EFFECT OF REACTION PATH AND INITIAL DISTRIBUTION ON MOLECULAR WEIGHT DISTRIBUTION OF IRREVERSIBLE CONDENSATION POLY M ERS H E N N KILKSON
Engineering Ttchnologj Laboratory, E. I. du Pont de sVemours & Co.. Inc., Wzlmington, Del. Some problems of irreversible polymerization in batch and steady-state reactors have been solved b y the use of a discrete transform. Analytic solutions for arbitrary initial distribution have been obtained in all cases treated and relationships between the moments of initial and product distributions have been developed. Distributed residence time causes marked broadening of the distributions. Chain stoppers counteract the broadening effects. The results imply that viscosity i s often not a suitable control vaiiable, especially when the reacior configuration i s ill defined. In certain systems viscosity may remain essentially unchanged while the number-average chain length increases 1 0-fold.
majority of work on prediction of molecular weight ’ . (hft$Tl) of polymers has been accomplished by statistical approaches. \'cry formidable success has been achieved via this avenue by researchers such as Flory ( 6 ) , Stockmaier (9):a n d others following them, bur the statistical method inherently precludes incorporation of a reaction path other than the case of “equal probability of reaction of all Yimilar chemical groups,” such as exists in a homogeneous batch reactor, with a few exceptions. T h e molecular weight distribution is regarded simply as a state function of a few parameters such as extent of reaction and stoichiometry. Often this is not a great shortcoming. For example, many polymerization systems are so highly reversible that they rcxnain throughout the reaction in a n equilibrium state in all practically encountered reactor configurations. Irreversible polymerizations and systems involving irreversible side reactions, ho\vever, are highly dependent on path. T h e study of the path effects is all the more important because viscosity, \vhich depends on the shape as well as mean value of the molecular weight distribution, is one of the primary means of controlling polymerization reactions. T h e alternative approach. kinetic formulation, permits the direcL incorporation of reaction path a n d is relatively straightfor\\fard. Ivhereas the statistical approach requires a great deal of ’.intuitive insight” and is therefore prone to errors in complex cases. Hoivever. the kinetic formulation requires integration of the infinite set of rate equations describing the generation and decay of the individual chain length species. I n some cases the set of equations can be solved by solving three or four of the first members of the set and deriving a recursion formula by induction. as was done by Flory ( 6 ) ,but these cases are rare and the existence of a simple recursion formula is required. Saito (8)solved several cases of chain scissioiiing a n d crosslinking by approximating the discrete distribution of chain lengths with a continuous variable. Zeman and Amundsen ( 70) similarly utilized continuous variable approximation and employed a generating function to solve some problems in addition polymerization. Several others have also employed the continuous approximation concept, which often results in HE
Tdistribution .
considerable mathematical simplification. However, a continuous approximation is the least accurate in the short chain length region and the short chains are, as a rule, the most preponderant a t all times during the reaction. Condensation rates are proportional to the number frequency of molecules, so that the continuous approximation may introduce a significant error in the predictions. Abraham ( 7 ) proposed and presented an example of the use of a discrete transform, the Z-transform, for condensation of the infinite set of rate equations into a form amenable to integration and inversion. T h e method preserves the discrete nature of the distribution a t all times and derives additional benefits afforded by the integral transform, notably the relatively easy incorporation of a n arbitrary initial distribution and a concise presentation that simplifies the utilization of the solutions as elements in formulation of increasingly more complex problems. T h e developments reported in this paper are based on the use of the Z-transform and variants of it which. it is believed. have several desirable features. After initial formulation the transforms are normalized, resulting in marked mathematical simplification of the expressions to be integrated. T h e normalized transform is a generating function for probability or frequency distribution and is thus closely akin to the momentgenerating function used in statistics and the more general characteristic function. Association of the normalized Ztransform with these functions and drawing on the large body of information available concerning the properties of these functions allow short cuts in solution of some more complex problems-once a solution for an analogous. simpler problem has been obtained-by use of the statistical properties of the normalized Z-transform. It follows that several other discrete integral transforms are equally suited for solution of polymerization problems. T h e majority of texts on Z-transform. such as that of Kagazzini and Franklin ( 7 ) . are oriented toward process dynamics a n d control theory and. i t is felt. present little insight to this particular application beyond some formal mathema tics ; texts on the properties of Laurent’s series of a complex variablr VOL. 3
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( I ) are probably more helpful. Bharucha-Reid ( 2 ) presents numerous applications of discrete transforms to problrms of siriiilar nature. notably some biological processes. and is very helpful in both formulation and solution of problems of polymerization kinetics. Pertinent Definitions and Properties of Z-Transform
The Z-transform of a distribution of chain lengths { lli\ may be defined as
For convenience the tranaforrn C(t.7) is often denoted b k C L Dysignate (li) It follo\vs that the momepts of the normalized distribution are related t o the moments of the nonnormalized distribution by
a
and thus \\.here P, = concentration of chains of length T i . moles per unit volume. and t = an arbitrary transform variable. 'l'he following properties of the transform will be used immediately.
=
Ragazzini and Franklin (71 give more detailed mathematical illformation on the %-transfoj-in. .['he 2"-transform is a onesided Laurent's series in z. \\.hose coefficients are the concintrations P,. Standard textbooks on this series are very helpful in recogpizing propr.rties of the transform.
total concentratiop of molecules
For convenience P(1 ) is often denoted by If P, = P 7 , ( r ) .then
?
Irreversible Condensations in Batch (Plug-Flow) Reactor
Random Condensation of . l B Fnd Consider the condensation of bifunctional molc~culesof type -4---(BL4;-i 1 B in thr prcsencr of monof'unctional molecules or chain stoppers. .4 -(@,4-X,,- 1 . \\here .4.B c.esignate active end groups capable of reacting \\-ith each Qther to form a link [.lB) a1.d .I' is a n inert end group. Let the reaction betwecn 1: and B groups o c ( ~ i rirrevrriibl). accoldi?lg t6 second-order kilietics. \virh a rate constant independent of chain length. so that the over-411 decrease in concentration of end groups is given by
'I he kth moment of distribution .\Ichis defined as 'WAI
=
-rhus
'lhe weight- and number-averages of the Lhain length distribution. and .ij,. can be constructed frpm the moments (le)
Lvhere 0 = time and i;= rate constant. Designate a bifunctional molecule containing n .4R units b!- I , and it.; monofunctional counterpart (.lBj,,= I -4.Y by i,,,. 'l'he above condrnwtion mechanism will then be represented by
'Thr %-transform 0bq.s the convolution theorem analogously to the Laplace transform. If t\vo distributions { I , \ and { I , * } are independent. then An individual species I, may be formed from any pair of bifunctional molrcules \vhocc indices add up to n and destroyed by reaction \vith an>- molecule. regardless of length or kind. T h u s the rate equation for species I,, is given by since by defini[ion lo dors not exist. 'I'he normalized Z-transform is defined as lvhrre f',,
=
P , , ( r ) = concentrarion of 1,
P,!,= P O l ( ~=) concrntration of I, T
Thus
=
hi)
.I'he ?peck4 I,,, is crratrd b>-indicia1 addition of a bifunctional molecule and a monofunctional molecule and destro) ed by rraction with any bifunctional molecule ; accordingly
282
IBEC FUNDAMENTALS
'I'he rate constant. k. is defined in terms of unit concentration of end groups. Equations 5 and 6 define the infinite sets of rate equations describing the time dependency of the t\\-o distributions ( i n }and ; i n r ) . Multiplying Equation 5 by Z-" and summing over n
2
n=1
P.Z+
5
2 , = 1 PI
Identification of the meinbers of Equarion transform (Equation I ) >ields
.;*I
+5 i = 1
tributions and are very simply related even far divimilar feed distributions. It is convenient to change variable? before integration of Equations 14 and 16. Define
f
P(1.r) P( 1 .O)
=
~~
r)
in terms of the
Thus J indicates the fraction of original R-type ends remaining and /Yo denotes the ratio of chain stoppers to bifunctional molecules a t T = 0. Differentiatin5 Equation 18 and substituting for PI 1 . T ) and Px(1.d
Trancformation of Equation 6 in a similar manner lields for the distribution ( l n l l Dividing Equation 14 by 20 and integrating yields Evaluar'ing Equations 8 and 9 a t t = 1 )ields the rate equations for the total number of molecules of each kind Denote 1 1 ~~
-f
+ 'YO
q,'
=
f+
*.. 1 - q
'YO
+ x,
= ---
1
122)
'I'hus Clearly, the total number of the monofunctional moleculec may not change, xvhereas the number of bifunctional molecules decreases because of both internal condensation and change of class or kind by reaction n.ith monofunctional molecules. T h e normalized transform for the distribution of bifunctional molecules C, = C(2.r) is by definition.
0
q
< 1 +1 X"
l i m q = 1 -,f X*-O
Rearranging Equation 21 and inserting 22 yields
c, = Thus
6
C,,
(1 - q ) 1
-
(23)
qC2,
and consequently by Equation 17 1 bP(2.r) ac, ~- -~
b~
P(1.r)
-
br
C,
dP(1.r) __
P(1,r)
dr
(13)
Substituting Equations 8 and 10 in Equation 13 yields Integrating Equation 20 establiqhes time dependenc) of q(f.X,) TI
T h e normalized transform for the monofunctional molecules is defined as
\vhich reduces for
X, = 0 to I
f =
The time dependency of C,, is obrained analogously to Equation 14
-
1
(25a)
+ P(l.O) T
If the feed material is all monomer, then C,, =
2-l
and
Dividing Equation 14 by 16 and integrating yields C, ~~~. - C,,
-
C,"
~
CiIO
Identifying the coefficients of ZPaccording to definition of the transform (Equation 1 ) sho\vs the relative frequency of species l,?. as a function of q. to be
ivhere C,, and Ci,,, indicate the transforms of feed distributions of the t\vo classes of molecules. From Equation 17 we see that rhc distributions remain identical for identical feed disVOL. 3
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Equation 27 is of course the \vel1 known result of Flory ( 6 ) , often termed the "most probable" distribution. If the initial distribution is other than monomer and of complex nature. the relative frequency of species I , may be obtained by the inversion integral for the extraction of coefficients of Laurent's series (I)
As a rule. ho\sever. the individual coefficients are of little value if they d o not obey a reasonably simple recursion formula. in Lvhich case one may expect to be able to expand successfully the transform of a distribution such as Equation 2 3 and obtain the coefficients without reverting to the inversion integral. Recourse can be taken in such instances to expressing the distribution in terms of its moments or. more often. in terms of the averages coxstructecl from these moments. The moments may be readily obtained by logarithmic differentiation of the transform and b>-evaluating the derivatives a t the limit z -+ 1 . as rhobvn by Equations I C and Id. For example. first differentiation of Equation 2 3 yields
out by Flory. T h e fact that this ratio approaches 2 a t high degrees of conversion has long been used for qualitative comparison of any given distribution against the "most probable" distribution. Equation 33 demonstrates the diminishing influence of the feed distribution on the shape of the product distribution \rith increasing degree of polymerization-i.e.. q + 1 (attainable only if .Yo = 0). Since the level of chain stoppers also determines the maximum degree of polymerization attainable\ it is seldom that .Yo exceeds 0.05. Even at X, = 0.05 the effect of the shape of the initial distribution becomes negligible at complete reaction. It is noteisorthy that the kth moment of the product distribution depends onl>- on the kth and lo\ver moments of the feed distribution (note Equations 30 to 3 5 ) . If a product polymer is to conform to the "most probable" type of distribution. the feed material must be either mono.mer or also a "most probable" distribution. Each distribution of the "most probable" type requires that it be describable by some q : y*. such that
Equating 37 and 23 yields -1
I
and thus
1 -q*go = ~-
where Proceeding in this manner yields the required moments for the averages of the distribution { I , )
1
im.z
+
+
+
1ri-m
+
__ .\if(&
=
'
z
]\pI +
P(1.0) [frn't'
l\fi(k
+ XOom,(':]
(36)
For monomeric feed of both classes .vv(0) = .vn(0) = 1 and, according to Equation 13. .YE = 1 q . as was pointed
.vn
284
I&EC FUND'AMENTALS
+
-
(3) (4)
1nz
(39) (40)
I,,,
-L.
(41 J
no reaction
Subscripts .iand y indicate again the presence of the kind of inert end groups on the molecule and chain length n indicates the sum of A Yor B X units in the chain. T h e monofunctional molecules A X and B I. will participate in further polymerization. However. the totally inert molecules X I- \sill, once formed. cease to participate in the reaction. Hence such a system is considerably different from the system involving only one kind of monofunctional molecule. The normalized transforms of the sets of rate equations for this system are given by
+
+
-
Asf decreases. the total concentration of hifunctional molecules decreases and eventually this class of molecules vanishes completely. The moment of the combined distribution. { 1 , I n * } , are given b>-the sum of the nonnormalized moments
"Tn(o)
- 1,
In-in
(33) Expressions 31 to 33 refer to the distribution of bifunctional molecules and. by 1'. also to the monofunctional molecules and the combined distribution. { 1, I,,}. if the feed distributions, { l n o }and { i n Z n lare . identical. The moments of distribution { / 7 r 7 } for the general initial condition are given bv
q
Condensation of A B in Presence of Two Dissimilar Chain Stoppers. Let the system described by Cquations 3 and 4 contain anorher type of monofunctional molecules of the general structure B-(L4B)nx71'. where Y is again an inert end group. There \vi11 now be four different kinds of molecules and the possible condensation reactions are described by 1,
and
-
1 ~
-
-
and the equations of change for the total number of molecules b\
.'""
= - p(l.T)[P(l.T)
dT
- - dP,(1 .T) -
dPV(1.T)
-
+
+ dP ( 1 T)
dr
dT
Pz(1.T)
+
Py(l.+)]
(46)
=
dT P,(l.T)
'
Py(l.T)
(47)
From Equations 42 through 4.5 it is evident that
Furthermore. genera! integration of Equation 42 shows distribution [ l n } to be of the form
where 1
-
\\here the integration or summation. respectively, is taken over the \\hole range in which x is defined. By definition l h . C, represents the Z-transform of the relative frequencies of { l , ) . Setting arbitrarily z = P - I and $(x) = x. transforms C, and m ( t ) become equivalent. l h e variable x is. in the case of polymerizations treated in this paper. the chain length. n. The large. \\-ell organized body of information on the moment-generating functions in texts such as Cramer's (5) makes such association particularly useful. O n e of the subtheorems on the properties of m ( t ) explains the nature of solution Equation 2.1. It has been shown that if m l ( t l ) m s ( t ) . . represent the moment-generating functions of the frequency distributions of some variables Y , . \ ? . etc.: and if the ,t are independently distributed. the moment-generating function for the distribution of the sum of k variates taken from these distributions is given by the product of the momentgenerating functions from \vhich these samples are taken. Thus
q is given by
m(X1 +
x*+
'.' X k ) ( t ) = mx,(t) ' m x , ( t )
'
'
mXk(t)
(53)
Consider solution 23
rhus. the three distributions in Equation 48 \vill%for monomeric feed material, be again of the form of the most probable distribution described b) Equation 3'. T h e nature of distribution {lri,,) is not a t once obvious Solution of the time and P ( 1 . T ) yields functions Pz(l.T). PP(1.7).
c,
=
(1 - q )
ac
C,,, = Cz,
'
(511
C,
Letting C,, = 2-l shows that Equation 5 1 is reasonable. T h e expansion of C,,, \vi11 no\\ not contain a 2 - l term, which simply states that dn l,,, molecule must contain at least two primary units-namely. X'4 - BIT
Statistical Properties of the Normalized Z-Transform
T h e particularly useful features of the normalized Z-transform C, become apparent through its close kinship to the moment-generating function used in statistics and the more general form of the moment-generating function^-the characteristic function. If some variable. X. possesses a relative frequency (probability) distribution given by f( x). then the moment-generating function, m ( t ) , of some function of x, say $(x). is defined as the expected value of e' E ( &" j } ; where t is an arbitrary variable (6) "'%
-
-
-
P
~
P
T h e factor ( 1 - q)qn-l represents the relative frequency (on a number basis) of molecules that are the result of joining of n molecules from the feed distribution T h e units are randomly picked from the feed distribution whose normalized To restate. they are independently transform is given by C,, and identicall) distributed. so that the transform of the length distribution, given that n units have joined together, is given by Therefore. solution 23 may be restated as [C,,]".
where T h e transform C,,, can be readily solved only if Cz0= C,,, = C z V o :C,,,, = 0. in which case it can be shown that
C,, 1 - qC2,
=
nc
[probability of having n original molecules joined together] X [the transform of the distribution for the sum of the n randomly joined molecules]
Such probabilistic reasoning can be used to generalize available solutions for monomeric feed conditions. as shown belo\v. Random Condensation of Unlike Components AL4 and BB Consider the condensation of t\vo bifunctional molecules, A A and B B , lvherc the condensation occurs via reaction between an A and a B end group. Such condensation produces three classes of mo!ecules. corresponding to the types of end groups on the molecule. Designate
(AA-BB).-l (AA-BB). (BB---AA).-I
rlA by I,, (both end groups A ) by I,, (one end group A , one end group B ) BB by in, (both end groups B ) (56)
Assuming second-order irreversible kinetics N ith rate constant independent of chain length. the rate equations for the individual species I n k . I n i , and I,, become
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Since Expressions 57 and 58 are analogous. we ma) rvrite a t once m
m
l
I n the above expression 7 = 2kO ( k defined in terms of endgroup concentrations). ‘I‘he above system was treated by Case (,?) by probabilistic methods of monomeric feed of classes e and k and in the absence of class j from the feed. ‘I‘he latter assumption is realistic, since class j is capable of self-polymerization and not normally found in this type of polymerization rxccpt as an intermediate. Case’s results can be manipulated to yield the folloiving recursion formulas for each of the three classes
cnfi= e,,
=
e,,
=
(1
-
4)qn-I
The first unit of thr transform ofj-type molecules represents the condensation of 1 k-type and 1 e-type unit. It is again accompanied by reducrion of the n-value. according to definiI n vie\v tions 56, and will therefore be given by z C Z ~., C,,,. of recursion formula 60 Lve can. by reasoning similar to the above. lvrite for the transform of j-type molecules
Logarithmic differentiation of Equations 62, 63. and 64 yields
(60)
Parameter q in this case is related to the number-average degree of polymerization “with respect to any given component.” Let
“T,
= =
-vnj=
number-average of ,4A units in k-type chains number-average of BB units in e-type chains number-average of .4.4-BB units in j-type chains By definition, mh(l) = - T n h , = 7n e , m j ( l ) = For = 1 and the above monomeric feed conditions mho(1)= expressions for 1 - q become
Then
I
, Thus q is analogous to definition 22. Each distribution of the k-, e-, and j-type units represents a “most probable” distribution. as \vas predicted by Flory and others even prior to the Lvork of Case. The recursion formula (Equation 6 0 ) can be used to develop the solutions for any arbitrary initial distribution { I n k o ) and {I,,,) by use of the probability generating function properties of C,. Consider the distribution { j n k } . T h e first member of the series in the transform is clearly (1 - q)C,B,, corresponding to the product of the probability that a given k-type molecule entering the system in the feed remains unreacted times the normalized transform of the length-distribution of k-type molecules in the feed. T h e probability that a k-type molecule contains n original (but not necessarily monomeric) k-type units is (1 - q)q“-l: and such a molecule \vi11 also contain, by Equation j6! n - 1 original e-type units. T h e length of a n e-type molecule is defined on is. I,, contains n BB units but only the basis of BB units-that n - 1 A.4 units. Thus the over-all length of such a k-type unit containing n original k-type units must be reduced by one unit for the presence of each of the n - 1 I,,, units contained in such a molecule; in terms of the transform of the distribution this can be accomplished by multiplying this term by z n - ’ . ‘The transform of the relative chain length distribution, if n original k-type units have joined, is therefore given by Crl;o[~Czeo‘ Czfioln-‘
(61)
which is in agreement ivith the definition of q for Equation 60. .4 straightforicard integration of the transformed rate equations is more rigorous. but more difficult. Such probabilistic reasoning can save a considerable amount of time by furnishing solutions \chich can then be proved or disproved. I n fact. the above problem was first solved by this method and the results Lvere then proved consistent ivith rate Equations 57 to 5’9, The time function. q. as well as the time dependency of the total concentrations oi each class. P k ( l z ~ )Pe(1.7), . and P j ( l . ~ )must . of course be obtained by integration, but do not require the solution of the transformed equations for an arbitrary z . The time functions are given by
P,(1.7)
I--?=--
=
1 1 - Rv2
[ l - R ] [ 1- Rv2] [l - Ru’]
where
Summation of Equation 61 over-all values of n yields the transform of the k-type units.
czfi =
(1 - q)jCzk,
+ ~[zCze,I
;t7 . .
,
I,
2% =
Pn(l.0)- P,(l.O); 1 > v = exp\(-A
or 286
[cz,oI*
l&EC
FUNDAMENTALS
’ 71
0
'l'hat is. class k is arbitrarily chosen to designate the majority component. l ' h e transforms of the sets of rate equations and the proof of compatibility of Equation 57 to 59 u i t h solutions 62 to 64 are sho\vn in the Appendix. Step-by-step Condensation. T h e last item concerning polymerization in batch reactors is polymerization b>- the me c ha i i isin c
where I,, = an active molecule of length n and I , * = a monomeric unit capable of reacting Lvith the active unit l,l. 'I his rnechanism was discussed in detail by Abraham ( 7 ) as an exarnplr of the use of the /?-transform to solve polymerization problems. Generalizing his result to an arbitrary- initial distribution of / l , L } and normalizing the transform yields for the transform of active molecules
c, = c20 e x p ( ( z - ' '
where
Y =
P * 0 P,
= =
-
1 ) v ( I - f,}
(77)
P,,* P,, concentration of monomer a t T = 0 concentration of active molecules, invariant in time
f
= P*(I.T)
Fi action
,+ is given
+
Irreversible Condensation Polymerization in SteadyState Reactors with Distributed Residence Time
Polymerizations in Continuous Stirred Tank Reactor. T h e continuous stirred tank reactor represen1j the qiniplrst and best knoivn euample of a steady-state reactor i v i r h distributed residence time. given by the distribution function
P*(l.Oi (77j
b\
f I he number- dnd
Coupled with the use of an integral transform. xvhich i a alrnust mandatory for successful solution of the rate equations. thr kinetic treatment leads at once to solutions in terms of generalized initial distributions. such as Equations 2 3 and 31 to 33. 7'he general solutions provide considrrablc insight lvhich is helpful in application of probabilistic reasoning to more complex problems. such as the general solution for 1:1: BB condeniation described above. and provide tht. eleinent.; in formulation of models for more complex reactor confisuration.;. 'I'he subsequent sections deal Lvith come stcady-statc reactors with distributed residence time. Such s)stems sho\v a clear advantage of the kinetic approach over the probabilistic reasoning. 'l'hr. mechanisms of polymerization5 to be treated have hreri alrcad!- discussed in connection \vith batch polymerizations.
=
expj-i),
M eight-averages
'
(73)
T )
wheref(0)
relative frequency of fluid elements exiting a t age 0 9 = residence h i e (age) of a fluid particle 6 = mean residence time of fluid in reactor
for i,! 1 are given b\
in = Tn(o) + u(i
(74)
- f)
(75)
Foi monomeric feed Equations 74 and 75 reduce to the expressions given b\ Abraham. which she\\ that
hiechanism '1 represents the chain groivth step of several of polymerization mechanisms. including some ionic and free radical systems. In polymerizarions proceeding bymechanism 71 each successive "event" or reaction occurring in a system containing rnolecules \%ithmaxirnum length j will producc, a distribution \vith maximum length j 1 ; in polymerization by previously described mechanisms such as .? and 4 the distribution of the nrxt generation \vi11 have a length 2 j . It t h u s seem? appropriate to term mechanism 71 a "step-bystel)" polymerization and the previous mechanisms as "random" 1)olymr.rizations.
+
Batch Polymerizations
Hoinogeneous batch polymerization problems are relatively amenable to solution via probabilistic reasoning. by virtue of singlt~-valueJresidence time in such a system. Given a rate coilstant inJelwndent of chain length. as has been assumed throughout this paper. such qyctems ob?)- the rule of "equal rcactivity of any end group." ' I h e principal advantage of kinetic ireatmrnt of such problems lies in the relative1)- straightfon\-ard mannrr in ivhich the rate Tquations can be formulated. ivherea.; probabilistic reasoning requires a great deal of "intuitive skill" and iq therefore more prone to errors in logic.
=
POLYMERIZATION O F .1R IS PRESENCE O F ilS. 'l'he randorri polymerization specified by mechanisms .? and 4 are coiisidered first. Let the stirred tank have a volume L 7 and let tht. rate of volume flow in and out of the vesiel be Q. 'I'he seta of material balances for the individual members of I i,, I and (in,} are given. according to Equations 5 and 6. by
(79)
Let p = Q k l . Since I T 'Q = 0 = mean residence time in the reactor. represents the inverse of a reduced mean residence time. 1 ' 7 . I hus 9 +. m indicates infinitely short mean residence and no reaction. ivhereas p-+ 0 indicates infinitel!- long mean residence time and complete reaction. HoLvever. for conveniencc subscript o is still used to denote feed distribution; an infinitcly long residence time is denoted by. " : ]B Thus. for example, the transforms of the distribution [ l n z } are denoted ac
__
P*(z./3) =
2 Pnrz-"
VOL. 3
=
PA,:
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r-] lim
fir= B,
NOVEMBER
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287
+ Pa
~Pzzo
'
Pz - P,, B '
BPzr =
o
(83)
The moments of the distributions can be readily evaluated from Equations 82 and 83. 1he moments of { l n }are given by
The behavior of these distributions for monomeric feed is shoiLn in Figures 1 to 3 Figure 1 s h o w the number-average chain length of [I, 1 . A\-n. as a function of the extent of reaction f . defined as
and the ratio
and of { i n z }by
X, = p ,
,'f
For .Yo> 0 it is seen that .f, passes through a maximum a t a rather - low over-all degree of polymerization (at f = 0.06, or l\-li = 17) and regresses to 1: in accordance with Equation 99 and the stipulation of monomeric feed for both classes. 'The location of the maximum is little affected by X,. but the height decreases with increasing level of chain stoppers. Figure 2 sho\vs .f, 185. f . Lvhich increases monotoiically with -7,. ?-he over-all \\-eight-to-number average ratio unlike - -\'% .I,,. sho\vn in Figure 3. also passes through a maximum if A', > 0 and culminates near 2 a t complete reaction. 'Thus the system behaves in marked contrast to batch polymerization for the same mechanism. The distributio,i becomes extremely wide in the absence of chain stoppers. However. the preseiice of chain stoppers acts to narrow the distributioii. ana distributiorLs become dissimilar. - - of classes { i n ]and .vw .Tnis also highly dependent on the exte.it of reaction. 'I'he reason for such behavior becomes most obvious upon examination of the system describe.: helo\\-. STEP-BY-STE.~ POLYMERIZATIOZ. Application of the Ztrai-,sform to solve the set of material balances describing polymerization accorLing to mecha,iism 71 in a continuous stirred tank reactor is straighrforivard. The normalized transform of the distribution of active species exiting the stirred tank can be shoivn IO be given by
& :21,(2
=
Mzo(2'
+ .Mo(2' + 2@[,M,:')]*+ 2M,(',M (1) P + P ~ I S + + ~ I ~ 3+pr
(89)
? h e moments of the combined distribution may be obtained by addition of the nonnormalized moments. Av%kl J f \ K
+ .Lf,'e'
'I'he nature of these cistributions !\-ill be assessed in terms of the ratio .yw, .yn,using the "most probable" or Flory distribution. for \\-hich this ratio approaches 2 a t high degrees of pol>-merization. as a reference. In absence of chain stoppers{ i n } tends toward infinite disthat is, P, = 0-cistribution persity as the residence time goes to infinity. hm @-O
--a:
{flirt
(90)
.\-?)
(95)
T h e presence of chain stoppers-i.e.. P, > 0-has a profound effect on the nature of the distribution. The limiting values of the moments as 7 m of the distribution of bifunctional molecules. { l n } .>ield the relationships -+
The fraction of incoming monomer leaving the reactor. f . is related to holdup time and initial concentration of active molecules bv D*
T h a t is: as the reduced mean residence time approaches infinit>-.distribution { l n ] regresses to the feed distribution, { l n o } . Of course 3 -+ 0 implies b>- Equation 84 also that -+ 0. so that this class eventually vanishes. From the moments of { l n z ] a t the limit 13 +. 0 )\-e obtain
The moments of the normalized distribution exitine; reactor are established from Equation 95 7n"
(93)
2
=
m,
+a
and
and where
Consequently. for monomeric feed of thr active molecules
288
l&EC FUNDAMENTALS
(9')
\vhich increases monotonically u i t h value lim a+=
cy
a n d attains the limiting
!!: = .Tn
(100)
Comparison of Equation 100 \vith 90 sho\vs that the step-bystep polymerization is much less intiuenced by residence time distribution than random polymerization. T h e above results also clarify the action of chain stoppers in random polymerization as counteragents to the distending effects of distributed residence time for random polymerizations. At high mean holdup times in the stirred tank the total concentration of bifunctional molecules assumes a verv low level. according to Equation 84, while the concentration of monofunctional molecules remains unaffected by holdup time. l ' h e chain gro\vth of class { l n ] is a second-order process bvith respect IO 8, whereas the gro\vth of class (in,) is first-order in class { in! \vi11 tend to extend both and 8,. Thus. if 8 -merizarion problems. 'l'hc: ability of the transform to handle convolutions extends its applicability to second-order '.random" condensationa~--av('rv valuable asset, since such +steins are practicall) ver!. cornmolt but difficult to handle mathematically. I h e atatiatic.il properties of the normalized transform add insight as \vel1 .is provide mathematical simplification. ~ l ' h ee a s of incorpor 1tion of an arbitrary initial distribution and the concise nature of the transformed solutions lead to a convenient prqrehsion from the simple to the more complex problems. usins; th:: elementary solutions as building blocks for the more involved problems. T h e chemical mechanisms of polymerizations treated fall into t\vo distinct categories : ,.random" and "step-by-step"
/
E n OVERALL NUMBER-AVERAGE CHAIN LENGTH IO 675 168 200 I V L I L L I _ I L L - l 01 001 0001
' IO f
, FRACTION OF B-ENDS REMAIkING UNREACTED
Figure 8. Random condensation in plug-flow reactor with recycle
where
O v e r - a l l w e i g h t - a v e r a g e v s f a t two levels of recycle r a t e
P*(at 4) =
2.5
I
I
~-P,*
c:, + FC, -~
i
2.0-
-
~
'Iz
1i-F
\ 1
12
F= 2
Evaluation of moments yields the follo\ving relationships :
"f, =
.fn(Oj
+ v[l
I"'1
Rn REACTION)
C O (M P E LT'+
arid C,, is. by material balance
c,,
'
MAXIMUM
F =
-/'I
I
F = 0 I
10
100
A
1000
N , , N U M B E n - A V E R A G E CHAIN LENGTH
Figure 9. Step-by-step polymerization in plug-flow reactor with recycle
and. for
.in.+
cc
Weight-to-number a v e r a g e ratio
VOL.
3
VI.
n u m b e r - a v e r a g e of active molecules
NO. 4
NOVEMBER 1 9 6 4
291
condensations. T h e reactor configurations include batch or plug-flow reactors and t\co steady-state reactors \sith distributed residence time. In all systems treated it has been shown that a moment of a given order of the product distribution can be expressed in terms of the same and lotser order moments of the feed distribution. Thus: for'example, O)= P e ( 1 , 7 ) - Pe(l,O)
(11A)
Define
-
Since ,i-w is closely related to viscosity, it folloics that viscosity is a very insensitive means of following the degree of polymerization in such systems. If the residence time distribution in a given reactor is unknown or variable. then viscosity alone is nor a sufficient means of controlling the reaction; under all circumstances it appears desirable to develop the relationships between "YE and residence time distribution for any given reaction system. if this response is to control the reaction. .4 very \side distribution has been predicted in several instances. T h e developments refer to irreversible polymerization and the existence of even a relatively moderate rate of degradation reaction will sharp1)- limit the width of the distribution. Certainly one ~ s o u l dnot expect to see often. because of either reverse reaction or the presence of trace quantities of chain stoppers. Appendix.
Condensation of AA
+ BB
s
=
Pk(1:0) - Pc(l,O)
u
=
expiI
(12A)
--AT]
and choose class k to be the majority component, so that R 5 1, A 2 0. Substituting Equations 11-4 and 12.4 into 4 4 and integrating yield
(1 3'4) which. \Then substituted into 11.4, yields a t once (14'4)
Adding t\cice Equation 4A to 6.4
T h e nonnormalized transforms of sets 57 to 59 are given by
\chich upon integration and elimination of Equation 13.4 yields
292
I&EC
FUNDAMENTALS
Pk(l.7)
according to
Solutions 62 to 64 can now be proved compatible u i t h the transforms of rate equations 8.4 to 10'4 and the time depenclency- of q developed. Differentiate the probabilistic solution (Equation 62) u i t h respect to T.
T h e equivalence can no\\. be readily proved by substitution of the "probabilistic" solutions (62 to 64) into 24.4. T h a t is, substitution shows that
T h e equivalence of Equation 62 with 8A and 6 3 with 9A is a t once obtained through differentiation of 62 and 63 and substitution of 19.4 for 9. where Acknowledgment
Combining solution 6-4 bvith the transform of the rate Equation 8.3, shows that
Nomenclature
Dividing Equation 16A by l7A and isolating C,, -q' c2,
=
T h e author acknowledges the assistance of \Y. J. Schaffers in proof of compatibility of Equations 57 to 59 with 62 to 64 and for solution of Equation 45. He similarly thanks \Y. H. Abraham for his helpful suggestions on revision of the paper.
reactive end groups Z-transform of distribution j C,} fraction of end groups unreacted, 0 5 / 5 1 rate constant (defined a t location) molecule of chain length n kth moment of distribution kth derivative of P ( z ) with respect to In(t-1) kth moment of normalized distribution Cn1 moment-generating function of variable ( X ) number-average chain length weight-average chain length concentration of a monomer capable of reacting with active molecule I , concentration of I , Z-transform of set of concentrations P n )(defined a t location) flow rate parameter indicative of extent of reaction (defiled a t location) molar ratio of feed components (defined a t location) a n arbitrary transform variable volume of reactor inactive end groups mole fraction of chain stoppers in feed (defined at locations) arbitrary transform variable
(jZC,koCze,
(1 - q ) P , ( l . r ) + ~
+
T1 - qzC,koCze,lP,(l,r) (18.4)
By definition 1 C,, may not contain terms without z. Therefore, if the probabilistic solutions are correct, it must be that
Substitution of Equation 15A for P,( 1.r) and integration yield
[ I - R ] [ 1 - Rv']
1 - q =
[1 -
R[1 or Let
=
2
[l -
RP]*
PI'
(20AA)
Rv]*
= fC,ko
'
c,,,
P(1.r)= P Differentiate the probabilistic solution (Equation 64) with respect to T : -y? c,, =TI-> + q ([Il -- 9)[2]* 921' -~
GREEKSYMBOLS a = P*/P = Q kV = 1 / r P T
8
Substitution of Equation 19A into 21 A yields
$(x) X P
Expression 22A must be equivalent to the transforms of a normalized set of rate equations 10A: if the probabilistic solutions are to be correct. Comparison of the two expressions reduces this requirement to proof of the equality
By use of time functions 13A. 14A. l 5 A . and 20AA we obtain
~ _ -_ 1 _[ l _ Rv]' _ ~- 1_. _-1 [P,]' 4 R[1 - v]' 4 9
PbPe
so that the equivalence requirement (Equation 23A) reduces to
= k6 = reduced time (defined a t location) = time
a singlevalued normalized function of x an arbitrary variable = Po*in main text, exp[ -AT] in Appendix = =
literature Cited E N G . CHEM.FUNDAMENTALS 2, 221 (1963). (2) Bharucha-Reid, A. T., "Elements of the Theory of Markov Processes and Their Applications." McGraw-Hill. New York. 1960. (3) Case. L. C., J . Polymer Scz. 29, 455-95 (1958). (4) Churchill, R. V.. "Complex Variables and Applications," McGraw-Hill. New York, 1960. (5) Cramer, H., "Mathematical AMethodsof Statistics," Princeton University Press, Princeton, N. J.. 1915. (6) Florv. P. J., "Principles of Polymer Chemistry," Cornell University Press. Ithaca, N. Y., 1953. (7) Ragazzini, .J. R.. Franklin. G. F., "Sampled-Data Control Systems," McGraw-Hill. New York, 1958. (8) Saito. J.. J.Phys. Soc. Jnpun 13, 198 (1958). (9) Stockmaier, LV. H.. J . Chrm. Phys. 12, 325-31 (1944). (10) Zernan, R., Amundsen. N. R.. A.Z.Ch.E. J.9, S o . 3, 297-302 (1963).
(1) .4hraham, bv. H.. IND.
RECEIVED for review November 13. 1963 ACCEPTLO August 3. 1964 VOL.
3
NO. 4
NOVEMBER
1964
293
