AB), + (AB),,,. A more general representation in this paper A(BA),+A(BA),
will be discussed
I& A(BA),+,+ r
A.
(1)
In this notation the free A species may be obviously related to the A unit on the ends of the molecule (as in the polyesterification example cited earlier) or may be only a condensation by-product (e.g., water in the polymerization of a nylon “salt”). Both forward and reverse rate constants are associated with Eq. (1) or, equivalently, the reverse process may be expressed in terms of an equilibrium constant, which for our purposes can be defined as K = k,/k,. Except for reversibility, the usual simplifying assumptions are made: (a) Perfect mixing in the reactor. (b) Isothermal reaction. (c) Kinetics follow the stoichiometry of the reaction, i.e., second-order, with no
126
Reversible polycondensation in a semi-batch reactor
dependence of reaction rate on chain length. (d) No change of volume during reaction. If species A(BA), is denoted by SI and the concentration of S, by C, (the subscript zero denotes the free A molecules) then the net rate of formation of each species may be written?
These reactions when written in general form A(BA),+
k=l 0
r
N
J$j-l)C,-2
c.
A(BA),
(5)
have no distinguishable “forward’ and “reverse” direction. Hence, as a first approximation, we might expect that the rate constants in both directions are equal and, also, k; = kP However, we are not required to specify this second condition. The rate of formation of S1 due to rearrangement is given by
-x C&j-k - 2C* ‘% Ck k=l
A(BA), $ A(BA),+.-,+ k;
-I,
Ckl}
k=i+l
j=1,2,. r,, = 4k{y
Ck y k=l
C+[
g
I=1
(k-
r; = 4k; $ $ [C,Cj+k_l-CjCk]
* - ,N
l)Ck]}
k=l
(2)
j=
1,2,...,N
I=1
with
k-l
[clcj+k-l
-
c
jck]
=
0
(6)
for where N is the maximum chain length in the system. N --* m in a physical system but may be any finite value for a truncated mathematical model. The equations are written as functions of N in order to retain closed material balance properties even with severe truncation.
i.e., the combination of indices j+k-1 stricted to the region, 1 < j+ k - 1 s N.
is re-
C. Material balances
B. Rearrangement Flory [ l] discusses the possibility of rearrangement among the various species. Such rearrangement does not result in the generation of A nor affect the number average chain length+ of the system
Xk =
2jxj= i jCj/,tl Cj
5=1
f=l
A total of N + 1 differential material balance relationships are required to describe the reaction in any type of reactor. For a semibatch reaction system:
dco
z=
ro-j$jQo
(7)
(3) dCj -=rjr; dt
but will affect higher moments such as the weight average degree of polymerization (4) tNomenclature largely follows that of previous authors with exceptions noted. Note that k, is the rate constant which would be determined by analyzing the reaction of monofunctional homologs. The use of such a basic rate constant with the multiplier (four) to account for bifunctionality of both reacting species follows Tanford[ lo]. $The primes in Eqs. (3) and (4) are used to indicate that the mole and weight fractions are with respect to polymeric species only.
j=
1,2, . . . ,N
where me is reactor total pressure assuming vapor-liquid equilibrium, V reactor volume, Q” volumetric vapor outflow rate. The reactor pressure can be expressed by a form of Raoult’s law 7rlJ=&$x0 = p,*Cd 2 c, (8) j=o where p$ is the pure component vapor pressure PThese conditions are required to maintain closed material balances for a truncated set of mathematical expressions, i.e., N relatively small.
127
D. A. MELLICHAMP
of S,, at the reactor temperature, x0 is the mole fraction of S,. In a semi-batch polymerization reactor the vapor outflow rate is ordinarily adjusted to maintain a programmed pressure in the reactor. This adjustment in Q” can be accomplished mathematically by defining an auxiliary expression (algebraic or differential) (9)
QOO) = Q%r,;rr(t)]
D. Initial conditions Initial conditions for the batch reactor may be written generally C,(t = 0) = C,(O)
j=
O,l,. . . ,N.
(10)
The choice of initial conditions for a reversible reaction system raises interesting questions concerning physical realizability. Previous investigators assumed that pure monomer (S,) is present in the reactor at start-up. Pure monomer is of course a non-existent idealization. There must always be a distribution of polymeric species present except in very dilute solutions. In commercial polyester production, for example, “monomer” is the term used to describe an equilibrium mixture of polymeric species with a ratio of A units to B units approximately equal to two. The number average chain length is actually greater than one and depends on the equilibrium temperature (pressure). In subsequent discussions we will assume a starting mixture in chemical and physical equilibrium (at reaction temperature) with A : B = 2.0. These two conditions are given by
where t(O) is the initial molar density of the mixture. The initial conditions given by Eqs. (1 l), (12), and (13) are not in a particularly useful form. Conversion to the individual C,(O) will be effected after dimensional analysis. E. Conversion of equations to dimensionless form A dimensionless concentration is defined as follows:
so that
The rate expressions describing polymerization become R,=4
y ~x~~~-2*j~*~-~[(j-l)*, I k-l k=l -2
k=l
l-l
with
(j+1xm
(12)
B(O)=
if
=2-o.
j+k-l>
N
G
1
‘+k-1-c
There is one additional condition where i. CXO)= 6(O)
(17)
Ro= 4
5 C,(O)
gojcj(o)
II
)ck
1,2,...,N
j=
J-O so
i k-1+1
where Rj = rj/kz&2(0). The rate expressions describing rearrangement (11) become
JWo(0) - r(O)
A(O)
(14)
5 = G/ $.W C,(O) = W(0)
(13) 128
,-
4
RJ kit*(o)
Reversible polycondensation
(ki is set equal to k, for convenience And for the semi-batch reactor
in a semi-batch reactor
or
at this point).
MO) = j$I (j$=
I?,--QII,
(19)
l)AJ(O) = P(0).
Similarly from Eq. (22)
dhJ ==Rj+R;
2 A,(O)= 1 -A,(O)
1-1
j=l,2,...,N where
P*QO
1
A,(O)=p(O) = I-I,(O).
0 = kfs$(O)t.
j=
(20)
3 MO)
I=0
N
1,2,...Jv.
Ro
=
4([~,M+$,(~- l)AdO)]} =0 [l-p(o)]z-p15$=0
(22)
The system is initially in chemical and physical equilibrium so that (a) II,(O), the equilibrium initial pressure, is related to K, the equilibrium constant (b) the Flory-Schultz (“most probable”) distribution holds. The most probable distribution is expressed as a function of the extent of reaction p and we note that
(j-l)A,.
(27)
Alternatively, the system chemical ,equilibrium constant may be used. In the appendix Eq. (17) is shown to reduce to the following expression
(21)
p=,il
(26)
A,(O) = [1 -P(o)12PYo)
The system initial conditions are also expressed in terms of dimensionless quantities
5 A,(O) = 1. 1=0
(25)
Knowledge of the system equilibrium pressure at t = 0 is sufficient to calculate the species concentrations
He = TAP*
= MO)
= 1 -p(O).
And from Eq. (20)
Q=RTV/&(O)
ho(O)
(24)
(23)
or P(O)
=s
COLLAPSING THE MATERIAL
v/K
(28)
N DIFFERENTIAL BALANCES
If we are willing to characterize the reacting system by average properties (_%k,A: B, etc.), a great reduction in the order of the system can be obtained. This “collapsing” of the N species equations to three simultaneous nonlinear equations involves minor approximations and no simplifying assumptions. Start with the semi-batch reactor expressions
From Eq. (2 1)
$=4(1%: Ak y A,-;[ i 2=1 k=l 129 C.E.S.
Vol. 24 No. I-1
(k-
~)A~]]--Q (29)
D. A. MELLICHAMP
can be expressed as balances on the threeSmolecular forms of the A species. Defining
-0
;[(j-L)&--2
i k=j+l
+
“Free A”:
AF=Ao
“Internal A”:
AI= i (j--1)X, j=l
“Terminal A”:
AT = 2 i
Ak]
5 i [Azb+k-z-bbl}
k=l Z=l
t3’)
(34)
A,
j=l
with the double sum in Eq. (30) subject to the restriction 1 G j+ k - I s N. Assuming N large (i.e., ljrnit Ak= 0) Eq. (29) is shown to be
and substituting in (3 l), (32), and (33) we have
equivalgnt to the following expressiont: $=
4{[ $
Ak]‘-$[
i
(k-
l)&]}-a.
k=l
(31)
The form of Eq. (31) suggests that we look for eCWtiOUS
defining
the
quantities
d(A,) -=Rc-Q d8
(35)
d(AI) - R d8 ’
(36)
d(A,)=_2R d@
g Ak and
C
(37)
k=l i,
with dimensionless
(k-l)hk.
If both sides of Eq. (30) are summed over j from 1 to N the following expression is obtained after rearrangement:
Rc=
reaction rate
(A~)~-&4(&h
(38)
Initial conditions for these equations are given by the following: A,(O) = AR(O) = A,(O) = II(O) = G Again, both sides of Eq. (30) are multiplied by (j- 1) and summed overj to obtain:
AT(O)=~[~-AX~(O)] =2[1-II(O)]
d/K
(39)
=-&. (40)
N
-2[jz
(j-l)S
11
.
(3 3,
In both operations the contributions from R; sum to zero identically. We note at this point that Eqs. (31), (32), and (33) constitute a solvable system of equations. These equations can be rewritten in terms of the extent of reaction p. Alternatively, they
weight of the
iiT;= 2+1 A
(41)
and the ratio A: B by A -= B
tDeve1opment of the collapsed equations is detailed in tht Appendix.
The number average molecular system is given by
>’
(42)
The ratio A: B is ordinarily used to obtain laboratory characterization of low molecular
130
Reversible polyumdensation
weight polymer (p < O-90) with molecular weight (often estimated as a function of viscosity) used for higher extent of reaction. An additional relation is required to determine reactor pressure (in order to adjust vapor outflow rate according to Eq. (9)):
2AF
b=
AT+
final reactor pressure of 0.1 II, was chosen (F = 0.1 corresponds to p = 0.9 assuming full equilibrium). The low final reaction extent limits this study to the early stages of polymerization in order to eliminate the need to handle prohibitively large systems of equations. Commercial polymerization catalysts specifically designed for this polyesterification system yield forward reaction rates typically on the order of 1 l/g mole min. Hence @,= 30 corresponds to a reaction time of 5-10 min. The removal of 90 per cent of the excess A (glycol) in this period of time constitutes a rather severe forcing of the reaction system, subject as it is to constraints on the maximum rate of heat transfer and vapor entrainment. Table 1 presents a summary of reaction conditions and appropriate choices for Q(O), D1, and Dz in Eq. (47) to maintain II, within specified limits, of II( @) (+ 2%) over the entire time of reaction.
(43)
2Ap’
It should be noted that Eqs. (34)-(43) can also be obtained from an “overall” analysis of the reaction . ..AT+AT . . . . .
5 .....A,...+AF.
The factor four in Eq. (38) results from the assumed bifunctionality of AF and the fact that the concentration of sites for reverse reaction is double the concentration ofA,. NUMERICAL
in a semi-batch reactor
SOLUTION
Table 1. Reaction conditions and parameters in numerical soIutions
The semi-batch reaction Eqs. (29) and (30) [or alternatively (35)-(38)] are written explicitly in terms of the vapor outflow rate Q. Most semi-batch polymerizations are controlled by programming reactor pressure. In these studies a linear variation in pressure was used l-I(@) = l-I(O) - [rI(O) -FrI(o)l;
Q(0) = 0.1 /IF(O) = ;I r 2;; AT(O)= K = F* = 0.1 rI(0) =
(44) T
where F is the fraction of the initial pressure desired at the end of reaction, @T. An auxiliary integral equation corresponding to Eq. (9) was used to define Q(6)) as follows Q(o)
= Q(0) +D,(l&--II)
+I& 1 (II,-l-I)dO. 0 (45)
A specific reacting system was chosen to keep the numerical studies close to physical reality: the di-ethylene glycol ester of terephthalic acid is the reacting monomer. For purposes of discussion a reaction temperature of approximately 230°C is appropriate, and under these conditions the equilibrium constant is O-79 and Q(O), 4.3. A
A,(O) = O-47056 l-05886 0.79 0.47056
Solution of the species equations (IV+ 2 differential equations corresponding to Eqs. (29), (30) and (45)) was accomplished by means of a fourth order predictor-corrector routine which reduced computing times significantly over a similar order Runge-Kutta method. Reaction system order (values of N) of 10, 20, 30 and 40 were investigated. Figure 1 presents curves of x,& number average degree of polymerization, versus 8 for these cases. Figure 2 contains similar plots of A : B. In both figures plots of xh and A : B obtained from the collapsed equations ((35)-(38)) are given. It is seen that solution of the species equations for higher N tends toward the solution obtained from the collapsed equations. Hence, as expected, the collapsed equations furnish a much more efficient means of estimating number average properties such as &.
131
D. A. MELLICHAMP
6
I
0
0
I
I
I
t
I
20
IO
30
9 Fig. 1. Number average degree of polymerization as a function of reaction time.
For the reacting system under investigation, chain lengths up to 50 or 60 would have to be accounted for to obtain equivalent levels of accuracy with the species equations. One interesting aspect seen from the solution of the species equations is the importance of the contribution of rearrangement to the overall rate of reaction. At conditions of chemical equilibrium the chemical reaction equations describing rearrangement can be obtained by combination of the chemical equations describing the polymerization reactions, hence do not constitute a mathematically independent set of relationships. The Flory-Schultz distribution, derived from equilibrium considerations,
Aj = (1 -
j=
p)“p’-’
(46)
1,2,.
reduces the rate expression describing rearrangement (Eq. (20)) to zero identically. R; = 4(1 -P)~
$ 5 [(p’-1) (ti+k-1-l) k=l
I=1
- (p’-1) (pk-‘)I
z
0.
Results from numerical studies of the semibatch system indicate that the distribution of concentrations is always close to one which makes the Rj = 0 even when the reacting 132
Reversible polycondensation in a semi-batch reactor
A:B 1.4
1.2
1.o 0
20
IO
30
Fig. 2. Ratio of A : B vs. reaction time.
system is relatively far from equilibrium. Previous theoretical work by Abraham[4] indicates that this result obtains approximately for a flow reactor with pure monomer feed (h, = 1) and assumed irreversible kinetics. We would expect the approximation to be much better with a reversible model and system feed consisting of a distribution of the S,. The application of the “normal” distribution to non-equilibrium situations allows a gross simplification in the calculation of weight average polymer properties and higher moments of distribution of the polymeric species. The collapsed equations can be used to calculate the extent of reaction p and Eq. (46) to estimate the distribution of the A+ Table 2 summarizes the course of reaction as calcuiated from solution of the collapsed equations. Figure 3 presents
Table 2. Extent ofreactionand reactor pressure vs. time (solution of collapsed equations) Dimensiontess time, 8 0 5 10 15 20 25 30
Dimensionless pressure, II 0.470% 0.39997 o-32938 0.25880 0.18822 0.11764 0*04706
Extent of reaction, P
0.470% 0.53804 O-61079 o@M6 o-76038 0*83518 0.90492
the mole fraction distribution versus time. Discrete points were obtained by solution of the species equations with N = 40; the continuous lines were obtained from the values ofp in Table 2 assuming a normal distribution. Coincidence up 133
D. A. MELLICHAMP
10-l
I
o-s
XI
IO-=
I
o-4 0
IO
20
30
8
Fig. 3. Mole fraction distribution of polymeric species vs. reaction time.
to 8 > 25 is obtained; at this point the accuracy of the species equations breaks down. Graphs of the species weight fractions (Fig. 4) and weight average degree of polymerization (Fig. 5) are also presented for comparison. In all of these studies the truncation of the species equations to some manageable level (ZV finite) leads to much greater error in the calculated concentrations than does the assumption of a normal distribution (operation near equilibrium) coupled with the collapsed or overall material balance equations. Use of these overall equations for the calculation of number average properties is rigorous in any event.
The engineer faced with the mathematical description of a practical polymerization problem, i.e., polymerization in a reactor which neither approaches plug flow nor perfect mixing conditions, and with final values of x.& equal to several hundred, finds that the assumption, even, of instantaneous chemical equilibrium is minor compared to the assum.ptions required to handle the complexities of mixing and diffusional mass transfer. The application of the overall or collapsed reaction rates to the partial differential equations required to describe a real polymerization reactor would appear to be a more fruitful approach and one which is open for further study. 134
Reversible polycondensation in a semi-batch reactor
20
IO
30
0 Fig. 4. Weight fraction distribution of polymeric species vs. reaction time.
CONCLUSIONS
1. The N + 1 species material balance equations describing reversible polycondensation (both polymerization and rearrangement reactions) can be collapsed to three ordinary but non-linear differential equations. These equations are exact in the sense that they can be used to determine number average polymer properties rigorously. 2. Numerical solution of the species equations applied to the case of semi-batch reaction shows that a high order system (N large) must be
treated to obtain accurate estimates of species distributions. Because of the high degree of mathematical interconnection in the species equations, an approximately normal distribution of the species obtains, even when the reacting mixture is relatively far from chemical equilibrium. Hence the collapsed equations can be used with an algebraic relation to estimate distribution of the species with a resulting great reduction in computational effort. The extension to the CSTR has not been made in this paper; however, work by other authors 135
D.A. MELLICHAMP
16
12
8
4
0
0
-’
30
20
IO
8 Fig. 5. Weight average degree of polymerization vs. reaction time.
indicates that a nearly normal distribution is obtained in several cases of interest. The application of the collapsed equations to these cases is also valid. Adcnowkdgment -The author wishes to express his appreciation to the University of California, Santa Barbara, Research Committee for their generous donation of computer time.
B
cj &,Ds F
NOTATION
A AF AI AT
molecular unit in the ABABAB... polymer chain dimensionless concentration of “Free A” dimensionless concentration of “Internal A” dimensionless concentration of “Terminal A”
K N
Q Q 136
molecular unit in the ABABAB... polymer chain concentration of the species with chain lengthj empirical constants in the equation generating vapor outflow rate (Eq. (45)) fraction of initial reactor pressure at the end of the reaction reaction equilibrium constant maximum chain length in the mathematical polymerization model volumetric flowrate of SOremoved from the reactor dimensionless volumetric flowrate of&
Reversible polycondensation in a semi-batch reactor
Rj
sj T V
xi xty j,k,l,m,w
kt
k; kr P * PO
rj
dimensionless reaction rate of the collapsed system (overall rate of reaction) dimensionless rate of generation of species with chain length j by polymerization dimensionless rate of generation of species with chain length j by rearrangement species with chain length j absolute reaction temperature volume of reacting mass number average degree of polymerization (chain length) weight average degree of polymerization summation indices forward polymerization rate constantt rearrangement rate constantt reverse polymerization rate constant? extent of reaction pure component vapor pressure of So at reaction temperature, T
rate of generation of species with chain length j by polymerization rj' rate of generation of species with chain length j by rearrangement r time 4 weight fraction of polymeric species with chain length j (So not included) mole fraction of species with chain length j mole fraction of polymeric species with chain length j (S, not inciuded) dimensionless reaction time dimensionless time at end of reaction dimensionless reactor pressure (programmed) dimensionless reactor pressure calculated from vapor-liquid equilibrium relationships dimensionless concentration of species with chain length j molar density of reactants reactor pressure (programmed) reactor pressure calculated from vapor-liquid equilibrium relationships
tAll rate constants expressed in terms of monofunctional reactants. REFERENCES
[I] FLORY P. J., Principles of Polymer Chemistry. Cornell University Press 1953. r2i BIESENBERGERJ. .4..A.I.ch.E.Jl1965 11369. i3j ROOTSAERT W. J. M. and Van de VUSSE J. G., Chem. Engng Sci. 1966 211067. 141 ABRAHAM W. H., Chem. Engng Sci. 1966 21327. VI KILKSON H., Ind. Engng Chem. Fundls 1964 3 283. [6] ZEMAN R. J. and AMUNDSON N. R., Chem. Engng Sci. 1965 20 33 1. [7] ZEMAN R. J. and AMUNDSON N. R., Chem. Engng Sci. 1965 20637. 183 SMITH N. H. and SATHER G. A., Chem. Engng Sci. 1965 29 15. [9] TADMOR Z. and BIESENBERGER J. A., Ind. Engng Chem. Fundls 1966 5 337. [lo] TANFORD C., Physical Chemistry of Macromolecules. Wiley 196 1.
APPENDIX COLLAPSING
THE N DIFFERENTIAL BALANCES
Starting with the expression for A0
we note that MASS
(A. 1) subject to
k+l-1
P N.
D. A. MELLICHAMP Then if pit
Ak= 0 there is little loss in accuracy if Eq. (29)
+ 2 ; A,AIN-,_,+ 5 A,Aw-r I-N l-N-1
is rewritten as:
--,A,-2A,A2-3A,As
......*................... -NA,AN
-2AzA,-3X,X2-4ArAJ
.............-NA.A.-.-(N-l)A.A.v
To completely collapse the species equations, expressions for N N X Akand z (k - l)Ak are required. h=l Summing Eth sides of Eq. (30) over all j -(N-l)AN-,Al-NANAt-(N-l)AN-IA5.........-2A.+,A
N - ANAN.
..............................
-NAd,-(N-l)
After expanding on 1 and factoring -2
(j-
l)Aj-2
= (A, - A,)(A, + 2x2 + 3X, . . . . . . . . . . . . . . . . . . . . . . . ..+NAJ
i Ak b-f+, I
+ (A2- A,)(2A, + 3A, + 4As . . . . . . . . . . .+NA+,+(N-HAN) + (A3- AsH3A,+ 4X2+ 5X3. 1. . . . .+(N-l)A,,+(N-2)AJ
+ i 5 bA,+r-t - Wd k-1 bl
(A.3)
I
with [A,A,+k-I- A&l = 0 for j+k-I>
N
c ‘+k--l-z
1.
f(AN_l - A&((N
- I)A, + NAz+ (N - l)A,. . .+ 3A,+_1+2AN)
+(A.v-A,&NA,+(N-])A*+
Proceeding with pairs of terms
(N-2)A,.
. . . . .+21,-,+X,)
= 0. Hence the complete expression is written
-&[ iAj]=-4{[
it=j+1 5A/c (2) jiU--Ihj--2 j=* =i,U-1)&--2 :Ici1)Aj =-ji ti-115 (3)
ji i,
(j-
iAjr-+[
+A#
Both sides of Eq. (32) are now multiplied by the quantity 1) and then summed over allj
&-I)$=$ *1
.[
j-l I &.j-l)Aj
$, [AtAj+k--~ - A&l
(expanding j and k and collecting only the sums along the diagonals)
(A.3 with
[AA+H - AjA,] = 0 = i AlAS++ 2 *iI AlAs-1 + . ..+ (N - 1) ;$; AlAN+ z I=1 N N + N z AIAN-~+l + (N - 1) z AlA.+-I+P + ..
k-1
64.4)
for
c
j+k-l> ‘+k-1-c
t-4
138
N
1.
Reversible polycondensation
in a semi-batch reactor
Again proceeding with pairs of terms
+ . . . + (2N - 3)
2
A&+,
+ (N - 1) i
1-N-l N--1 =
;a
N-l
a-1 =
;a
N--l
b;,
ti+k-
lhk-
ij
?&A,-3AlAY-4AfAJ._...-NAIAN_,--(N-
- ~AxA,-8A&
N-j
x J-1
-l2A&h,--5A&-
z O’+k--Zhk k==*
AlAm._<
I-N
l)ArAN
- 10AaAs. . . - 2(N - l)A,A,, - 2(N - 2)As 18A&. . . . . . . . . .-3(N-3)ArAN
N-J AJ ;,
hk
s
‘:
-(N-l)NA,vA,-(N-lXN-DANA,-(N-l)(N-2)A,A, ,~(i-l)*Aj-i?
(j-l)
i,
,$, _
. . - (N - l)ANAN.
At
=jI
[(j-lP-O’-2)U-I)15
Expanding the positive terms on 1 partitioning the negative terms {Row (j)=f[Row (j-l)-Column (j)]}, and factoring the common term from each row
=i
ci--lpY
= (A,-A1)[Ak+3As+6A,+.
..+(N-2;N-1)XN_-1+
W-WAN 2 A,+3A2+6Aa++....++
(N-
l)NA.v_, +
1
(N-1)N
2”~
for
j+k-I '+k-I<
G
>
N
+(A,-&)
1.
(again we expand j and k and collect the summations on ! along the diagonals) = i
l-l
A1A3_-( + 3 i
I-1
i
+. . (N-
I-1
AlAs_,+n +
i
(N- “:” +‘) i
(N-2)(N+UAN 2
AIAN-,+,
Hence we obtain
AIA,,T--(+s
R&mrmGLes dquations d%quilibre de matCriau decrivant la polycondensation reversible, a la fois polymerisation et r&urangement. sont dCrivCes et appliqudes au cas special de la reaction en demilots. Les solutions des equations N + 1 pour des valeurs de N allant jusqu’a 40, sont prtsentbes. On provoque l’arr6t des equations a une strie de trois equations differentielles ordinaires non-lit&tires et on montre qu’elles sont rigoureuges pour le calcul des propri6tes du nombre moyen. Finalement, les equations arri?tees sont accouplbs avec une fonction normale de distribution pour le calcul de la repartition des esp&ces et des proprietes de poids moyen, ces resultats se comparant favorablement aux solutions obtenues a partir des equations individuelles d’tquilibre de mat&au. Zusammenfassung- Die Materialbilanzgleichungen, welche die reversible Polykondensation, und zwar sowohl die Polymerisation als such die Umlagerungsreaktionen beschreiben, werden abgeleitet und auf den Spezialfall “Semi-batch” Reaktion angewandt. Es werden Lijsungen der N+ 1 Gleichungen fur Werte von N bis zu 40 berichtet. Die Gleichungen werden auf einen Satz von drei nicht linearen gewohnlichen Differentialgleichungen reduziert, und es wird gezeigt, daL3sit bei der Berechnung von Zablendurchschnittseigenschaften streng zutreffen. Schliefllich werden die reduzierten Cileichungen mit einer normalen Verteilungsfunktion verbunden, urn die Verteilung der Spezies und der Gewichtsdurchschnittsmerkmale zu berechnen. Diese Ergebnisse waren im Vergleich mit den Losungen aufgrund der einzelnen Materialbilanzgleichungen sehr zufiiedenstellend. 139
3
+;A~-A~)[~A~+~~~+(~+~:(~-~)A~
AtAd++ 6 2 A,A5_,+ . . . y
+y
%+~Az+~OAS+ . . . . . . .+
1
1
l)AN
Pergamon Press.
Printed in Great Britain.
Reversible polycondensatlon in a send-batch reactor DUNCAN A. MELLICHAMP Department of Chemical and Nuclear Engineering, University of California, Santa Barbara (First received 5 February 1968; in revisedform 20 May 1968) Ah&r& -The material balance equations describing reversible polycondensation, both polymerization and rewmngement, are derived and applied to the special case of semi-batch reaction. Solutions of the N + 1 equations for values of N up to 40 are presented A collapsing of the equations to a set of three non-linear ordinary daferential equations is directed and shown to be *orous for the calculation of munber average properties. Finally, the collapsed equations are coupled with a normal distribution function to calculate distribution of the species and weight average properties, these results comparing very favorably with solutions obtained from the individual material balance equations. INTRODUCTION IN THE past
several years considerable attention has focused on the theoretical analysis of polymerization reactions. Recent kinetic work has evolved from the predominantly statistical approach employed by earlier workers[l]. In the continuing attempt to obtain better estimates of the distribution of reacting species, use has been made of generating functions[2,3], Ztransforms [4,5], the continuous variable tech-
volatility or insolubility of the A molecules to withdraw them from the reacting mass. Polyesterification processes, for example, utilize the relatively high vapor pressure of the glycol constituent of monomer vis-h-vis the polymer chains themselves to force the polymerization reactions to completion. As an illustration, poly(ethylene terephthalate) is produced by a polycondensation of the diester of ethylene glycol and terephthalic acid:
nique[6,7], and the faster computers which are now available to solve the system equations [8,9]. One important class of polymerization reactions, polycondensation, is accurately characterized only through the use of reversible kinetic expressions. For example, the condensation of ABA type monomer is described by a sequence of reactions of the form.
HOCHzCHzO
0
+HOCHCHO 0 2
HFt 0
ABA+ ABA # ABABA+ ABABA+
A
ABA # ABABABA+ 0 0 0
A(BA), + A&A).
0
HOCHzCHzO A
# A(BA),+, + A
where the A molecule which is split off in the polymerization reaction is not inert but can react reversibly to break down developed polymer chains. Commercial polymerization processes of this nature typically make use of the relative
+ HOCH2CHeOH. Pressures down to several millimeters Hg absolute are required to drive the reaction to the higher molecular weight polymer suitable for commercial films or fibers. Most published treatments of polycondensation systems start with irreversible mechanisms, either a priori, by neglecting the A type molecule split off in the reaction, or by assuming the A 125
D. A. MELLICHAMP
molecule is some conveniently innocuous species which desires not to participate further in the reaction scheme. In an attempt to introduce a more realistic model, Abraham[4] utilized an extension of the Z-transform approach to the reversible case; however, he was unable to obtain an analytic solution to the transformed partial differential equation. This paper discusses alternate methods of analyzing a reacting polymer system. The rate expressions for all polymeric species are written. This analysis differentiates between polymerization reactions and chain rearrangement, i.e., reversible reactions of the form A(BA), + A(BA), @ A(BA),+
forming reactors required to obtain high molecular weights in continuous flow systems. The results of this work indicate strongly that the assumption of the Flory-Schultz distribution to estimate weight average molecular weight, second and higher moments, or polydispersity functions introduces much less error than the assumption of an irreversible reaction mechanism. In particular, there appears to be little error introduced for batch reactions (1) where physical constraints (reactant carryover at high vapor flow rates) limit the rate at which polymerization can be carried out, or (2) in high molecular weight reactions which are mass transfer limited to the extent that instantaneous point chemical equilibrium of the reactants can be assumed.
A(BA),+,_,
which under certain conditions can be the major mechanism of species redistribution. The entire set of N equations is truncated to some manageable level and solved (together with necessary vapor-liquid equilibrium expressions) by brute force techniques using the digital computer. The particular case of semi-batch reaction is investigated at some length. The set of N reaction equations is “collapsed” by means of standard summation techniques to a set of three nonlinear differential equations. Although these equations are not amenable to analytical solution and do not contain explicit information about the concentration of the individual species they do furnish an easy way to solve for the system number average molecular weight as a function of reaction time. From an engineering point of view this is a primary objective since the most important polymer in particular intrinsic viscosity, properties, are primarily functions of the number average molecular weight. However, one result of this study has been to show that the Flory-Schultz distribution can be coupled with the “collapsed’ equations to correlate accurately results obtained by solution of the individual species reactions. Hence the collapsed equations can be used with a very real saving in computation time to describe approximately a given reactor system: batch, series of CSTRs, or the film
DESCRIPTION
OF REACTION
SYSTEM
A. Polymerization Smith .and Sather[8] and Abraham[4] both considered condensation of homologs of the bifunctional molecule AB (AB), + (AB), + (AB),,,. A more general representation in this paper A(BA),+A(BA),
will be discussed
I& A(BA),+,+ r
A.
(1)
In this notation the free A species may be obviously related to the A unit on the ends of the molecule (as in the polyesterification example cited earlier) or may be only a condensation by-product (e.g., water in the polymerization of a nylon “salt”). Both forward and reverse rate constants are associated with Eq. (1) or, equivalently, the reverse process may be expressed in terms of an equilibrium constant, which for our purposes can be defined as K = k,/k,. Except for reversibility, the usual simplifying assumptions are made: (a) Perfect mixing in the reactor. (b) Isothermal reaction. (c) Kinetics follow the stoichiometry of the reaction, i.e., second-order, with no
126
Reversible polycondensation in a semi-batch reactor
dependence of reaction rate on chain length. (d) No change of volume during reaction. If species A(BA), is denoted by SI and the concentration of S, by C, (the subscript zero denotes the free A molecules) then the net rate of formation of each species may be written?
These reactions when written in general form A(BA),+
k=l 0
r
N
J$j-l)C,-2
c.
A(BA),
(5)
have no distinguishable “forward’ and “reverse” direction. Hence, as a first approximation, we might expect that the rate constants in both directions are equal and, also, k; = kP However, we are not required to specify this second condition. The rate of formation of S1 due to rearrangement is given by
-x C&j-k - 2C* ‘% Ck k=l
A(BA), $ A(BA),+.-,+ k;
-I,
Ckl}
k=i+l
j=1,2,. r,, = 4k{y
Ck y k=l
C+[
g
I=1
(k-
r; = 4k; $ $ [C,Cj+k_l-CjCk]
* - ,N
l)Ck]}
k=l
(2)
j=
1,2,...,N
I=1
with
k-l
[clcj+k-l
-
c
jck]
=
0
(6)
for where N is the maximum chain length in the system. N --* m in a physical system but may be any finite value for a truncated mathematical model. The equations are written as functions of N in order to retain closed material balance properties even with severe truncation.
i.e., the combination of indices j+k-1 stricted to the region, 1 < j+ k - 1 s N.
is re-
C. Material balances
B. Rearrangement Flory [ l] discusses the possibility of rearrangement among the various species. Such rearrangement does not result in the generation of A nor affect the number average chain length+ of the system
Xk =
2jxj= i jCj/,tl Cj
5=1
f=l
A total of N + 1 differential material balance relationships are required to describe the reaction in any type of reactor. For a semibatch reaction system:
dco
z=
ro-j$jQo
(7)
(3) dCj -=rjr; dt
but will affect higher moments such as the weight average degree of polymerization (4) tNomenclature largely follows that of previous authors with exceptions noted. Note that k, is the rate constant which would be determined by analyzing the reaction of monofunctional homologs. The use of such a basic rate constant with the multiplier (four) to account for bifunctionality of both reacting species follows Tanford[ lo]. $The primes in Eqs. (3) and (4) are used to indicate that the mole and weight fractions are with respect to polymeric species only.
j=
1,2, . . . ,N
where me is reactor total pressure assuming vapor-liquid equilibrium, V reactor volume, Q” volumetric vapor outflow rate. The reactor pressure can be expressed by a form of Raoult’s law 7rlJ=&$x0 = p,*Cd 2 c, (8) j=o where p$ is the pure component vapor pressure PThese conditions are required to maintain closed material balances for a truncated set of mathematical expressions, i.e., N relatively small.
127
D. A. MELLICHAMP
of S,, at the reactor temperature, x0 is the mole fraction of S,. In a semi-batch polymerization reactor the vapor outflow rate is ordinarily adjusted to maintain a programmed pressure in the reactor. This adjustment in Q” can be accomplished mathematically by defining an auxiliary expression (algebraic or differential) (9)
QOO) = Q%r,;rr(t)]
D. Initial conditions Initial conditions for the batch reactor may be written generally C,(t = 0) = C,(O)
j=
O,l,. . . ,N.
(10)
The choice of initial conditions for a reversible reaction system raises interesting questions concerning physical realizability. Previous investigators assumed that pure monomer (S,) is present in the reactor at start-up. Pure monomer is of course a non-existent idealization. There must always be a distribution of polymeric species present except in very dilute solutions. In commercial polyester production, for example, “monomer” is the term used to describe an equilibrium mixture of polymeric species with a ratio of A units to B units approximately equal to two. The number average chain length is actually greater than one and depends on the equilibrium temperature (pressure). In subsequent discussions we will assume a starting mixture in chemical and physical equilibrium (at reaction temperature) with A : B = 2.0. These two conditions are given by
where t(O) is the initial molar density of the mixture. The initial conditions given by Eqs. (1 l), (12), and (13) are not in a particularly useful form. Conversion to the individual C,(O) will be effected after dimensional analysis. E. Conversion of equations to dimensionless form A dimensionless concentration is defined as follows:
so that
The rate expressions describing polymerization become R,=4
y ~x~~~-2*j~*~-~[(j-l)*, I k-l k=l -2
k=l
l-l
with
(j+1xm
(12)
B(O)=
if
=2-o.
j+k-l>
N
G
1
‘+k-1-c
There is one additional condition where i. CXO)= 6(O)
(17)
Ro= 4
5 C,(O)
gojcj(o)
II
)ck
1,2,...,N
j=
J-O so
i k-1+1
where Rj = rj/kz&2(0). The rate expressions describing rearrangement (11) become
JWo(0) - r(O)
A(O)
(14)
5 = G/ $.W C,(O) = W(0)
(13) 128
,-
4
RJ kit*(o)
Reversible polycondensation
(ki is set equal to k, for convenience And for the semi-batch reactor
in a semi-batch reactor
or
at this point).
MO) = j$I (j$=
I?,--QII,
(19)
l)AJ(O) = P(0).
Similarly from Eq. (22)
dhJ ==Rj+R;
2 A,(O)= 1 -A,(O)
1-1
j=l,2,...,N where
P*QO
1
A,(O)=p(O) = I-I,(O).
0 = kfs$(O)t.
j=
(20)
3 MO)
I=0
N
1,2,...Jv.
Ro
=
4([~,M+$,(~- l)AdO)]} =0 [l-p(o)]z-p15$=0
(22)
The system is initially in chemical and physical equilibrium so that (a) II,(O), the equilibrium initial pressure, is related to K, the equilibrium constant (b) the Flory-Schultz (“most probable”) distribution holds. The most probable distribution is expressed as a function of the extent of reaction p and we note that
(j-l)A,.
(27)
Alternatively, the system chemical ,equilibrium constant may be used. In the appendix Eq. (17) is shown to reduce to the following expression
(21)
p=,il
(26)
A,(O) = [1 -P(o)12PYo)
The system initial conditions are also expressed in terms of dimensionless quantities
5 A,(O) = 1. 1=0
(25)
Knowledge of the system equilibrium pressure at t = 0 is sufficient to calculate the species concentrations
He = TAP*
= MO)
= 1 -p(O).
And from Eq. (20)
Q=RTV/&(O)
ho(O)
(24)
(23)
or P(O)
=s
COLLAPSING THE MATERIAL
v/K
(28)
N DIFFERENTIAL BALANCES
If we are willing to characterize the reacting system by average properties (_%k,A: B, etc.), a great reduction in the order of the system can be obtained. This “collapsing” of the N species equations to three simultaneous nonlinear equations involves minor approximations and no simplifying assumptions. Start with the semi-batch reactor expressions
From Eq. (2 1)
$=4(1%: Ak y A,-;[ i 2=1 k=l 129 C.E.S.
Vol. 24 No. I-1
(k-
~)A~]]--Q (29)
D. A. MELLICHAMP
can be expressed as balances on the threeSmolecular forms of the A species. Defining
-0
;[(j-L)&--2
i k=j+l
+
“Free A”:
AF=Ao
“Internal A”:
AI= i (j--1)X, j=l
“Terminal A”:
AT = 2 i
Ak]
5 i [Azb+k-z-bbl}
k=l Z=l
t3’)
(34)
A,
j=l
with the double sum in Eq. (30) subject to the restriction 1 G j+ k - I s N. Assuming N large (i.e., ljrnit Ak= 0) Eq. (29) is shown to be
and substituting in (3 l), (32), and (33) we have
equivalgnt to the following expressiont: $=
4{[ $
Ak]‘-$[
i
(k-
l)&]}-a.
k=l
(31)
The form of Eq. (31) suggests that we look for eCWtiOUS
defining
the
quantities
d(A,) -=Rc-Q d8
(35)
d(AI) - R d8 ’
(36)
d(A,)=_2R d@
g Ak and
C
(37)
k=l i,
with dimensionless
(k-l)hk.
If both sides of Eq. (30) are summed over j from 1 to N the following expression is obtained after rearrangement:
Rc=
reaction rate
(A~)~-&4(&h
(38)
Initial conditions for these equations are given by the following: A,(O) = AR(O) = A,(O) = II(O) = G Again, both sides of Eq. (30) are multiplied by (j- 1) and summed overj to obtain:
AT(O)=~[~-AX~(O)] =2[1-II(O)]
d/K
(39)
=-&. (40)
N
-2[jz
(j-l)S
11
.
(3 3,
In both operations the contributions from R; sum to zero identically. We note at this point that Eqs. (31), (32), and (33) constitute a solvable system of equations. These equations can be rewritten in terms of the extent of reaction p. Alternatively, they
weight of the
iiT;= 2+1 A
(41)
and the ratio A: B by A -= B
tDeve1opment of the collapsed equations is detailed in tht Appendix.
The number average molecular system is given by
>’
(42)
The ratio A: B is ordinarily used to obtain laboratory characterization of low molecular
130
Reversible polyumdensation
weight polymer (p < O-90) with molecular weight (often estimated as a function of viscosity) used for higher extent of reaction. An additional relation is required to determine reactor pressure (in order to adjust vapor outflow rate according to Eq. (9)):
2AF
b=
AT+
final reactor pressure of 0.1 II, was chosen (F = 0.1 corresponds to p = 0.9 assuming full equilibrium). The low final reaction extent limits this study to the early stages of polymerization in order to eliminate the need to handle prohibitively large systems of equations. Commercial polymerization catalysts specifically designed for this polyesterification system yield forward reaction rates typically on the order of 1 l/g mole min. Hence @,= 30 corresponds to a reaction time of 5-10 min. The removal of 90 per cent of the excess A (glycol) in this period of time constitutes a rather severe forcing of the reaction system, subject as it is to constraints on the maximum rate of heat transfer and vapor entrainment. Table 1 presents a summary of reaction conditions and appropriate choices for Q(O), D1, and Dz in Eq. (47) to maintain II, within specified limits, of II( @) (+ 2%) over the entire time of reaction.
(43)
2Ap’
It should be noted that Eqs. (34)-(43) can also be obtained from an “overall” analysis of the reaction . ..AT+AT . . . . .
5 .....A,...+AF.
The factor four in Eq. (38) results from the assumed bifunctionality of AF and the fact that the concentration of sites for reverse reaction is double the concentration ofA,. NUMERICAL
in a semi-batch reactor
SOLUTION
Table 1. Reaction conditions and parameters in numerical soIutions
The semi-batch reaction Eqs. (29) and (30) [or alternatively (35)-(38)] are written explicitly in terms of the vapor outflow rate Q. Most semi-batch polymerizations are controlled by programming reactor pressure. In these studies a linear variation in pressure was used l-I(@) = l-I(O) - [rI(O) -FrI(o)l;
Q(0) = 0.1 /IF(O) = ;I r 2;; AT(O)= K = F* = 0.1 rI(0) =
(44) T
where F is the fraction of the initial pressure desired at the end of reaction, @T. An auxiliary integral equation corresponding to Eq. (9) was used to define Q(6)) as follows Q(o)
= Q(0) +D,(l&--II)
+I& 1 (II,-l-I)dO. 0 (45)
A specific reacting system was chosen to keep the numerical studies close to physical reality: the di-ethylene glycol ester of terephthalic acid is the reacting monomer. For purposes of discussion a reaction temperature of approximately 230°C is appropriate, and under these conditions the equilibrium constant is O-79 and Q(O), 4.3. A
A,(O) = O-47056 l-05886 0.79 0.47056
Solution of the species equations (IV+ 2 differential equations corresponding to Eqs. (29), (30) and (45)) was accomplished by means of a fourth order predictor-corrector routine which reduced computing times significantly over a similar order Runge-Kutta method. Reaction system order (values of N) of 10, 20, 30 and 40 were investigated. Figure 1 presents curves of x,& number average degree of polymerization, versus 8 for these cases. Figure 2 contains similar plots of A : B. In both figures plots of xh and A : B obtained from the collapsed equations ((35)-(38)) are given. It is seen that solution of the species equations for higher N tends toward the solution obtained from the collapsed equations. Hence, as expected, the collapsed equations furnish a much more efficient means of estimating number average properties such as &.
131
D. A. MELLICHAMP
6
I
0
0
I
I
I
t
I
20
IO
30
9 Fig. 1. Number average degree of polymerization as a function of reaction time.
For the reacting system under investigation, chain lengths up to 50 or 60 would have to be accounted for to obtain equivalent levels of accuracy with the species equations. One interesting aspect seen from the solution of the species equations is the importance of the contribution of rearrangement to the overall rate of reaction. At conditions of chemical equilibrium the chemical reaction equations describing rearrangement can be obtained by combination of the chemical equations describing the polymerization reactions, hence do not constitute a mathematically independent set of relationships. The Flory-Schultz distribution, derived from equilibrium considerations,
Aj = (1 -
j=
p)“p’-’
(46)
1,2,.
reduces the rate expression describing rearrangement (Eq. (20)) to zero identically. R; = 4(1 -P)~
$ 5 [(p’-1) (ti+k-1-l) k=l
I=1
- (p’-1) (pk-‘)I
z
0.
Results from numerical studies of the semibatch system indicate that the distribution of concentrations is always close to one which makes the Rj = 0 even when the reacting 132
Reversible polycondensation in a semi-batch reactor
A:B 1.4
1.2
1.o 0
20
IO
30
Fig. 2. Ratio of A : B vs. reaction time.
system is relatively far from equilibrium. Previous theoretical work by Abraham[4] indicates that this result obtains approximately for a flow reactor with pure monomer feed (h, = 1) and assumed irreversible kinetics. We would expect the approximation to be much better with a reversible model and system feed consisting of a distribution of the S,. The application of the “normal” distribution to non-equilibrium situations allows a gross simplification in the calculation of weight average polymer properties and higher moments of distribution of the polymeric species. The collapsed equations can be used to calculate the extent of reaction p and Eq. (46) to estimate the distribution of the A+ Table 2 summarizes the course of reaction as calcuiated from solution of the collapsed equations. Figure 3 presents
Table 2. Extent ofreactionand reactor pressure vs. time (solution of collapsed equations) Dimensiontess time, 8 0 5 10 15 20 25 30
Dimensionless pressure, II 0.470% 0.39997 o-32938 0.25880 0.18822 0.11764 0*04706
Extent of reaction, P
0.470% 0.53804 O-61079 o@M6 o-76038 0*83518 0.90492
the mole fraction distribution versus time. Discrete points were obtained by solution of the species equations with N = 40; the continuous lines were obtained from the values ofp in Table 2 assuming a normal distribution. Coincidence up 133
D. A. MELLICHAMP
10-l
I
o-s
XI
IO-=
I
o-4 0
IO
20
30
8
Fig. 3. Mole fraction distribution of polymeric species vs. reaction time.
to 8 > 25 is obtained; at this point the accuracy of the species equations breaks down. Graphs of the species weight fractions (Fig. 4) and weight average degree of polymerization (Fig. 5) are also presented for comparison. In all of these studies the truncation of the species equations to some manageable level (ZV finite) leads to much greater error in the calculated concentrations than does the assumption of a normal distribution (operation near equilibrium) coupled with the collapsed or overall material balance equations. Use of these overall equations for the calculation of number average properties is rigorous in any event.
The engineer faced with the mathematical description of a practical polymerization problem, i.e., polymerization in a reactor which neither approaches plug flow nor perfect mixing conditions, and with final values of x.& equal to several hundred, finds that the assumption, even, of instantaneous chemical equilibrium is minor compared to the assum.ptions required to handle the complexities of mixing and diffusional mass transfer. The application of the overall or collapsed reaction rates to the partial differential equations required to describe a real polymerization reactor would appear to be a more fruitful approach and one which is open for further study. 134
Reversible polycondensation in a semi-batch reactor
20
IO
30
0 Fig. 4. Weight fraction distribution of polymeric species vs. reaction time.
CONCLUSIONS
1. The N + 1 species material balance equations describing reversible polycondensation (both polymerization and rearrangement reactions) can be collapsed to three ordinary but non-linear differential equations. These equations are exact in the sense that they can be used to determine number average polymer properties rigorously. 2. Numerical solution of the species equations applied to the case of semi-batch reaction shows that a high order system (N large) must be
treated to obtain accurate estimates of species distributions. Because of the high degree of mathematical interconnection in the species equations, an approximately normal distribution of the species obtains, even when the reacting mixture is relatively far from chemical equilibrium. Hence the collapsed equations can be used with an algebraic relation to estimate distribution of the species with a resulting great reduction in computational effort. The extension to the CSTR has not been made in this paper; however, work by other authors 135
D.A. MELLICHAMP
16
12
8
4
0
0
-’
30
20
IO
8 Fig. 5. Weight average degree of polymerization vs. reaction time.
indicates that a nearly normal distribution is obtained in several cases of interest. The application of the collapsed equations to these cases is also valid. Adcnowkdgment -The author wishes to express his appreciation to the University of California, Santa Barbara, Research Committee for their generous donation of computer time.
B
cj &,Ds F
NOTATION
A AF AI AT
molecular unit in the ABABAB... polymer chain dimensionless concentration of “Free A” dimensionless concentration of “Internal A” dimensionless concentration of “Terminal A”
K N
Q Q 136
molecular unit in the ABABAB... polymer chain concentration of the species with chain lengthj empirical constants in the equation generating vapor outflow rate (Eq. (45)) fraction of initial reactor pressure at the end of the reaction reaction equilibrium constant maximum chain length in the mathematical polymerization model volumetric flowrate of SOremoved from the reactor dimensionless volumetric flowrate of&
Reversible polycondensation in a semi-batch reactor
Rj
sj T V
xi xty j,k,l,m,w
kt
k; kr P * PO
rj
dimensionless reaction rate of the collapsed system (overall rate of reaction) dimensionless rate of generation of species with chain length j by polymerization dimensionless rate of generation of species with chain length j by rearrangement species with chain length j absolute reaction temperature volume of reacting mass number average degree of polymerization (chain length) weight average degree of polymerization summation indices forward polymerization rate constantt rearrangement rate constantt reverse polymerization rate constant? extent of reaction pure component vapor pressure of So at reaction temperature, T
rate of generation of species with chain length j by polymerization rj' rate of generation of species with chain length j by rearrangement r time 4 weight fraction of polymeric species with chain length j (So not included) mole fraction of species with chain length j mole fraction of polymeric species with chain length j (S, not inciuded) dimensionless reaction time dimensionless time at end of reaction dimensionless reactor pressure (programmed) dimensionless reactor pressure calculated from vapor-liquid equilibrium relationships dimensionless concentration of species with chain length j molar density of reactants reactor pressure (programmed) reactor pressure calculated from vapor-liquid equilibrium relationships
tAll rate constants expressed in terms of monofunctional reactants. REFERENCES
[I] FLORY P. J., Principles of Polymer Chemistry. Cornell University Press 1953. r2i BIESENBERGERJ. .4..A.I.ch.E.Jl1965 11369. i3j ROOTSAERT W. J. M. and Van de VUSSE J. G., Chem. Engng Sci. 1966 211067. 141 ABRAHAM W. H., Chem. Engng Sci. 1966 21327. VI KILKSON H., Ind. Engng Chem. Fundls 1964 3 283. [6] ZEMAN R. J. and AMUNDSON N. R., Chem. Engng Sci. 1965 20 33 1. [7] ZEMAN R. J. and AMUNDSON N. R., Chem. Engng Sci. 1965 20637. 183 SMITH N. H. and SATHER G. A., Chem. Engng Sci. 1965 29 15. [9] TADMOR Z. and BIESENBERGER J. A., Ind. Engng Chem. Fundls 1966 5 337. [lo] TANFORD C., Physical Chemistry of Macromolecules. Wiley 196 1.
APPENDIX COLLAPSING
THE N DIFFERENTIAL BALANCES
Starting with the expression for A0
we note that MASS
(A. 1) subject to
k+l-1
P N.
D. A. MELLICHAMP Then if pit
Ak= 0 there is little loss in accuracy if Eq. (29)
+ 2 ; A,AIN-,_,+ 5 A,Aw-r I-N l-N-1
is rewritten as:
--,A,-2A,A2-3A,As
......*................... -NA,AN
-2AzA,-3X,X2-4ArAJ
.............-NA.A.-.-(N-l)A.A.v
To completely collapse the species equations, expressions for N N X Akand z (k - l)Ak are required. h=l Summing Eth sides of Eq. (30) over all j -(N-l)AN-,Al-NANAt-(N-l)AN-IA5.........-2A.+,A
N - ANAN.
..............................
-NAd,-(N-l)
After expanding on 1 and factoring -2
(j-
l)Aj-2
= (A, - A,)(A, + 2x2 + 3X, . . . . . . . . . . . . . . . . . . . . . . . ..+NAJ
i Ak b-f+, I
+ (A2- A,)(2A, + 3A, + 4As . . . . . . . . . . .+NA+,+(N-HAN) + (A3- AsH3A,+ 4X2+ 5X3. 1. . . . .+(N-l)A,,+(N-2)AJ
+ i 5 bA,+r-t - Wd k-1 bl
(A.3)
I
with [A,A,+k-I- A&l = 0 for j+k-I>
N
c ‘+k--l-z
1.
f(AN_l - A&((N
- I)A, + NAz+ (N - l)A,. . .+ 3A,+_1+2AN)
+(A.v-A,&NA,+(N-])A*+
Proceeding with pairs of terms
(N-2)A,.
. . . . .+21,-,+X,)
= 0. Hence the complete expression is written
-&[ iAj]=-4{[
it=j+1 5A/c (2) jiU--Ihj--2 j=* =i,U-1)&--2 :Ici1)Aj =-ji ti-115 (3)
ji i,
(j-
iAjr-+[
+A#
Both sides of Eq. (32) are now multiplied by the quantity 1) and then summed over allj
&-I)$=$ *1
.[
j-l I &.j-l)Aj
$, [AtAj+k--~ - A&l
(expanding j and k and collecting only the sums along the diagonals)
(A.3 with
[AA+H - AjA,] = 0 = i AlAS++ 2 *iI AlAs-1 + . ..+ (N - 1) ;$; AlAN+ z I=1 N N + N z AIAN-~+l + (N - 1) z AlA.+-I+P + ..
k-1
64.4)
for
c
j+k-l> ‘+k-1-c
t-4
138
N
1.
Reversible polycondensation
in a semi-batch reactor
Again proceeding with pairs of terms
+ . . . + (2N - 3)
2
A&+,
+ (N - 1) i
1-N-l N--1 =
;a
N-l
a-1 =
;a
N--l
b;,
ti+k-
lhk-
ij
?&A,-3AlAY-4AfAJ._...-NAIAN_,--(N-
- ~AxA,-8A&
N-j
x J-1
-l2A&h,--5A&-
z O’+k--Zhk k==*
AlAm._<
I-N
l)ArAN
- 10AaAs. . . - 2(N - l)A,A,, - 2(N - 2)As 18A&. . . . . . . . . .-3(N-3)ArAN
N-J AJ ;,
hk
s
‘:
-(N-l)NA,vA,-(N-lXN-DANA,-(N-l)(N-2)A,A, ,~(i-l)*Aj-i?
(j-l)
i,
,$, _
. . - (N - l)ANAN.
At
=jI
[(j-lP-O’-2)U-I)15
Expanding the positive terms on 1 partitioning the negative terms {Row (j)=f[Row (j-l)-Column (j)]}, and factoring the common term from each row
=i
ci--lpY
= (A,-A1)[Ak+3As+6A,+.
..+(N-2;N-1)XN_-1+
W-WAN 2 A,+3A2+6Aa++....++
(N-
l)NA.v_, +
1
(N-1)N
2”~
for
j+k-I '+k-I<
G
>
N
+(A,-&)
1.
(again we expand j and k and collect the summations on ! along the diagonals) = i
l-l
A1A3_-( + 3 i
I-1
i
+. . (N-
I-1
AlAs_,+n +
i
(N- “:” +‘) i
(N-2)(N+UAN 2
AIAN-,+,
Hence we obtain
AIA,,T--(+s
R&mrmGLes dquations d%quilibre de matCriau decrivant la polycondensation reversible, a la fois polymerisation et r&urangement. sont dCrivCes et appliqudes au cas special de la reaction en demilots. Les solutions des equations N + 1 pour des valeurs de N allant jusqu’a 40, sont prtsentbes. On provoque l’arr6t des equations a une strie de trois equations differentielles ordinaires non-lit&tires et on montre qu’elles sont rigoureuges pour le calcul des propri6tes du nombre moyen. Finalement, les equations arri?tees sont accouplbs avec une fonction normale de distribution pour le calcul de la repartition des esp&ces et des proprietes de poids moyen, ces resultats se comparant favorablement aux solutions obtenues a partir des equations individuelles d’tquilibre de mat&au. Zusammenfassung- Die Materialbilanzgleichungen, welche die reversible Polykondensation, und zwar sowohl die Polymerisation als such die Umlagerungsreaktionen beschreiben, werden abgeleitet und auf den Spezialfall “Semi-batch” Reaktion angewandt. Es werden Lijsungen der N+ 1 Gleichungen fur Werte von N bis zu 40 berichtet. Die Gleichungen werden auf einen Satz von drei nicht linearen gewohnlichen Differentialgleichungen reduziert, und es wird gezeigt, daL3sit bei der Berechnung von Zablendurchschnittseigenschaften streng zutreffen. Schliefllich werden die reduzierten Cileichungen mit einer normalen Verteilungsfunktion verbunden, urn die Verteilung der Spezies und der Gewichtsdurchschnittsmerkmale zu berechnen. Diese Ergebnisse waren im Vergleich mit den Losungen aufgrund der einzelnen Materialbilanzgleichungen sehr zufiiedenstellend. 139
3
+;A~-A~)[~A~+~~~+(~+~:(~-~)A~
AtAd++ 6 2 A,A5_,+ . . . y
+y
%+~Az+~OAS+ . . . . . . .+
1
1
l)AN