Chemical Engineering Science, 1965, Vol. 20, pp. 15-23. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Polycondensation in a continuous, stirred-tank reactor N.
H. SMITH* and G. A. SATHER
Chemical Engineering Department, University of Wisconsin, Madison, Wisconsin (Received 24 September 1963; in revised form 8 June 1964) Abstract-The purpose of this paper is to analyse transient and steady-state polycondensation in a continuous, stirred-tank reactor as an example of large, complex kinetic systems. The approach used is to effect a partial analytical solution of the exact rate equations supplemented by a numerical solution by digital computer. By dimensional analysis a single dimensionless group is obtained, which is related to the polymer yields and molecular weight distribution for both transient and steady-state operation of an isothermal reactor with constant feed composition.
INTRODUCTION BECAUSEof more efficient computing methods, there has been growing interest in analysing industrially realistic kinetic systems, i.e. systems with a large number of species reacting according to complex kinetic rate laws. Examples of such work may be found in the comprehensive study of free radical polymerization by LIU and AMUNDSON[l, 21. In a recent study of the transient behaviour of continuous, stirred-tank reactors by digital computer, condensation polymerization was suggested as a prime example of a complex kinetic system . The purpose of this paper is to present the analysis of transient and steady-state polycondensation in a continuous stirred-tank reactor by analytical and numerical methods. In a practical sense this study is to provide design information on steady-state conversion as well as on the economically sign& cant periods of start-up and shut-down for a reaction mechanism commonly encountered in the chemical industry. More generally the approach may be applied to problems requiring the solution of large numbers of simultaneous, non-linear differential equations. The reaction mechanism of polycondensation considered in this paper applies to an equimolar mixture of two bifunctional reactants, e.g. ethylene glycol (BB) and succinic acid (AA), or to a single bifunctional compound with the ability to condense with itself, e.g. o-hydroxydecanoic acid
(AB). For the latter type of compound the reaction scheme is AB + AB = ABAB ABAB + AB = ABABAB
. . &I,
In these equations the small molecule normally split out in each reaction has been omitted. There is substantial evidence to indicate that each reaction step follows second-order kinetics . The present analysis does not take into account the reverse reactions and is restricted to isothermal operation. The approach adopted is to (i) start with the rigorous rate expressions for each polymeric species, (ii) apply dimensional analysis to determinate important variables, (iii) find exact solutions for concentration as a function of time for a single stirred-tank reactor insofar as this is feasible and (iv) complete the solution by means of approximate numerical methods used in conjunction with a digital computer. Although ZEMAN and AMUNDSON have developed an analytical method based on generating functions to solve for polymer concentration for certain cases of free radical polymerization, such an approach has not been found useful in the case of poly-condensation . Batch polycondensation has been treated analytically from the
* Present address: Esso Research and Engineering Co., Florham Park, N.J.
N. H. SMITHand G. A. SATHER
standpoint of statistical mechanics  and kinetic theory .
Although the rigorous formulation of the rate equations is given in terms of individual polymeric species, in most cases it is impossible to determine concentrations by quantitative chemical techniques. It is possible, however, to measure certain average quantities derived from the individual concentrations. These quantities, which may be correlated with important physical properties of the polymer, are given in terms of C,, the molar concentration of x-mer, where x is the number of primary structural units in the polymer molecule (also called the degree of polymerization, D.P.), or alternatively,
T = V/Q
e = t/T
where T is the hold-time and 0 is the reduced time (number of hold-times elapsed). Equation (7) may now be written
m = 1,2, . . . 00 (11)
The steady-state mass balance may be obtained from equation (11) by deletion of the derivative term
The mole-fraction of x-mer is given as
where Ci is the (constant) inlet concentration of species m. It is convenient to introduce the following definitions :
The weight fraction of x-mer (molecular weight distribution) w, is defined as follows:
- AW, = - QAC,,, = Q(C; - C,,,)
molecular weight of polymer molecule molecular weight of monomeric unit
-AW, + Vr,,,
where C,,, is the molar concentration of species m within the reactor at time t, AW is the net molar outflow of species m and r,,,is the net rate of formation of species m by chemical reaction per unit volume. By assuming perfect mixing which implies that the outlet concentration is equal to the internal concentration, the following may be written:
m = 1,2, . . . 00
In order to complete the description of the reaction system the kinetic term r,,,must be specified for the polycondensation mechanism given by equation (1). The rate expressions for each species, derived by DOSTAL and RAFF, and discussed by TANFORD,are given in terms of molar concentrations by the following algorithm [7, 81:
The distribution of molecular weights is charao terized by the first and second moments as follows :
r,=-kc, These two quantities are called the number and weight-average degrees of polymerization, respectively.
k/2 c C,C,_, - kc,,, xwCi s=l i=l m = 2, 3, . . . co (13)
In these equations the Principle of Equal Reactivity is used, which states that the rate constant k is approximately independent of chain length . It is also necessary to state the initial conditions in the case of unsteady-state operation. For example,
An unsteady-state mass balance about a continuous, stirred-tank reactor of volume V with constant volumetric flow rate Q may be written for each reactive species m as follows: 16
Polycondensation in a continuous, stirred-tank reactor
in the case of start-up, it is reasonable to have the initial condition that there is pure solvent in the reactor : m = 1,2, . . . co
C,(O) = 0
For the case in which the reactor initially contains polymer solution at the inlet concentration, initial conditions would be m = 1,2, . . . co
C,(O) = c;
Cy = constant > 0 m = 2,3, . . . co
By combining equation (13) with equation (11) subject to equation (16), the general expression for transient operation is given as follows:
dcl_cO_c x-1 dCm x=
- kTC, C,C,_,
. . . 00
The steady-state equations may be obtained setting the derivatives equal to zero.
The initial conditions become A*(O)= constant
The basic dektitions given by equations (3) and (4) may be rewritten as follows:
Dimensional analysis therefore results in the isolation of the parameters of the system into a single group, i.e. k, = J,,#’ ; N ; A,dW (27) The parameter N may be considered to be a measure of the effect of chemical reaction on the concentration profiles (“kinetic effect number”). When N is large, the reaction terms tend to predominate over the flow terms; when N is small, the flow terms become important. The limiting cases are batch operation when N + co (actually T -+ COas Q --, 0) and non-reactive blending or purging when N + 0. PARTIALANALYTICAL SOLUTION A. Transient operation
1, is defined as
A, = c&y
N, = LJa
By definition ~=~
A dimensionless concentration follows :
N = kTC:
In actual practice the feed solution to a polymerization reactor usually contains only monomer (and catalyst); hence, the following simplification may be made :
c; = 0
where N is a dimensionless group defined as follows :
In their present form equations (21) and (22) are non-linear and, hence, appear insoluble analytically. However, if an explicit expression for a(0) were obtained, equation (21) would be linear in rll and could be solved for n,(e). Since no term in the quadratic sum
Substitution of equations (19) and (20) into (17) and (18) followed by algebraic simplikation yields the following dimensionless equations :
in equation (22) can contain lj where j 2 m, the mth equation will always be linear in rZ, provided explicit expressions for A# < m) are available. In principle it is, therefore, possible to solve for each subsequent 1, by successively substituting the result for A,_, into equation (22). First, an expression for u(0) is required. When equation (22) is summed on m from 2 to co, the sum is added to equation (21), and equation (20) 17
N. H. SMITHand G. A. SATHER
is applied, the result is f
For monomer initially present [L,(O) = l-01, equation (36) may be solved to yield
= 1 - ci + $N 2 mil &$,,-, - Ncl’ m=2
Denoting the double sum by (DS), it may be expanded as follows: (DS) = g j=l =tl
A; + j=l
( > 2
W) = l-[2bN’fJ--62+1--N’(1-b)2]exp(--N’B)-_b2exp(-22N’8) N’[l - b exp (- we)12 (38)
The dimer equation (equation 22 with m = 2) is
(2% as follows:
Substitution of equation (29) into (28) gives du z=1-u-fNu2
For the initial condition of pure solvent [or(O)= 1 - exp( - NV) 1 - d exp(-NV) I
where N’ = J(1 + 2N)
and d = (1 - N’)/(l + N’)
n(e) = m(e) - n,(e)
For the case of monomer initially present at the inlet concentration [a(O) = l-01 1 - (b/d)exp( - N’(Y) 1 - b exp( - N’0) I
As mentioned previously, this equation is linear in I, because L,(e) and c#J) are known explicitly. The solution to this and equations of higher Y.His impractical because of the large number of mathematical manipulations required. At this point an approximate numerical method becomes desirable. It should be indicated that polycondensation reactions are often followed by measuring monomer concentration (m = 1) or by gravimetrically analysing the products of-condensation (m > l), given by rc(6) where (40)
Although analytical solutions for the higher species become formidable, it is possible to derive a simple expression for the number-average D.P., 5?,,(e). By equations (5), (19) and (20)
where b=l-N’+N l+N’+N
Equation (31) may be substituted into the monomer equation (equation 21) to yield the linear equation
1 - exp( - NV) 1 - d exp( - NV) (36)
which may readily be solved for the initial condition of pure solvent to give the following result: ue
The summation, which is the number of monomeric units in the reactor at time 6, is independent of chemical reaction and, in fact, equals L,(6) if no chemical reaction were to occur. The result for the initial condition of pure solvent is given by
1 - exp(-8)
where [email protected]
) is given by equation (3 1). For monomer initially present the result is
zn(e) = -L
1 - [2d~v’tl- d2 + l]exp( -NV) - d2 exp(-2N’O) N’[l - d exp( - N’8)]2 (37)
a(6) where [email protected]
) is given by equation (34). 18
Polycondensation in a continuous, stirred-tank reactor NUMERICALSOLUTION
B. Steady-state operation The steady-state dimensionless concentration of total species a and monomer A1are obtained directly from equations (31), (34), (37) and (38) for 6 = co :
To find higher I,(co) equation (22) with the derivative set to zero is rearranged as follows: &,,=*N
m = 2, 3, . ..cc
(1 + Na)
Equation (46) may be restated by successive elimination of A1starting with equation (45) as follows: Nm-1 I
The coefficient in equation (48) is given explicitly by g = _(-1).1.3.5...(2;-3) m m!
m = 1,2, . . . co
(50) The steady-state molecular weight averages are given by (51) ~&o-)-N++
A fourth-order RtmgeKutta iterative procedure developed by GILL for digital computation was used to calculate the concentration profiles A, by equations (21) and (22) (a CDC 1604 computer was used) [9, lo]. The method proved rapid, efficient and accurate for solving large numbers of simultaneous, non-linear differential equations, and is sufficiently general to handle variable temperature and feed composition. By inspection of equations (20), (21) and (22), it is apparent that the number of equations (and terms in the sum a) are unlimited. Thus, some truncation method is necessary in order to use a numerical method. One possibility is to generate a large number of equations at the start of the computation, e.g. 500, and solve this number at each increment in 0. This method was employed in the treatment of free radical polymerization in batch reactors [l]. Because of the physical nature of polymerization one would expect to find in a start-up situation almost all monomer, some dimer, less trimer, etc., and virtually no higher species early in the run. Only as time progresses would the number of higher species become significant. A solution using a fixed number of equations would tend to be wasteful of computations early in the run and would also run the risk of introducing serious truncation error if the upper bound on the number of equations were approached. The technique employed here is to introduce a variable truncation limit M which is low at the start of a run but which is increased if M2&, a quantity to which ZWis sensitive, exceeds q, a specified tolerance limit. Thus, the number of equations to be solved is automatically adjusted to M as the number and numerical importance of species increases.
The former result is obtainable from either equation (42) or (43) at 8 = co. The latter rather interesting result can be obtained by direct evaluation of the sums in the definition of f, using equations (47) and (50):
&SULTS AND D~scuss~o~i
In order to analyse the effect of N on the product distributions Q) and w,(e), and the average quantities Z&3) and &,@), six start-up runs (pure solvent initially present) were simulated by computer at N-levels of 0.1, 1, 3, 10, 33 and_lOO. In Fig. 1 are shown the curves of dimensionless concentration vs. reduced time AX(e)for species with x values
A convergent power series results which reduces to the simple expression given by equation (52). 19
N. H. SMITHand G. A. SATHER
of about 100-1000 min. At this level of N about 78 per cent of the monomer is converted at steadystate; the yields of polymer are quite low and rapidly diminish with X. For example, at 8 = 10.0, AZ0has reached only OGOO56and L,, (not shown) has become equal to 0.00007. The iterative nature of the polymerization process is shown in Fig. 1. The monomer (x = 1) rapidly reaches a maximum and begins to decline in response to the consecutive formation of higher species. Each species shows a similar maximum although, as x increases, the peaks are flatter. There is also a progressive lag in the appearance of each species because of the second-order nature of the rate expressions. It is clear that continuous, stirred-tank operation is not suited to producing polymer with a large proportion of higher molecular weight species. In Fig. 2 are shown the weight distributions of species with x equal to 1, 2, 5, 10, 20 and 50 as a function of reduced time at a value of N equal to 100. The weight distributions reach a maximum which becomes less sharp with increasing X. In general, there is a low proportion by weight of each subsequent species indicating the broad and unfavourable molecular weight distributions attainable even at an unrealistically high value of N. To characterize the entire set of curves of w,(e), it is possible to construct an envelope curve which
8 FIG. 1. Dimensionless concentration vs. reduced time at N = 10. Numbers on curves are x-values.
of 1 (monomer), 2, 5, and 10 and for a value of N = 10. Of the values of N used, this is the most physically reasonable since Cy will ordinarily lie in the range O-l-lOM, and k usually falls in the range O*Ol-O*l (M-min)-i at normal operating temperatures (200”(Z), which implies a hold-time T
8 FIG. 2.
Weight fractions vs. reduced time for N = 100. Numbers on curves are x-values. 20
Polycondensation in a continuous, stirred-tank reactor
N=lO N=3 N=O.I
3 4 5 6 8 FIG. 5. Number average degree of polymerization vs. reduced time for several values of N.
3. Envelope curves showing the maximum weight fraction of various species vs. reduced time for several values of N. Numbers refer to x-values. FIG.
6 8 IO 8 FKL 6. Weight average degree of polymerization vs. reduced time for several values of N.
8 4. Weight fraction of monomer vs. reduced time for several values of N. 21
N. H. SMITHand G. A. SATHER
passes through the maxima of each species. At each maximum a node is introduced to specify the D.P. of the species involved. Three such curves are shown in Fig. 3 in which the left-most curve has been constructed from Fig. 2. The effect of decreasing N is to shift the envelope to the right, indicating a broadening and a lowering of each peak with reduced time. As N approaches zero, the envelope approaches the line wl,Ux = 1-O with a single node at 8 = 0. As N increases, the number of nodes on each envelope increases over any given time interval. As a measure of monomer conversion, the weight fraction of monomer wl, is plotted against reduced time 8 for various values of N in Fig. 4. The effect of increasing N is to reduce the unreacted monomer in the steady-state product and the time required to reach steady-state, c.f., with N = 100, steady-state is reached at about 0 = 3; but withN= l,atabout8= 5. The effect of N on the quantities 2” and Z,,, are shown in Figs. 5 and 6. In both cases the curves are shifted upward by increasing the value of N. However, the higher the molecular weight averages are, the broader the distribution becomes. The commonly used measure of distribution width is the ratio of weight to number-average D.P. At steady-state this ratio is E,/X” =
(N’ - 1)
As this ratio becomes larger, there tends to be an increase in the relative amounts of species above and below the mean. For a value of N = 100, this ratio is 13.2 for the continuous case as compared with 2.0 for batch operation at complete conversion . As N decreases, lower ratios are possible, e.g. at N = 10, the ratio is 3.74 at steady-state but the penalty is a low molecular weight product (2, = 11-Oand I?, = 2.79). In cases where narrow molecular weight polymer is required, continuous polycondensation is less desirable than batch. In preparing Fig. 6 it was observed that a significant condensation of information could be achieved by correlating the quantities (2, - 1)/N vs. time 8 as shown in Fig. 7. In addition to the initial condition of pure solvent [L,(O) = 01, the case of monomer initially present [Al(O) = l-01 is shown. These curves fit the computed results to about 1 per cent over the range of N from O-1 to 100. The approach to steady-state is first order for the latter case and second order for the former, showing the strong dependence of non-linear systems on initial conditions. For those cases in which there are both analytical and numerical solutions a direct comparison of results is possible in order to assess the accuracy of the numerical procedure (see Table 1). These results show agreement within 1 per cent indicating that the numerical method with q = 0.005 is adequate in predicting transient and steady-state concentration profiles and average molecular weights
Reduced correlations for ftu vs. 6 with &(O) = 0 and x1(0) = 1.0.
for polycondensation reactor. Table 1.
1 1 1 1 2 5 10 75
d gm k M
0.2 0.5 1.0 co al co co a,
0.162 0.237 0.227 0.218 0.0519 000977 0.00261 0%000515
Inlet molar concentration of species with D.P. ofm Molar concentration of species with D.P. of m Degree of polymerization Double sum Parameter defined by equation (33) Constant defined by equation (48) Kinetic rate constant Value of x at which truncation occurs in numerical calculation Dimensionless group (kinetic effect number defined as kTCl”) Parameter defined by equation (32) Mole fraction of species with D.P. of x Volumetric flow rate Rate of formation of species with D.P. of m by chemical reaction per unit volume F;te-time (= V/Q)
Comparison of exact and computer results
in a continuous stirred-tank
in a continuous,
0.162 0.237 0.227 0.218 0.0520 0 00986 0.00261 [email protected]
: Reactor volume W?lI Molar flow rate of species with D.P. of m WV&Weiaht fraction of snecies with D.P. of m x Number of monomer units in a polymer chain fn Number-average D.P. Weight-average D.P. & a Sum of dimensionless concentrations from 1 to 03 A Difference 17 Tolerance limit in computer programme de fining acceptable truncation of number of species to be integrated Reduced time (= t/T) Dimensionless concentration of species with D.P. of m (= C&IO) n- Dimensionless concentration of condensation products
Acknowledgement-Assistance in the form of a fellowship for N. H. SMITHfrom Standard Oil Company of California and of a generous grant of computer time from the University of Wisconsin Research Committee is gratefully acknowledged.
NOTATION AR , . . . General symbols for chemical species b Constant detined by equation (35)
:;; [31 r41 ‘,z; 171 181 191 WI
LIU S. and fbiUNDSON N. R., Rubber Chem, Tech. 1961 34 955. LN S. and AMWDSON N. R., Chem. Engng. Sci. 1962 17 797. SMITHN. H., Ph.D. Thesis, University of Wisconsin 1963. J?LORY P. J., Princigles of Polymer Chemistry. Cornell University Press, Ithaca, N.Y. 1953. ZEMANR. and AM&D.&N N. R., Amer. In.&. Gem. Engrs. J. 1963 9 297. SMOLUCHOWSKI M. V.. 2. Phvs. Chem. 1917 92 129. DOSTAL H. and RAFP k., 2. phys. Gem. 1936 32 117. TANFORDC., Physical Chemistry of Macromolecules. John Wiley, New York, 1961. GILL S., Proc. Camb. Phil. Sot. 1951 47 96. LAPIDUS L., Digital Computation for Chemical Engineers pp. 82-130. McGraw-Hill, New York 1962. R&atm6-Le sujet de cet article est l’analyse des stades transitoire et permanent de la polycondensation dans un reacteur agite a alimentation continue, comme exemple des systemes cinetiques complexes. Le pro&de utilid consiste a effectuer une resolution analytique partielle des equations exactes de vitesse et de la completer par une resolution num6rique a l’aide d’un calculateur digital. Par l’analyse dimensionnelle, un groupe simple sans dimension est obtenu, faisant intervenir les rendements en polymeres et la distribution de leur poids moKculaire pour les regimes transitoires et permanents d’un r&cteur isotherme avec alimentation de composition constante.