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Computer modeling of infinite reaction sequences: A chemical lumping
Chemical Engineering Science. Vol. 40. No. Printed in Great Britain.
IO. pp. 1843-1849.
1985.
OOOS-2509/85 53.00+0.00 Pergamon Press Ltd.
COMPUTER MODELING OF INFINITE REACTION SEQUENCES: A CHEMICAL LUMPING MICHAEL
FRENKLACH
Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, (Received
6 Augusr
U.S.A.
1984)
Abstract-A lumping method is presented for modeling a system described by an infinite number of consecutive sequences composed of generally reversible reactions wirh an irreversible step connecting the sequences. The lumping technique is based on the similarity in molecular structure and reactivity and is not restricted to a specific reaction order or type, representing the reaction mechanism exactly. The lumping also satisfies the Wei and Kuo condition for exactness. The method is illustrated by an example on detailed modeling of soot formation in hydrocarbon pyrolysis. Computer implementation of the technique is’ simple.
INTRODUCTION
There is a variety of natural phenomena and industrial processes whose mechanisms are composed of a large number of consecutive reaction steps. Their number can be SO large that the reaction sequence should, for practical purposes, be treated as an infinite one. Examples of such systems can be drawn, for instance, from the fields of polymerization (Holland and Anthony, 1979), polymer degradation (Ederer et al., 1981), biochemistry (Biebricher et al., 1983) and combustion (Frenklach et al., 1985). The dynamics of infinite polymerization reaction systems has attracted attention in 196Os, when it was assessed by primarily analytical mathematical methods (Gold, 1958; Kyner et al., 1959; Goodrich, 1961; Murdoch, 1961; Liu and Amundson, 1962; Abraham, 1963; Zeman and Amundson, 1963; Holland and Anthony, 1979). In these treatments, the polymerization process is described by a relatively simple, lumped reaction mechanism, assuming each step irreversible. Similar approximations have been also used in more recent works (e.g. Jaisinghani and Ray, 1977; Ray, 1981; Schmidt and Ray, 1981; Hamer et al., 1981; Villermaux and Blavier, 1984; Blavier and Villermaux, 1984). Although this approach may be sufficient for certain practical applications, it is not suitable when a more detailed description of the process is necessary, as, for instance, the case in combustion (Frenklach et al., 1985). The infinite nature of the reaction sequence obviously requires lumping; only in very simple (and physically unrealistic) cases, like consecutive monomolecular irreversible reactions, can the exact solution for time varying concentrations, resolved for all the species, be obtained. The subject of lumping has been addressed by a number of researchers (Prater et al., 1967; Wei and Kuo, 1969, Kuo and Wei, 1969; Hutchinson and Luss, 1970; Luss and Hutchinson, 1971; Bailey, 1972; Liu and Lapidus, 1973). Wei and Kuo (1969) presented a general lumping analysis for a system of a finite number of monomolecular reactions and derived the conditions for the exact lumping of this system, meaning that the dynamics of the lumped ,:RS40: 1”-L
classes can be exactly described by first-order kinetics. Bailey (1972) extended their analysis to continuous mixtures. Liu and Lapidus (1973) generalized the analyses of Wei and Kuo (1969) and Bailey (1972) to include time-dependent rate coefficients in both discrete and continuous mixtures. Luss and Hutchinson developed lumping methods for the cases of many parallel irreversible reactions of first (Hutchinson and Luss, 1970) and nth (Luss and Hutchinson, 1971) order. The basic presumptions in the above works on lumping were the following: (1) it is practically impossible to know all the details of the reaction mechanism and associated rate coefficients and (2) it is practically impossible or at least inconvenient and may be unnecessary to resolve the dynamics for all the reaction species. Although these conditions remain true for most processes of interest, there appear to be reaction systems for which not all of the presumptions hold. Formation of soot in hydrocarbon combustion presents an example of such a situation. The main growth of soot mass is a polymerization process, for which the initiation reactions and the reactions involved in the individual polymerization step must be modeled in detail: a great number of reactive intermediates must be considered and discriminated, the principle of detailed balancing must be obeyed for all the reactions, and no steady-state approximation can be assumed (Frenklach er al., 1985). Of course, the state-of-the-art in the modeling of gas-phase kinetics allows one to describe the mechanism in terms of elementary reactions and to reliably estimate all the required rate coefficients and equilibrium constants (Cardiner, 1984). Still, in order to complete the modeling of soot formation to the point that its results can be compared with experiment, the number of polymerization steps must be very large, practically infinite. A lumping, therefore, is necessary. A lumping method, developed to account for the total amount of polymer (soot) formed, was suggested recently (Frenklach and Gardiner, 1984). The method is not restricted to a specific order or a type ofreactions
1843
1844
MICHAEL
involved, representing the reaction mechanism exactly. The basis for lumping is similarity in molecular structure and reactivity rather than mechanistic or mathematical considerations. It will be shown in this paper that the lumping satisfies the Wei and Kuo’s condition (Wei and Kuo, 1969) for exactness. The approach is applicable not only for computation of a total polymer mass, as was discussed in the previous paper (Frenklach and Gardiner, 1984), but, more generally, it can be used to evaluate the time-dependent polymer distribution. This constitutes the subject of this paper. STATEMENT
OF THE
PROBLEM
system, which is the subject of the following analysis, is schematically represented in Fig. 1. All chemical reactions of the system are grouped into initiation and polymerization blocks, the polymerization blocks being repetitive sets of generally reversible reactions. There are no restrictions on the reactions within these blocks: they can be reversible or irreversible, consecutive or parallel, of any reaction order, with temperature-dependent or, more generally, timedependent rate coefficients, etc. Thermal and volumetric effects can be also included. There are two principal requirements: The
(11 The reaction steps connecting the initiation and
(2)
polymerization blocks and the polymerization blocks themselves must be irreversible; that is, the change in free energy for these steps must be so large that the rate of the reverse direction can be practically neglected compared to that of the forward direction. (There can be a number of the connecting steps between two blocks; all of them, however, must be irreversible). The rate coefficients (and equilibrium constants) for similar chemical reactions in polymerization blocks must be identical; that is, the rate coefficients can depend on various variables, like temperature, time, monomer concentration, etc., but not on the sequential polymerization block number.
These two requirements actually guide the grouping of the reactions into the blocks rather than limit the generality of the method. Indeed, for a system of consecutive reversible reactions to proceed with a significant conversion, there must be a step with large reaction affinity, i.e. an essentially irreversible reaction
FRENKLACH
(Boudart, 1983; Frenklach and Gardiner, 1984; Frenklach et al., 1985) and, therefore, the first requirement must necessarily be satisfied. The second requirement is easily satisfied if, for instance, the mechanism is constructed from elementary reactions (Frenklach and Gardiner, 1984; Frenklach et al., 1985). It should be noted that the reactions are grouped into the polymerization blocks based on the similarity in chemical properties and, therefore, a single polymerization step may include addition of several monomer units (see Frenklach and Gardiner, 1984; Frenklach et al., 1985). The formal mathematical description of the introduced system is given in Frenklach and Gardiner (1984). However, in order not to bury the simplicity of the method in the awkwardness of the notations and indices, the mathematical development here is given for a system
which represents the main features discussed above and yet has sufficiently simple rotation. Species Al, B1, C, and D, constitute polymerization block 1; species A,, B,, C2 and D, polymerization block 2; etc. Species A,, AZ,. . are similar in that the rate coefficients of their similar reactions (e.g. A, --* B,, A, + B,, etc.) are identical. The same is true for B,, B,, _ _ . , Cl, C2, . _ _ and DlrD2.. . . . The reactions are assumed to be of the first order; however, the rate coefficient can depend on time, monomer concentration, etc. It is assumed here that reactions B + C and D + A are the monomer addition steps. It is also assumed that m, is the number of monomer units in A,. r,, is the rate of irreversible formation of Al from the initiation block. The objective is to determine the weight distribution of polymer molecules Pi, or equivalently (Holland, 1979), the distribution of the number of monomer units j. A distribution is completely characterized by specifying the moments of the distribution function (Hudson, 1963). Thus, the stated objective is to determine the moments of J, where J is a discrete random variable of the number of monomer units and can have values ofj = 1,2,3, . . . The rth moment of .I is defined as Cj’pj P:=E{J’} =‘cp,_ (2) j Usually, one is interested only in the first few moments (the first, second and third), because they are sufficient to define the basic features of the distribution function (Johnson and Leone, 1977): mean variance
Fig. 1. Schematic diagram of a reaction system composed of an infinite number of consecutive sequences of reversible reactions (polymerization blocks) with an irreversible step
.
..
connectingme sequences.
skewness
P = P;
(3)
&=&-(p;)2
(4)
y = cc; - 3P(; P; + 2(&Y cl’
(5)
Thus, the following analysis is extended to the third moment of J for system (1).
Computer modeling of infinitereactionsequences MATHEMATICAL
and
DEVELOPMENT
kinetics of system (1) is defined by the following system of differential equations: The
dAt ~ = r. - &,A, dt dB,
-
dt
dC, -
dt
dD,
-
dt
dAz ~ dt dBz dt
+ k,,Dr
(6Al)
= k.bA,
-kk,,B,
- k&B1 + k,&,
(6Bl)
= k&B,
-
- kedCl + k,D,
(6’3)
k,,C1
= kcdCl - kdcD1 - k,,D,
(6Dll
= k,,D,
(6A2)
d, =
5 i=
i’D, 1
for r = 0, 1, 2, eq. (8) can be rewritten as = morO+ k&b, -kcbcO + k,do.
$(Gpj) j
(10)
In order to determine b,, for example, we add eqs (6Bl), (6B2), etc. and thus obtain dbo ~ = k.bao - kb.bo - ktibo + k&o dt
(1lW
with the zero initial condition. Similarly, we obtain -
k,,A2
+ k,,B2
dco
= k,,A,
- k,,B,
- k,,B,
+ k,,C,
(6B2)
where k,, is the rate coefficient of reaction a + /3 and the initial values of all dependent variables are zero. Adding all eqs (6) together, we obtain -$A,+B~+c~+D~+A~+B~+
._.)=ro
or
$ (E
pj)
=
(7)
r0
~
= k,b,
-
ddo -
= k,c,
- k,d,
dt
..
dr
with the initial condition
da0 -= dt
+(mo+2)A2+
. . .] = more + LB1
(1 ld)
(1 la)
+ kcbcO -k&do
is added to account for the monomer mass consumed in polymerization reactions (6). Solving the combined set of differential equations, referred to as the firstorder model, simultaneously, c Pi and CjP, and, hence, according to (2), the &St momem ,u(; are determined at any given reaction time. To determine c j2P,, for the second moment, eqs (6) are multiplied b; the squares of the corresponding numbers of monomer units and added together, which results in
- kbCl + LPI
+ kid% - k&z
k&do
(llc)
all with zero initial conditions. Equations (7), (10) and (11) are combined with the differential equations describing the kinetics of the initiation reactions, to one of which, the equation for the monomer, a term - k,b,
+ moB,+(mo+l)C,+(mo+l)D,
-
r. - &.a0 + kb,bo + k&o,
C Pj , = o = 0.
Multiplying eqs (6) by the’corresponding number of monomer units, that is, multiplying eq. (6Al) by mo, (6Bl) by mo, (6Cl) by m, + 1, (6Dl) by m. + 1, (6A2) by m, + 2, (6B2) by m, + 2, etc. and adding them together, we obtain
kcbcO - kcdcO + kdcdo
and
i
&,A,
1845
+ kia&
&C&4,
or
+ m:B,
+ (m. + l)‘C,
+ (m. + l)*D,
. . .]=m~ro+[(mo+
+(mo+2)‘A2+
l)* -mi](k,B,
-kk,bCr)+[(mo+2)2-(~o+I)21kdrD1+[(mo+3)2
+ ha f,
Di
(8)
again with the zero initial condition. Introducing the notations ar =
b, =
2 i’Ai i=1 F
i’Bi
i=l
c,
=
fJ
.
i’C,
- (mo + 2)*1 (k&G - k&2)
- (me + 3)*lk,+,D2
. . . or $
(9)
+ [(mo + 4)*
(X.i” Pj)= mfro + (2mo + l)(k,B1 i + (2mo + 3)k,D, -
k&z)
-k&r)
+ (2mo + 5)(k,B,
+ (2mo + 7)k,,D,
MICHAEL
1844 =
mar0 + 2mo (k&b,
+ 2
k,,c,-,) + 2mo k,d,
-
(4i - 3)(k,Bi
- k,,,C,)
i=l
+ g
i=1
(4i-
l)kd&
or mgro + (2mo - 3)(Lbo
- Leo)
+ (2mo - l)k.&o + 4(&b,
- k,,c,)
+ 4k,,,d,.
(12)
The differential equation for b,, for example, is obtained by multiplying eqs (6Bl) by i and adding them together, that is
dbl
-
= k,,a, - k,,b,
dt
- k&b, + kcbcl
_
(13b)
Similarly, we obtain dcl
-
dt
ddl ~ dt
= k,b,
- k,,c,
- kcdc, + k&d,
(1W
= k,c,
-k&d,
- k,d,
VW
and da, ~ = r. - k,baI + kb,,bl + 4. dt = r. - k,*aI + khbl
2
(i + l)Di
i=l
+ k, ( i$,
iDi + igl
FRENKLACH
and Gardiner, 1984), will be further considered here. A recent study (Frenklach et al., 1985) suggested that soot formation in that system can be described as a large number of initiation reactions followed by a repetitive sequence of primarily reversible steps with an irreversible step connecting the sequences (c. Fig. 1). It is crucial for the description of the initiation reactions and the reactions within the repetitive sequences to be kinetically accurate; for instance, no reverse reaction can be neglected or steady-state assumed. The main polymerization route can be schematically presented as
where the first index of P is the polymerization block number and the second index defines different chemical species. The rate coefficients, which are dependent on concentrations of various species (like hydrogen atoms, C2H radicals, acetylene molecules), are defined in Frenklach and Gardiner (1984). r. is the rate of irreversible formation of PI, 1 from the initiation block of reactions. The entity of interest is the total number of carbon atoms, N, accumulated in polymer species Pi,. It is assumed that a molecule of PI, 1 contains 24 carbon atoms and two carbon atoms are added in each ofthereactionsP,,2 + Pi,J, Pie4 + Pi,SrPi.4 + Pi,9,Pi.6 + Pi+ 1.1. The objective is * pi.79 pi.8 + Pi.12 and pi.8 to determine the dynamics of the distribution of N. The moments of N, considering N a discrete random variable with values n = 24, 26, 28, . . , are defined similarly to (2) as 2
Di)
p; E E(N’}
or
da,
-=ro-kk,,aI+k,,b,+k,(d,+do).
dt
I
second moment ~5. Following similar procedures, one can easily derive the differential equations to solve for higher order moments.
(14)
i=l
pij
’
j
where nij is the number of carbon atoms in a molecule of Pi,_ Following the approach of the previous section, the differential 1,2,
equations
for
izl C nTipij, j
3, are
$(i$,FPU)=rO
d ( iz1 F nijpi,> =
24ro+2(R~,J+R~,s+R~,9 +R:.,+R:,,
EXAMPLE The example on soot formation during acetylene pyrolysis, introduced in the previous paper (Frenklach
j
C C
(13a)
Adding eqs (12) and (13), all with zero initial conditions, to the first-order model and simultaneously solving the combined set of the differential equations, which will be referred to as the second-order model, determines cj2Pi and, hence, according to (2), the
1 “TjPij
= ‘=;
$(,$lF~&fi,)
=
242rO+68R?.2+WR!i,5
+R:.12)
r
=
0,
1847
Computer modeling of infinitereactionsequences + R:, 9) + 84R:,
, + 92 (R;,
COMPUTER
I
+R~,12)+32(R:,j+R:,5+R:,9 +R&,+R:,,
243r, $($,I$IGfij)=
+R:,u)
+ 1736Rz.a +2168(R:,, (17) +3176(R:,,
+R:,g)+2648R:,,
1632R:,,+
+R:,&+
1824(R:,,
+R:,.)+2016R&,+2208(R&, +R:.12)+384(R~.s+R:.5+R:,9 +R:,t+Rii,~
+R:,~z)
where:
%* S5
= =
k,.S;
5
-
kj.jS;.
(15) (16)
i’pij
i=l
k 7.6 --k
and
IMPLEMENTATION
described method is particularly well-suited for computer implementation, if one notes that the differential equations describing the kinetics of reaction system The
I,8 = 0.
The differential equations for S;, j = 1, 2, . . _ , 12, r = 0, 1,2, are developed similarly to the procedures used for derivation of eqs (11) and (13). The equations forj # 1 can be obtained simply if PI+, are replaced by S; in differential equations describing the kinetics of the first polymerization block [e.g. c$ eqs (6Bl), (1 lb) and (13b)]. The equations for S; take form [cf: eqs (lla) and (13a)]:
dSy -= dr
r,-kk,,zS~+k,.,S~+k,,,Sl
dS: -= dt
r,--k,.,S:+k,,,S:+k,,,(S~+S~)
dS: -= dt
ro-kkl.?S:+kz,,S~+k,.,(S,2+2S~+s80).
are exactly eqs (11). Most computer codes for chemical kinetics input reactions in a natural chemical language, creating the reaction matrix and differential equations internally (CBme, 1983; Gardiner, 1984). Therefore, specifying only the first polymerization block, as detailed as required, with the last reaction in the sequence forming the initial species of the sequence [e.g. as in (17)], the eqs (11) are automatically created. Organized in this manner, the computer code exactly represents all the mass balances; that is, the exact consumption of monomer in polymerization reactions (and other species, like hydrogen atoms or C2H radicals in the previous example) are taking into account automatically. Including eqs (7) and (11) forms a first-order model. The high-order models are formed using the already created reaction matrix. DISCUSSION
OF THE
da -= dt
TECHNIQUE
-Ka
that for a
(18)
to be exactly lumpable, that is dP
Computational Fig. 2.
LUMPING
Wei and Kuo (1969) demonstrated monomolecular reaction system
Tic=
results for this example are given in
where
-Ran
P=Ma,
(19) (20)
the necessary and sufficient condition is MK = KM,
._
(21)
where: 48
32
Reactmn
time /
Y
ms
Fig. 2. Mean, standarddeviationand skewnessof the number of carbon atoms in soot polymer as a function of reaction time. The initialconditionsare those used in Frenklach and Gardiner (1984), i.e. temperature is 1600 K and initial concentration of acetylene is 4.0 x lo- ’ mol/cm3.
a is the vector of all the reaction species, a^is the vector of lumped species, K is the matrix of rate coefficients for the full system, & is the matrix of rate coefficients for the lumped system, M is the lumping matrix. The authors stated this condition for systems of a finite size with constant rate coefficients. Liu and Lapidus (1973) demonstrated that condition (21) can also accommodate time-dependent rate coefficients. It can be shown, for instance by induction, that condition (21) is also applicable to systems of an infinite size if the matrix M, that satisfies eqs (18)-(21), can be found. For system (l), for example, the con-
1848
MICHAELFRENKLACH
dition for exact lumping is satisfied if matrix M is assigned as follows
NOTATION a
P
A, B, C, D a, b, c, d E(
>
fr j K B k M
where 1 in both vectors a and a^is a dummy variable, introduced to accurately represent the kinetics of systems (1) and (17) in matrix form, i.e. in order for eqs (18) and (19) to reproduce eqs (6) and (1 l), respectively. It can be easily then verified that eq. (21) holds, that is
mo N
M
K 0
1
0
0
0
0 0 0 0
1
0
0
0 0 0
1 0 0
0 1 0
vector of reaction species, defined in (22) vector of lumped species, defined in (22) reaction species and their concentrations concentration moments, defined in (9) expectation of { } polymerization block number discrete random variable of the number of monomer units number of monomer units, the value of J matrix of reaction rate coefficients for the full system, a; defined in (23) matrix of reaction rate coefficients for the lumped system, a^;defined in (23) reaction rate coefficient lumping matrix, defined in (22) number of monomer units in A, discrete random variable of the number of carbon atoms
0
0
0
0
1
0
0 0 1
0 0 0
1 0 0
._.
0
0
0
r,,
-k,,
...
0
k,,
... ...
0 0
0 0
kbe 0
0
0
0
...
k -
kbc)
(Lb:
0
0 k - (k,$: k,,) k cd 0
0 0 k, -(kk+kdo) k del
0 0 0 0 0 -kk,,
... ...
... ::: _._
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _ . _
B 0
r0 0 0
0 -
kab 0k,, 0
M
0
k - (kbobk 0 k,
kbr)
k!; - (kcb + kc,,)
CONCLUSIONS
A lumping method is presented for modeling a system described by an infinite number of consecutive sequences composed of generally reversible reactions with an irreversible step connecting the sequences. The method is applied for determination of timedependent polymer distribution by computing the moments of the distribution function. The lumping technique is based on the similarity in molecular structure and reactivity and is not restricted to a specific order or a type of reactions involved, representing the reaction mechanism exactly. At the same time, the lumping satisfies the Wei and Kuo condition for exactness. The computer implementation of the method is simple. Acknowledgement-This work was supported by NASALewis Research Center, Grant No. NAG 3-477.
n
P R
r0
s
Subscripts a, b, c, d
number of carbon atom, the value of N polymer and its concentration defined in (15) rate of irreversible formation of A, or P dA&ed in (16) reaction time reaction species skewness of a random variable mean of a random variable rth moment about zero of a random variable variance of a random variable standard deviation of a random variable
reaction species polymerization block number
Computer modeling of infinite reaction sequences
j
number
of monomer
j,j’
reaction
species
r a, B
order reaction
Superscripts r
order
units
within
a polymerization
block species
REFERENCES
Abraham W. H., 1963, Path-dependent distribution of molecular weight in linear polymers. Ind. Engng Chem. Fundam. 2 221-224. Bailey J. E., 1972, Lumping analysis of reactions in continuous mixtures. Chem. Engng J. 3 52-61. Biebricher C. K., Eigen M. and Gardiner W. C. Jr., 1983, Kinetics of RNA replication. Biochemistry 22 25-2559. Blavier L. and Villermaux J., 1384, Free radical polymerization engineering-II. Modeling of homogeneous polymerization of styrene in a batch reactor, influence of initiator. Chem. Engng Sci. 39 101-l 10. Boudart M., 1983, Thermodynamic and kinetic coupling of chain and catalytic reactions. J. phys. Chem. 87 27862789. Came G. M., 1983, The use of computers in the analysis and simulation of complex reactions, in Modern Methods in Kinetics, Comprehensiue Chemical Kinetics (Edited by Bamford C. H. and Tipper C. F. H.), Vol. 24, Chapter 3. Elsevier, New York. Ederer H. J., Basedow A. M. and Ebert K. H., 1981, Modelling of polymer degradation reactions, in Modefling of Chemical Reaction Systems (Edited by Ebert K. H. er al.), pp. 189-2 15. Springer, Heidelberg. Frenklach M. and Gardiner W. C. Jr., 1984, Representation of multistage mechanisms in detailed computer modeling of polymerization kinetics. J. phys. Chem. 88 6263-6266. Frenklach M., Clary D. W., Gardiner W. C. Jr. and Stein S. E., 1985, Detailed kinetic modeling of soot formation in shock-tube pyrolysis of acetylene. Twentieth Symp. (Inc.) on Combustion, The Combustion Institute, Pittsburgh, in press. Gardiner W. C. Jr., Ed., 1984, Combustion Chemistry. Springer, New York. Gold L., 1958, Statistics of polymer molecular size distribution for an invariant number of propagating chains. J. them. Phys. 28 91-99. Goodrich F. C., 1961, Molecular weight distribution and reaction sequence in polymerization kinetics. J. them. Phys. 35 2101-2107. Hamer J. W., Akramov T. A. and Ray W. H., 1981, The dynamic behavior of continuous polymerization reactors-II. Nonisothermal solution homopolymerization and copolymerization in a CSTR. Chem. Engng Sci. 36 1897-1914.
1a49
Holland C. D. and Anthony R. G., 1979, Fundamentals of Chemical Reaction Engineering, Chapter 10. Prentice-Hall, Englewood Cliffs, New Jersey. Hudson D. J., 1963, Lectures on Elementary Statistics and Probability. CERN, Geneva. Hutchinson P. and Luss D., 1970, Lumping of mixtures with many parallel first order reactions. Chem. Engng J. 1 129-I 35. Jaisinghani R. and Ray W. H., 1977, On the dynamic behaviour of a class of homogeneous continuous stirred tank polymerization reactors. Chem. Engng Sci. 32 81 l-825. Johnson N. J. and Leone F. C., 1977, Sratisrics and Experimenral Design in Engineering and the Physical Sciences. Chapter 3. Wiley, New York. Kuo J. C. W. and Wei J., 1969, A lumping analysis in monomolecular reaction systems. Analysis of approximately lumpable system. Ind. Engng Chem. Fundam. 8 124-l 33. Kyner W. T., Radok J. R. M. and Wales M., 1959, Kinetics and molecular weight distributions for unsteady-state polymerizations involving termination by chain transfer with the monomer. J. them. Phys. 30 363-368. Liu S. and Amundson N. R., 1962, Calculation of molecular weight distributions in polymerization. Chem. Engng Sci. 17 7977802. Liu Y. A. and Lapidus L., 1973, Observer theory for lumping analysis of monomolecular reaction systems. A.1.Ch.E. J. 19 467-473. Luss D. and Hutchinson P., 1971, Lumping of mixtures with many parallel Nth order reactions. Chem. Engng J. 2 172-l 77. Murdoch P. G., 196 1, Finite-difference transforms for application to stage by stage process. A.1.Ch.E. J. 7 526-529. Prater C. D., Silvestri A. J. and Wei J., 1967, On the structure and analysis of complex systems of first-order chemical reactions containing irreversible steps--I. General properties. Chem. Engng Sci. 22 1587-1606. Ray W. H., 1981, Dynamic behaviour of polymerization reactors, in Modelling of Chemical Reaction Systems (Edited by Ebert K. H. et al.), pp. 337-354. Springer, Heidelberg. Schmidt A. D. and Ray W. H., 1981, The dynamic behavior of continuous polymerization reactors-I. Isothermal solution polymerization in a CSTR. Chem. Engng Sci. 36 1401-1410. Villermaux J. and Blavier L., 1984, Free radical polymerization engineering-I. A new method for modeling free radical homogeneous polymerization reactions. Chem. Engng Sci. 39 87-99. Wei J. and Kuo J. C. W., 1969, A lumping analysis in monomolecular reaction systems. Analysis of the exactly lumpable system. Ind. Engng Chem. Fundam. 8 114-123. Zeman R. and Amundson N. R., 1963, Continuous models for polymerization. A.1.Ch.E. J. 9 297-302.
IO. pp. 1843-1849.
1985.
OOOS-2509/85 53.00+0.00 Pergamon Press Ltd.
COMPUTER MODELING OF INFINITE REACTION SEQUENCES: A CHEMICAL LUMPING MICHAEL
FRENKLACH
Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, (Received
6 Augusr
U.S.A.
1984)
Abstract-A lumping method is presented for modeling a system described by an infinite number of consecutive sequences composed of generally reversible reactions wirh an irreversible step connecting the sequences. The lumping technique is based on the similarity in molecular structure and reactivity and is not restricted to a specific reaction order or type, representing the reaction mechanism exactly. The lumping also satisfies the Wei and Kuo condition for exactness. The method is illustrated by an example on detailed modeling of soot formation in hydrocarbon pyrolysis. Computer implementation of the technique is’ simple.
INTRODUCTION
There is a variety of natural phenomena and industrial processes whose mechanisms are composed of a large number of consecutive reaction steps. Their number can be SO large that the reaction sequence should, for practical purposes, be treated as an infinite one. Examples of such systems can be drawn, for instance, from the fields of polymerization (Holland and Anthony, 1979), polymer degradation (Ederer et al., 1981), biochemistry (Biebricher et al., 1983) and combustion (Frenklach et al., 1985). The dynamics of infinite polymerization reaction systems has attracted attention in 196Os, when it was assessed by primarily analytical mathematical methods (Gold, 1958; Kyner et al., 1959; Goodrich, 1961; Murdoch, 1961; Liu and Amundson, 1962; Abraham, 1963; Zeman and Amundson, 1963; Holland and Anthony, 1979). In these treatments, the polymerization process is described by a relatively simple, lumped reaction mechanism, assuming each step irreversible. Similar approximations have been also used in more recent works (e.g. Jaisinghani and Ray, 1977; Ray, 1981; Schmidt and Ray, 1981; Hamer et al., 1981; Villermaux and Blavier, 1984; Blavier and Villermaux, 1984). Although this approach may be sufficient for certain practical applications, it is not suitable when a more detailed description of the process is necessary, as, for instance, the case in combustion (Frenklach et al., 1985). The infinite nature of the reaction sequence obviously requires lumping; only in very simple (and physically unrealistic) cases, like consecutive monomolecular irreversible reactions, can the exact solution for time varying concentrations, resolved for all the species, be obtained. The subject of lumping has been addressed by a number of researchers (Prater et al., 1967; Wei and Kuo, 1969, Kuo and Wei, 1969; Hutchinson and Luss, 1970; Luss and Hutchinson, 1971; Bailey, 1972; Liu and Lapidus, 1973). Wei and Kuo (1969) presented a general lumping analysis for a system of a finite number of monomolecular reactions and derived the conditions for the exact lumping of this system, meaning that the dynamics of the lumped ,:RS40: 1”-L
classes can be exactly described by first-order kinetics. Bailey (1972) extended their analysis to continuous mixtures. Liu and Lapidus (1973) generalized the analyses of Wei and Kuo (1969) and Bailey (1972) to include time-dependent rate coefficients in both discrete and continuous mixtures. Luss and Hutchinson developed lumping methods for the cases of many parallel irreversible reactions of first (Hutchinson and Luss, 1970) and nth (Luss and Hutchinson, 1971) order. The basic presumptions in the above works on lumping were the following: (1) it is practically impossible to know all the details of the reaction mechanism and associated rate coefficients and (2) it is practically impossible or at least inconvenient and may be unnecessary to resolve the dynamics for all the reaction species. Although these conditions remain true for most processes of interest, there appear to be reaction systems for which not all of the presumptions hold. Formation of soot in hydrocarbon combustion presents an example of such a situation. The main growth of soot mass is a polymerization process, for which the initiation reactions and the reactions involved in the individual polymerization step must be modeled in detail: a great number of reactive intermediates must be considered and discriminated, the principle of detailed balancing must be obeyed for all the reactions, and no steady-state approximation can be assumed (Frenklach er al., 1985). Of course, the state-of-the-art in the modeling of gas-phase kinetics allows one to describe the mechanism in terms of elementary reactions and to reliably estimate all the required rate coefficients and equilibrium constants (Cardiner, 1984). Still, in order to complete the modeling of soot formation to the point that its results can be compared with experiment, the number of polymerization steps must be very large, practically infinite. A lumping, therefore, is necessary. A lumping method, developed to account for the total amount of polymer (soot) formed, was suggested recently (Frenklach and Gardiner, 1984). The method is not restricted to a specific order or a type ofreactions
1843
1844
MICHAEL
involved, representing the reaction mechanism exactly. The basis for lumping is similarity in molecular structure and reactivity rather than mechanistic or mathematical considerations. It will be shown in this paper that the lumping satisfies the Wei and Kuo’s condition (Wei and Kuo, 1969) for exactness. The approach is applicable not only for computation of a total polymer mass, as was discussed in the previous paper (Frenklach and Gardiner, 1984), but, more generally, it can be used to evaluate the time-dependent polymer distribution. This constitutes the subject of this paper. STATEMENT
OF THE
PROBLEM
system, which is the subject of the following analysis, is schematically represented in Fig. 1. All chemical reactions of the system are grouped into initiation and polymerization blocks, the polymerization blocks being repetitive sets of generally reversible reactions. There are no restrictions on the reactions within these blocks: they can be reversible or irreversible, consecutive or parallel, of any reaction order, with temperature-dependent or, more generally, timedependent rate coefficients, etc. Thermal and volumetric effects can be also included. There are two principal requirements: The
(11 The reaction steps connecting the initiation and
(2)
polymerization blocks and the polymerization blocks themselves must be irreversible; that is, the change in free energy for these steps must be so large that the rate of the reverse direction can be practically neglected compared to that of the forward direction. (There can be a number of the connecting steps between two blocks; all of them, however, must be irreversible). The rate coefficients (and equilibrium constants) for similar chemical reactions in polymerization blocks must be identical; that is, the rate coefficients can depend on various variables, like temperature, time, monomer concentration, etc., but not on the sequential polymerization block number.
These two requirements actually guide the grouping of the reactions into the blocks rather than limit the generality of the method. Indeed, for a system of consecutive reversible reactions to proceed with a significant conversion, there must be a step with large reaction affinity, i.e. an essentially irreversible reaction
FRENKLACH
(Boudart, 1983; Frenklach and Gardiner, 1984; Frenklach et al., 1985) and, therefore, the first requirement must necessarily be satisfied. The second requirement is easily satisfied if, for instance, the mechanism is constructed from elementary reactions (Frenklach and Gardiner, 1984; Frenklach et al., 1985). It should be noted that the reactions are grouped into the polymerization blocks based on the similarity in chemical properties and, therefore, a single polymerization step may include addition of several monomer units (see Frenklach and Gardiner, 1984; Frenklach et al., 1985). The formal mathematical description of the introduced system is given in Frenklach and Gardiner (1984). However, in order not to bury the simplicity of the method in the awkwardness of the notations and indices, the mathematical development here is given for a system
which represents the main features discussed above and yet has sufficiently simple rotation. Species Al, B1, C, and D, constitute polymerization block 1; species A,, B,, C2 and D, polymerization block 2; etc. Species A,, AZ,. . are similar in that the rate coefficients of their similar reactions (e.g. A, --* B,, A, + B,, etc.) are identical. The same is true for B,, B,, _ _ . , Cl, C2, . _ _ and DlrD2.. . . . The reactions are assumed to be of the first order; however, the rate coefficient can depend on time, monomer concentration, etc. It is assumed here that reactions B + C and D + A are the monomer addition steps. It is also assumed that m, is the number of monomer units in A,. r,, is the rate of irreversible formation of Al from the initiation block. The objective is to determine the weight distribution of polymer molecules Pi, or equivalently (Holland, 1979), the distribution of the number of monomer units j. A distribution is completely characterized by specifying the moments of the distribution function (Hudson, 1963). Thus, the stated objective is to determine the moments of J, where J is a discrete random variable of the number of monomer units and can have values ofj = 1,2,3, . . . The rth moment of .I is defined as Cj’pj P:=E{J’} =‘cp,_ (2) j Usually, one is interested only in the first few moments (the first, second and third), because they are sufficient to define the basic features of the distribution function (Johnson and Leone, 1977): mean variance
Fig. 1. Schematic diagram of a reaction system composed of an infinite number of consecutive sequences of reversible reactions (polymerization blocks) with an irreversible step
.
..
connectingme sequences.
skewness
P = P;
(3)
&=&-(p;)2
(4)
y = cc; - 3P(; P; + 2(&Y cl’
(5)
Thus, the following analysis is extended to the third moment of J for system (1).
Computer modeling of infinitereactionsequences MATHEMATICAL
and
DEVELOPMENT
kinetics of system (1) is defined by the following system of differential equations: The
dAt ~ = r. - &,A, dt dB,
-
dt
dC, -
dt
dD,
-
dt
dAz ~ dt dBz dt
+ k,,Dr
(6Al)
= k.bA,
-kk,,B,
- k&B1 + k,&,
(6Bl)
= k&B,
-
- kedCl + k,D,
(6’3)
k,,C1
= kcdCl - kdcD1 - k,,D,
(6Dll
= k,,D,
(6A2)
d, =
5 i=
i’D, 1
for r = 0, 1, 2, eq. (8) can be rewritten as = morO+ k&b, -kcbcO + k,do.
$(Gpj) j
(10)
In order to determine b,, for example, we add eqs (6Bl), (6B2), etc. and thus obtain dbo ~ = k.bao - kb.bo - ktibo + k&o dt
(1lW
with the zero initial condition. Similarly, we obtain -
k,,A2
+ k,,B2
dco
= k,,A,
- k,,B,
- k,,B,
+ k,,C,
(6B2)
where k,, is the rate coefficient of reaction a + /3 and the initial values of all dependent variables are zero. Adding all eqs (6) together, we obtain -$A,+B~+c~+D~+A~+B~+
._.)=ro
or
$ (E
pj)
=
(7)
r0
~
= k,b,
-
ddo -
= k,c,
- k,d,
dt
..
dr
with the initial condition
da0 -= dt
+(mo+2)A2+
. . .] = more + LB1
(1 ld)
(1 la)
+ kcbcO -k&do
is added to account for the monomer mass consumed in polymerization reactions (6). Solving the combined set of differential equations, referred to as the firstorder model, simultaneously, c Pi and CjP, and, hence, according to (2), the &St momem ,u(; are determined at any given reaction time. To determine c j2P,, for the second moment, eqs (6) are multiplied b; the squares of the corresponding numbers of monomer units and added together, which results in
- kbCl + LPI
+ kid% - k&z
k&do
(llc)
all with zero initial conditions. Equations (7), (10) and (11) are combined with the differential equations describing the kinetics of the initiation reactions, to one of which, the equation for the monomer, a term - k,b,
+ moB,+(mo+l)C,+(mo+l)D,
-
r. - &.a0 + kb,bo + k&o,
C Pj , = o = 0.
Multiplying eqs (6) by the’corresponding number of monomer units, that is, multiplying eq. (6Al) by mo, (6Bl) by mo, (6Cl) by m, + 1, (6Dl) by m. + 1, (6A2) by m, + 2, (6B2) by m, + 2, etc. and adding them together, we obtain
kcbcO - kcdcO + kdcdo
and
i
&,A,
1845
+ kia&
&C&4,
or
+ m:B,
+ (m. + l)‘C,
+ (m. + l)*D,
. . .]=m~ro+[(mo+
+(mo+2)‘A2+
l)* -mi](k,B,
-kk,bCr)+[(mo+2)2-(~o+I)21kdrD1+[(mo+3)2
+ ha f,
Di
(8)
again with the zero initial condition. Introducing the notations ar =
b, =
2 i’Ai i=1 F
i’Bi
i=l
c,
=
fJ
.
i’C,
- (mo + 2)*1 (k&G - k&2)
- (me + 3)*lk,+,D2
. . . or $
(9)
+ [(mo + 4)*
(X.i” Pj)= mfro + (2mo + l)(k,B1 i + (2mo + 3)k,D, -
k&z)
-k&r)
+ (2mo + 5)(k,B,
+ (2mo + 7)k,,D,
MICHAEL
1844 =
mar0 + 2mo (k&b,
+ 2
k,,c,-,) + 2mo k,d,
-
(4i - 3)(k,Bi
- k,,,C,)
i=l
+ g
i=1
(4i-
l)kd&
or mgro + (2mo - 3)(Lbo
- Leo)
+ (2mo - l)k.&o + 4(&b,
- k,,c,)
+ 4k,,,d,.
(12)
The differential equation for b,, for example, is obtained by multiplying eqs (6Bl) by i and adding them together, that is
dbl
-
= k,,a, - k,,b,
dt
- k&b, + kcbcl
_
(13b)
Similarly, we obtain dcl
-
dt
ddl ~ dt
= k,b,
- k,,c,
- kcdc, + k&d,
(1W
= k,c,
-k&d,
- k,d,
VW
and da, ~ = r. - k,baI + kb,,bl + 4. dt = r. - k,*aI + khbl
2
(i + l)Di
i=l
+ k, ( i$,
iDi + igl
FRENKLACH
and Gardiner, 1984), will be further considered here. A recent study (Frenklach et al., 1985) suggested that soot formation in that system can be described as a large number of initiation reactions followed by a repetitive sequence of primarily reversible steps with an irreversible step connecting the sequences (c. Fig. 1). It is crucial for the description of the initiation reactions and the reactions within the repetitive sequences to be kinetically accurate; for instance, no reverse reaction can be neglected or steady-state assumed. The main polymerization route can be schematically presented as
where the first index of P is the polymerization block number and the second index defines different chemical species. The rate coefficients, which are dependent on concentrations of various species (like hydrogen atoms, C2H radicals, acetylene molecules), are defined in Frenklach and Gardiner (1984). r. is the rate of irreversible formation of PI, 1 from the initiation block of reactions. The entity of interest is the total number of carbon atoms, N, accumulated in polymer species Pi,. It is assumed that a molecule of PI, 1 contains 24 carbon atoms and two carbon atoms are added in each ofthereactionsP,,2 + Pi,J, Pie4 + Pi,SrPi.4 + Pi,9,Pi.6 + Pi+ 1.1. The objective is * pi.79 pi.8 + Pi.12 and pi.8 to determine the dynamics of the distribution of N. The moments of N, considering N a discrete random variable with values n = 24, 26, 28, . . , are defined similarly to (2) as 2
Di)
p; E E(N’}
or
da,
-=ro-kk,,aI+k,,b,+k,(d,+do).
dt
I
second moment ~5. Following similar procedures, one can easily derive the differential equations to solve for higher order moments.
(14)
i=l
pij
’
j
where nij is the number of carbon atoms in a molecule of Pi,_ Following the approach of the previous section, the differential 1,2,
equations
for
izl C nTipij, j
3, are
$(i$,FPU)=rO
d ( iz1 F nijpi,> =
24ro+2(R~,J+R~,s+R~,9 +R:.,+R:,,
EXAMPLE The example on soot formation during acetylene pyrolysis, introduced in the previous paper (Frenklach
j
C C
(13a)
Adding eqs (12) and (13), all with zero initial conditions, to the first-order model and simultaneously solving the combined set of the differential equations, which will be referred to as the second-order model, determines cj2Pi and, hence, according to (2), the
1 “TjPij
= ‘=;
$(,$lF~&fi,)
=
242rO+68R?.2+WR!i,5
+R:.12)
r
=
0,
1847
Computer modeling of infinitereactionsequences + R:, 9) + 84R:,
, + 92 (R;,
COMPUTER
I
+R~,12)+32(R:,j+R:,5+R:,9 +R&,+R:,,
243r, $($,I$IGfij)=
+R:,u)
+ 1736Rz.a +2168(R:,, (17) +3176(R:,,
+R:,g)+2648R:,,
1632R:,,+
+R:,&+
1824(R:,,
+R:,.)+2016R&,+2208(R&, +R:.12)+384(R~.s+R:.5+R:,9 +R:,t+Rii,~
+R:,~z)
where:
%* S5
= =
k,.S;
5
-
kj.jS;.
(15) (16)
i’pij
i=l
k 7.6 --k
and
IMPLEMENTATION
described method is particularly well-suited for computer implementation, if one notes that the differential equations describing the kinetics of reaction system The
I,8 = 0.
The differential equations for S;, j = 1, 2, . . _ , 12, r = 0, 1,2, are developed similarly to the procedures used for derivation of eqs (11) and (13). The equations forj # 1 can be obtained simply if PI+, are replaced by S; in differential equations describing the kinetics of the first polymerization block [e.g. c$ eqs (6Bl), (1 lb) and (13b)]. The equations for S; take form [cf: eqs (lla) and (13a)]:
dSy -= dr
r,-kk,,zS~+k,.,S~+k,,,Sl
dS: -= dt
r,--k,.,S:+k,,,S:+k,,,(S~+S~)
dS: -= dt
ro-kkl.?S:+kz,,S~+k,.,(S,2+2S~+s80).
are exactly eqs (11). Most computer codes for chemical kinetics input reactions in a natural chemical language, creating the reaction matrix and differential equations internally (CBme, 1983; Gardiner, 1984). Therefore, specifying only the first polymerization block, as detailed as required, with the last reaction in the sequence forming the initial species of the sequence [e.g. as in (17)], the eqs (11) are automatically created. Organized in this manner, the computer code exactly represents all the mass balances; that is, the exact consumption of monomer in polymerization reactions (and other species, like hydrogen atoms or C2H radicals in the previous example) are taking into account automatically. Including eqs (7) and (11) forms a first-order model. The high-order models are formed using the already created reaction matrix. DISCUSSION
OF THE
da -= dt
TECHNIQUE
-Ka
that for a
(18)
to be exactly lumpable, that is dP
Computational Fig. 2.
LUMPING
Wei and Kuo (1969) demonstrated monomolecular reaction system
Tic=
results for this example are given in
where
-Ran
P=Ma,
(19) (20)
the necessary and sufficient condition is MK = KM,
._
(21)
where: 48
32
Reactmn
time /
Y
ms
Fig. 2. Mean, standarddeviationand skewnessof the number of carbon atoms in soot polymer as a function of reaction time. The initialconditionsare those used in Frenklach and Gardiner (1984), i.e. temperature is 1600 K and initial concentration of acetylene is 4.0 x lo- ’ mol/cm3.
a is the vector of all the reaction species, a^is the vector of lumped species, K is the matrix of rate coefficients for the full system, & is the matrix of rate coefficients for the lumped system, M is the lumping matrix. The authors stated this condition for systems of a finite size with constant rate coefficients. Liu and Lapidus (1973) demonstrated that condition (21) can also accommodate time-dependent rate coefficients. It can be shown, for instance by induction, that condition (21) is also applicable to systems of an infinite size if the matrix M, that satisfies eqs (18)-(21), can be found. For system (l), for example, the con-
1848
MICHAELFRENKLACH
dition for exact lumping is satisfied if matrix M is assigned as follows
NOTATION a
P
A, B, C, D a, b, c, d E(
>
fr j K B k M
where 1 in both vectors a and a^is a dummy variable, introduced to accurately represent the kinetics of systems (1) and (17) in matrix form, i.e. in order for eqs (18) and (19) to reproduce eqs (6) and (1 l), respectively. It can be easily then verified that eq. (21) holds, that is
mo N
M
K 0
1
0
0
0
0 0 0 0
1
0
0
0 0 0
1 0 0
0 1 0
vector of reaction species, defined in (22) vector of lumped species, defined in (22) reaction species and their concentrations concentration moments, defined in (9) expectation of { } polymerization block number discrete random variable of the number of monomer units number of monomer units, the value of J matrix of reaction rate coefficients for the full system, a; defined in (23) matrix of reaction rate coefficients for the lumped system, a^;defined in (23) reaction rate coefficient lumping matrix, defined in (22) number of monomer units in A, discrete random variable of the number of carbon atoms
0
0
0
0
1
0
0 0 1
0 0 0
1 0 0
._.
0
0
0
r,,
-k,,
...
0
k,,
... ...
0 0
0 0
kbe 0
0
0
0
...
k -
kbc)
(Lb:
0
0 k - (k,$: k,,) k cd 0
0 0 k, -(kk+kdo) k del
0 0 0 0 0 -kk,,
... ...
... ::: _._
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _ . _
B 0
r0 0 0
0 -
kab 0k,, 0
M
0
k - (kbobk 0 k,
kbr)
k!; - (kcb + kc,,)
CONCLUSIONS
A lumping method is presented for modeling a system described by an infinite number of consecutive sequences composed of generally reversible reactions with an irreversible step connecting the sequences. The method is applied for determination of timedependent polymer distribution by computing the moments of the distribution function. The lumping technique is based on the similarity in molecular structure and reactivity and is not restricted to a specific order or a type of reactions involved, representing the reaction mechanism exactly. At the same time, the lumping satisfies the Wei and Kuo condition for exactness. The computer implementation of the method is simple. Acknowledgement-This work was supported by NASALewis Research Center, Grant No. NAG 3-477.
n
P R
r0
s
Subscripts a, b, c, d
number of carbon atom, the value of N polymer and its concentration defined in (15) rate of irreversible formation of A, or P dA&ed in (16) reaction time reaction species skewness of a random variable mean of a random variable rth moment about zero of a random variable variance of a random variable standard deviation of a random variable
reaction species polymerization block number
Computer modeling of infinite reaction sequences
j
number
of monomer
j,j’
reaction
species
r a, B
order reaction
Superscripts r
order
units
within
a polymerization
block species
REFERENCES
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Holland C. D. and Anthony R. G., 1979, Fundamentals of Chemical Reaction Engineering, Chapter 10. Prentice-Hall, Englewood Cliffs, New Jersey. Hudson D. J., 1963, Lectures on Elementary Statistics and Probability. CERN, Geneva. Hutchinson P. and Luss D., 1970, Lumping of mixtures with many parallel first order reactions. Chem. Engng J. 1 129-I 35. Jaisinghani R. and Ray W. H., 1977, On the dynamic behaviour of a class of homogeneous continuous stirred tank polymerization reactors. Chem. Engng Sci. 32 81 l-825. Johnson N. J. and Leone F. C., 1977, Sratisrics and Experimenral Design in Engineering and the Physical Sciences. Chapter 3. Wiley, New York. Kuo J. C. W. and Wei J., 1969, A lumping analysis in monomolecular reaction systems. Analysis of approximately lumpable system. Ind. Engng Chem. Fundam. 8 124-l 33. Kyner W. T., Radok J. R. M. and Wales M., 1959, Kinetics and molecular weight distributions for unsteady-state polymerizations involving termination by chain transfer with the monomer. J. them. Phys. 30 363-368. Liu S. and Amundson N. R., 1962, Calculation of molecular weight distributions in polymerization. Chem. Engng Sci. 17 7977802. Liu Y. A. and Lapidus L., 1973, Observer theory for lumping analysis of monomolecular reaction systems. A.1.Ch.E. J. 19 467-473. Luss D. and Hutchinson P., 1971, Lumping of mixtures with many parallel Nth order reactions. Chem. Engng J. 2 172-l 77. Murdoch P. G., 196 1, Finite-difference transforms for application to stage by stage process. A.1.Ch.E. J. 7 526-529. Prater C. D., Silvestri A. J. and Wei J., 1967, On the structure and analysis of complex systems of first-order chemical reactions containing irreversible steps--I. General properties. Chem. Engng Sci. 22 1587-1606. Ray W. H., 1981, Dynamic behaviour of polymerization reactors, in Modelling of Chemical Reaction Systems (Edited by Ebert K. H. et al.), pp. 337-354. Springer, Heidelberg. Schmidt A. D. and Ray W. H., 1981, The dynamic behavior of continuous polymerization reactors-I. Isothermal solution polymerization in a CSTR. Chem. Engng Sci. 36 1401-1410. Villermaux J. and Blavier L., 1984, Free radical polymerization engineering-I. A new method for modeling free radical homogeneous polymerization reactions. Chem. Engng Sci. 39 87-99. Wei J. and Kuo J. C. W., 1969, A lumping analysis in monomolecular reaction systems. Analysis of the exactly lumpable system. Ind. Engng Chem. Fundam. 8 114-123. Zeman R. and Amundson N. R., 1963, Continuous models for polymerization. A.1.Ch.E. J. 9 297-302.