Ind. Eng. Chem. Process Des. Dev. 1986, 25, 575-584
575
Design of Multiphase Gas-Liquid Polymerization Reactors with Application to Polycarbonate Polymerization Patrick L. Mllls Chemical Reaction Systems Group, Central Research Laboratory, Monsanto Company, St. Louis, Missouri 63 167
Polymerization reactions between small gaseous molecules and liquid-phase monomers in the presence or absence of a suitable catalyst have commercial importance in the polymer industry, yet the design and analysis of polymerization reactors for these systems have not received considerable attention. In this paper, a physical and chemical model for use in the analysis of gas-liquid-liquid stepwise condensation polymerizations carried out in a continuous-flow stirred-tank reactor and other related modes of operation is presented. The model accounts for finite mass-transfer resistances between the gas-liquid and partially miscible liquid-liquid phases and includes either plug-flow or complete backmixing as choices for the gas-phase flow pattern. Polymerization kinetics involving multiple reactions with a chain growth stopper are described, various limiting cases of the model are given, and methods for solving the model equations are summarized. The model equations are applied to the gas-liquid-liquid interfacial polymerization of bisphenol A to polycarbonate as an example of a complex system having industrial importance. Methods for solving the infinite system of model equations are discussed, and the effects of various model parameters on the polymer chain length distribution are examined for a particular limiting case.
I t is well-known that development of polymer-based materials for specific applications is largely an art vs. a science since it is often difficult to establish direct relationships between end-use product requirements such as impact resistance, flow properties, thermal stability, and electrical conductivity, to name a few, to molecular characteristics of the polymer such as molecular weight distribution, degree of chain branching, intrinsic viscosity, etc. Thus, it is not surprising that the effort required to develop new polymer materials can be significantly greater than that required for small molecules due to the difficulties associated with altering the molecular characteristics of macromolecules by novel chemistry and the labor associated with polymer product characterization. Once the polymer product has been identified, an additional challenge is to transform the laboratory-scale polymerization methodology to economical pilot and commercial-scale reactors in an efficient fashion without altering the polymer molecular characteristics and hence the polymer product performance. Additional details related to this important topic and the role of polymerization reaction engineering are available in various review papers (e.g., Chappelear and Simon, 1969; Ray and Laurence, 1977; Ray, 1983) and the references cited therein. As summarized by Ray and Laurence (1977), polymerization processes can be broadly classified according to whether the reaction medium is homogeneous or heterogeneous. Homogeneous processes include both bulk and solution polymerizations, while the primary heterogeneous polymerization processes include emulsion, suspension, and precipitation polymerization. Detailed discussion and examples of these are given in classical monographs such as those of D’Alelio (1952), Burnett (1954), and Bamford et al. (1958) and the more recent monographs of Odian (1981) and Ulrich (1982). Particular noteworthy is that reaction engineering for heterogeneous polymerizations involving reactions between a gas and liquid monomer that occur in the presence of a partially miscible liquid media and possibly a catalyst has not received considerable attention. Industrial examples of such systems include the production of poly(pheny1ene oxide) polymer by the oxidation of 2,6 dimethylphenol (e.g., Cooper et al., 1973),the polycondensation of various assorted biphenols with
phosgene gas to yield aromatic polycarbonates (Christopher and Fox, 1962; Odian, 1981; Fox, 1982), and various other systems which are partly summarized by Ulrich (1982). The expected trend of developing additional speciality polymers, some possibly based upon selective insertion or addition of small gaseous molecules such as CO, COz,SOz, NH3, etc., using transition- and noble-metal complexes (e.g.: Hallgren, 1978), suggests that an understanding of the factors that influence the polymer molecular characteristics, such as the polymer molecular weight distribution, will be important for design, scale up, and troubleshooting of these reactors. One objective of this paper is to present reactor design equations for polycondensation polymerization carried out in an isothermal continuous-flow stirred tank reactor for the heterogeneous case in which a gas and two partially miscible liquid phases are present. Another objective is to briefly summarize the solution methods for the resulting system of infinite equations and to demonstrate the use of the Fast Fourier Transform (FFT) as a technique for obtaining the polymer molecular weight distribution by inversion of the z-transformed model equations to the chain-length domain. Since the polymerization kinetic schemes for gas-liquid systems cannot be readily generalized, the multiphase polycondensation of bisphenol A to polycarbonate is selected as a design example since it has not been previously analyzed and has industrial significance. The effect of various model parameters on the polymer molecular weight distribution is examined for this system as an application of the proposed equations and solution methods for reactor design. Physical Description and Assumptions The physical system considered in this work is an isothermal stirred tank reactor, although the analysis can be readily extended to reactors having other flow patterns. In the general case, two partially miscible liquid phases that flow either continuously through the reactor or are maintained batchwise within the reador are contacted with a continuously flowing gas phase. Both of the liquid phases are assumed to be well-mixed and, for purposes of this discussion, are assumed to consist of an aqueous phase and an organic solvent phase that are partially miscible. Ini-
0196-4305/86/1125-0575$01.50/00 1986 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
tially, the reactor is assumed to contain both phases with the monomer and possibly dissolved gaseous reactant which may be distributed between the aqueous and organic solvent phases at some prescribed concentration that is consistent with liquid-liquid thermodynamic equilibrium. A polymer capping agent referred to here as a chain stopper may also be initially present in the reactor or added later during the course of reaction as a means of controlling the final molecular weight of the polymer. The chain stopper may also be distributed between the two liquid phases analogous to the monomer. The case considered here assumes that the physical properties and the phase equilibrium constants are independent of pressure and are linearly related to the composition for simplicity, although this is not a restriction and more rigorous thermodynamic-phase equilibrium models could be used. When the gas is added to the reactor, it is transported by gas-to-liquid mass transfer to the organic phase where chemical reactions between the gas and liquid monomer can occur. The products formed from these reactions are assumed to be nonvolatile species which are transported between the liquid-liquid interface to initiate a sequence of series-parallel condensation reactions. The polymerization reactions are assumed to occur in the organic solvent phase with negligible polymerization in the aqueous phase due to the unfavorable solubility of the oligomers in the aqueous phase. For purposes of defining the appropriate mass-transfer steps, the aqueous phase is assumed to be the dispersed phase while the solvent phase is assumed to be the continuous phase due to its greater liquid density and the use of a nearly equal ratio of phase volumes to prevent phase inversion. In addition, transport of the gaseous reactant to the aqueous phase will be assumed to occur indirectly through the organic phase; i.e., the gas bubbles are not in direct contact with the aqueous phase itself due to the significantly greater solubility of the gas in the continuous organic phase. A schematic of the mixing and mass-transfer steps between the gas and liquid phases in accordance with the above description is analogous to that given by Ramachandran and Chaudhari (1984) in the context of threephase slurry reactors. The model equations given below for the gas-phase flow pattern are developed for the extreme cases of plug flow and complete backmixing of the gas bubbles for comparison purposes. The effect of the gas-flow pattern on the performance of three-phase slurry reactors has been summarized by Ramachandran and Chaudhari (1984), except that the reaction stoichiometry was limited to small molecules and did not include gasliquid polymerization kinetics. This aspect is briefly examined in this work also but from the viewpoint of assessing how the gas-flow pattern affects the polymer molecular weight distribution. Reactor Performance Equations With the assumptions outlined above, the mass balances are summarized below where the concentrations of the principle monomer reactant, the gaseous reactant, and chain stopper are denoted by the symbols A,, Bo, and Do, respectively, while the remaining monomer and polymer species are denoted by Uj for j 2 0. The variable R is used to denote the rate of reaction per unit volume of either the aqueous or solvent phase (subscripts a and s, respectively).
-dUjs - dt
ujs,i -
Ujs
+ Ruj,s for j = 1,2,...,m
(5)
7,
Development of eq 1is based upon the following mass balance equation for the gaseous reactant Bog = Bog,g(Stgl) + BosHBo{l - g(Stg1))
(6)
The function g(Stgl) assumes the form g(Stgl) = exp(-Stglz/L) for plug flow of gas bubbles so that the gas concentration varies with the distance z above the gas inlet, while g(Stgl)= (1- St,&' for completely backmixed gas bubbles so that the gas concentration is constant. Key parameters which emerge include the Stanton number for gas-liquid transfer S t 1 the mean residence time of the aqueous and solvent prases ra and T,, a Stanton number for the aqueoussolvent interphase transport Nu,, various phase volume capacity factors @, and @,,, gas solubility HBo, and liquid-liquid-phase equilibrium constants Kuo, Definitions for these and the remaining variables appearing in eq 1-6 are given in the Nomenclature section. The above differences between plug flow and complete backmixing of the gas phase and its effect on reactor performance has been noted also by Ramachandran and Chaudari (1984) in the context of three-phase slurry reactors. It was pointed out that the choice of plug flow vs. complete backmixing of the gas phase may be important in these systems for intermediate values of Stgl,e.g., 0.5 I Stg15 10. Whether or not significant differences in the polymer molecular weight distribution might occur over the same range in the context of the current work has not been quantified, however. The expressions for the gas-liquid and liquid-liquid interphase mass transfer appearing in eq 1-6 are based upon two-film theory. The overall mass-transfer coefficient for transport of species Uobetween the solvent phase and the aqueous phase is given by K,,, and can be expressed as the following sum of resistances in the solvent-aqueous liquid film: (7)
The overall mass-transfer coefficient for transport of species Bo between the gas phase to the solvent phase is denoted by K B ~ ,and ~ , is given by the following sum of gas-film and liquid-film resistances:
- -1 -KBo,gs
1
kBo,gHBo
1 + -IZBo,l
(8)
It is assumed here that the equilibrium solubility of the gaseous reactant in the solvent phase can be expressed in terms of a Henry's law constant H B o = Bog/Bos,and the phase equilibrium distribution coefficient for species Uo in the aqueous and solvent phases is given by KUo = U O S I uoa.
The above representation of mass transfer across the liquid-liquid interface using two-film theory represents the
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
simplest approach for interphase transport of distributed solutes but serves as a starting point for other more sophisticated models. Other complications such as the presence of surface active agents and polymer species, nonuniform drop-size distribution, and reaction in the liquid films adjacent to the liquid bulk are but a few examples of what might be encountered for specific systems. A good discussion of these as applied to liquid-liquid mass transfer of carboxylic acids is given by Schugerl and Dimian (1980). The reader is referred to the references cited therein as well as those given in the more general review of Godfrey and Hanson (1982) for further details on this subject. Initial conditions to accompany eq 2-6 can be given in the following general form where j I 0:
uj, = uj, uj, = uj,
att=O
( t = 0)
(94
( t = 0)
(9b)
A typical example would be to specify the initial monomer, chain stopper, and dissolved gas concentrations with zero concentrations for the remaining species. In either case, specification of eq 9a and 9b must be within the bounds defined by the phase equilibrium distribution coefficient for a given component. When mass transfer of species between the partially miscible liquid phases occurs at a higher rate then the rate of chemical reaction in the bulk liquid, then the liquid phases may be assumed to exist at physical equilibrium. This situation was considered by Goldstein and Amundson (1965) for nonisothermal free-radical polymerization with partially miscible liquids in a CFSTR in the absence of a gas phase. The mass balance equations given by eq 1-5 then reduce to the forms
-- dB08
dt
-dUjs - dt
ujs,i - ujs
+ RUj,,for j
= 0,1,...,m
(11)
7,
uj,
Uj, = - for j = 0,1,...,m Kuj Initial conditions for eq 10 and 11 are given by eq 9b. The aqueous phase concentrations U,, are given by eq 12 in terms of the organic solvent-phase concentrations Uj, and the thermodynamic distribution coefficient K,. As in the previous case, an infiiite set of differential equations for the species concentrations is obtained. Omitting the first term on the right-hand side of eq 10 and 11 and replacing the term PgsB0/7, in eq 10 by &!&IB,/V , gives the design equations for a continuous flow of gas with the batch liquid, i.e., semibatch operation which is also of interest. Reactor Design Example The multiphase polymerization reactor design equations given in the previous section are applied below to a specific example, namely, the polycondensation of bisphenol A [2,2-bis(4-hydroxyphenyl)propane] to polycarbonate polymer by reaction with gaseous phosgene. The preparation of this outstanding commercial thermoplastic by this reaction using the so-called interfacial synthesis process is well-documented in various monographs (Christopher
577
and Fox, 1962; Schnell, 1964; Odian, 1981),review articles (Vernaleken, 1977; Fox, 1982), and other specific literature (e.g.: Wielgosz et al., 1972) from the perspective of chemistry, polymer physics, and commercial application to which the reader is referred for details. Attempts to provide a quantitative description of the polymerization reaction kinetics with proper accounting of transport effects for a particular reactor configuration have not been reported in the open literature. Application of the proposed reactor design equations using a comprehensive set of chemical reactions that describe the formation of the polycarbonate polymer is given here and represents the first time that a reaction engineering analysis has been reported for this system. a. Chemical Reactions. The various chemical reactions which are believed to occur when an aqueous caustic solution of bisphenol A is dispersed as liquid droplets within an organic solvent such as methylene chloride and contacted with bubbling phosgene gas have been best described by Noguchi (1963) and Robertson et al. (1973). Noguchi’s (1963) work is the most comprehensive since it includes a summary of the potential reactions, estimates of the relative reaction rates, and an estimate of the preferred phase in which the reaction might occur. Neither author discusses the reactions of a typical monofunctional chain stopper, such as phenol, with the other proposed reaction intermediates to form a mono- or dicapped polymer. Since polycarbonate polymerization in the presence chain stopper is the one of primary commercial interest, a set of chemical reactions that are consistent with the basic proposition of Noguchi (1963), but is extended to include the reactions of a typical monofunctional chain stopper, are used here. The final reaction network that can be developed is given in Table I where the various symbols used to denote various monomer and polymer species are summarized in Table 11. Included in Table I are estimates of the phase in which the various reactions occur which is based upon the work of Noguchi (1963) and solubility considerations of the monomers and growing oligomers. The first three propagation reactions given in Table I do not directly involve the chain stopper (species Do),while the reaction between phosgene (species Bo) and phenol (species Do) to yield phenol chloroformate (species E,) leads to the remaining propagation reactions where monocapped polymers (species D , and E,) are formed. Termination to the dicapped polymer occurs by condensation between the monocapped polymers species D, and E,. The rate constants assigned to the reactions in Table I are based upon the assumption that the polymer end groups have equal reactivity. Reactions between monofunctional monomers are assigned a rate constant k,, while reactions between monofunctional polymers are assigned a propagation rate constant k , and similarly for difunctional polymers. The hydrolysis reaction for phosgene (species Bo)decomposition to hydrochloric acid (species C,) is assigned a rate constant k,, while the remaining ones are assigned an end group reactivity of Itp. Use of unequal reactivity for various end groups such as that used by Gandhi and Babu (1979) is also possible, but this leads to additional complications that are not yet supported by experimental evidence and whose investigation lies outside the scope of this work. b. Reaction Rate Expressions. The chemical reactions given in Table I can be used to develop reaction rate expressions for each of the monomer and polymer species given in Table 11. These are listed in Table I11 where the reaction rates are defined per unit volume of either aqueous (subscript a) or organic (subscript 0)phase and are assumed to follow elementary behavior with respect
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
Table I. Polycarbonate Polymerization Reaction Network
+ Bo &.
Initiation C1
(solvent/aqueous)
A,
(solvent) (solvent) (solvent) (solvent/aqueous) (solvent) (solvent) (solvent) (solvent)
Propagation A m + Bn % Cm+n+, A,, + Cm 3 A,+, B , + C, 3 B,+, Do + Bo h E , D , + B, 3 E,+, C m + Dn h D m + n C m + En Em+" Am + En 3 ~ m + n + 1
m, n = 0,1,...,m except m = n = 0 m = 1,2 ...,m ; n = 0,1,...,m m = 1 , 2 ,...,m ; n = 0,1,...,m m = 0,1,...,m ; n = 0,1,...,m
(solvent/aqueous) (solvent) (solvent)
Hydrolysis Bo 4C, B, % C, k An-l E,, k D ,
n = 1 , 2 ,...,m n = 1,2,...,m
(solvent)
Dn + Em
m, n = 0,l ...,m except m = n = 0 m = 1,2,...,m ; n = 0,1,...,m m = 1,2 ,...,m ; n = 0,1,...,m
Termination Fm+n
Table 11. Monomer and Polymer Species for the Polycarbonate Polymerization Reaction Network
m = 0,1,...,m ; n = 0,1,...m Table 111. Reaction Rate Expressions for Polycarbonate Polymerization a. Aqueous Phase ( m = 0)
where R = CH3
symbol A,, B" C, Do E,
monomer HO-R-OH c1-CO-c1 H-C1 Ph-OH Ph-OCO-C1
symbol A, B, C, D, E, F,
polymer HfO-R-OCO j,O-R-OH Cl-OCfO-R-0COjnCl HfO-R-OCO j,Cl HfO-R-OCOjO-Ph Cl-OCfO-R-OCOj,O-Ph Ph-OCOfO-R-OCOf,O-Ph
to the reaction order. Also, the kinetic rate constants k , and k,, which are associated with the growing polymer chains, are assumed to be independent of the polymer chain length n as a first approximation. c . Solution of the Model Equations. Substitution of the reaction rate expressions given in Table I11 into eq 1-5 or eq 10 and 11 gives an infinite set of differential equations whose solution yields the concentrations of the various monomer and polymer species as a function of time, polymer chain length j , and the model parameters. Complete sets of the equation are given in the final forms needed for numerical solution in sections A-C of the supplementary material. Some remarks on the methods used to solve the infinite system and some special considerations related to the computational aspects are summarized below. Two independent numerical methods were used in this work to solve these equations as part of a more detailed study on the computational aspects associated with determining the entire polymer molecular weight distribution for comparison to the gel permeation chromatographic data. These methods included (i) direct numerical integration and (ii) inversion of the z-transformed model equations. Details associated with these are given elsewhere (Mills, 1986) so that only a summary to clarify some important issues will be given here. 1. Solution by Direct Numerical Integration. To use direct numerical integration, a maximum chain length N must be specified which reduces the infinite set of differential equations to a finite set that can be readily solved by using one of several available techniques for the solution of initial-value ordinary differential equations such as Gear's method (Gear, 1971). The maximum chain length N must be chosen so that the remainder associated
with truncation of the infinite summations of species concentrations leads to negligible error when evaluating the derivatives. Methods for a priori estimation of the series truncation error are not available so that resortment to numerical experimentation is required by using various values of N in the practical applications of this method. CPU time requirements for the current problem can be significant since 6N + 10 derivative evaluations are required at each time step where N = 100-150 for a typical design calculation. A computer program that implements this approach using fourth-order Runge-Kutta corresponding to the case described by eq 10-12 is available upon request. Programs for other cases represent minor modifications from this case and have a similar structure. This general technique of direct integration was first extensively used by Liu and Amundson (1961) and still remains a popular method, although it does have certain limitations. 2. Solution by z Transforms. Application of the z-transform method to polymerization systems was first illustrated by Kilkson (1964). Since then, it has been used to analyze various other polymerization systems (e.g.: Villermaux and Blavier, 1984) and represents an alternate approach to direct numerical integration. The z transform
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 579
of a discrete sequence of polymer concentrations Ujis given by m
Z{UjJ=
E z-wj
j=O
= U(z,t)
(13)
Multiplying eq 5 by z-j, forming the sum for j = 1,2...,a, and adding the result to eq 3 gives the following z-transformed equation for species U: dUs(z,t) Osi(z,t)- Us(z,t) -dt
Equation 14 can be developed for each polymer species in the polycarbonate reaction network given earlier in Table I1 where z(RUj,,) denotes the z transform of the reaction rates listed in Table 111. Evaluation of eq 14 as z 1 also gives the differential mass balances for the total molar concentration of each species or zero moments of the polymer chain-length distribution. Differential equations for higher moments can also be developed from the following general equation for the kth moment of the polymer chain-length distribution for species U (Kilkson, 1964):
-
m
A summary of the z-transformed equations and the differential equations for the polymer moments are given in sections D-F in the supplementary material since these are quite lengthy. Although the z-transform method has been widely used to obtain the polymer MWD moments for many systems, inversion of the z-transformed equations to yield the polymer MWD has been claimed to be impossible (e.g.: Villermaux and Blavier, 1984). Chen and Spencer (1968) developed a method for inverting the z-transformed equations to obtain the polymer MWD, but no error analysis was given. In addition, this method has been apparently overlooked in many papers that followed. Application of this method to the current problem requires the evaluation of 25 derivatives at each time step. This is significantly less than the 600-900 derivative evaluations required by direct numerical integration and results in greater computational efficiency since no finite sums or convolution product sums are required. The detailed equations required to implement this method for this example are given in section E of the supplementary material. A detailed error analysis of the method with examples is given by Mills (1986). It suffices to say here that this method can provide results within the rounding error of the computer with less computational effort than direct numerical integration and is the preferred approach. d. Polymer Statistical Parameters. Various polymer statistical averages, such as the number, weight, and zaverage molecular weights, can be determined from the moments of the polymer chain-length distribution. In the equations that follow, the monomer species A,, Bo, ...,F, are excluded since these do not contribute to the polymer mass. The number average molecular weight M , can then be expressed in the form
In the above expression, Mu,denotes the molecular weight of the monomer species U,, M denotes the molecular weight of the repeating unit (0-R-OCO), pou denotes the zeroth moment, and plU denotes the first moment as defined by eq 15. The polymer weight average molecular weight M w is defined according to
Equation 17 for M whas a functional dependence on the second moment of the various polymer species in addition to the same parameters that appear in the expression for
Mn. The polydispersity index is often used by polymer chemists as a measure of the deviation of the polymer MWD from the Schultz-Flory most probable distribution and is given by the following ratio of weight-to-number average molecular weights M W
P=Mil Besides the above polymer MWD averages, another parameter of interest is the weight fraction of polymer whose chain length is j . This is given by m1
XI= -
5mk
(19)
k=l
The quantity m, in eq 19 denotes the total weight of the polymer whose chain is j where the polymer consists of species A,B,...,E Tn] = A~sMA] + B ] f i B ] + c]sMC] + D]sMD] + E]sME] + FpMF~ (20)
All the parameters given by eq 16-19 can be determined from experimental gel permeation chromatography (GPC) data, the methodology of which has been described in standard texts (Slade, 1975) and symposia proceedings (e.g.: Provder, 1984). Parameter estimation techniques can then be used to identify the unknown polymerization reaction rate constants since the remaining variables such as mass-transfer coefficients and phase equilibrium constants can be independently estimated from mass-transfer correlations and thermodynamic relations, respectively. Results and Discussion Reactor design calculations were performed to compare the solution techniques and to briefly investigate the role of various model parameters on design variables. Although a general development was presented for continuous flow of the gaseous reactant and liquid monomer, the principle emphasis here is upon the semibatch operation since this mode of operation is the one most often used in practice for polycarbonate polymeriation (Christopher and Fox, 1962; Schnell, 1964; Fox, 1982; Ulrich, 1982). The results given below correspond to the case of finite gas-liquid mass-transfer resistance with negligible liquid-liquid mass-transfer resistance as described by eq 10-12. Results for finite liquid-liquid mass-transfer resistance and comparisons between semibatch vs. continuous operation are given elsewhere (Mills, 1986). Selection of the physico-
580
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 1.0
0 75
1
0 60
*
.. '5 c
0
08
4-
z 0
t E
045
x
030 0 15
06
0 00
z
t = 60
w
0
60 E
5
08
5
04
0
5
m-
2 I-
p
06 120
X
04 02
02
00 12 00
0
100
200
300
400
500
600
: 09
-3
X
REACTION TIME, t
0- 0 6
Figure 1. Total molar concentration of species A , B, and C as a function of batch time. Parameters: A,& = 0) = 1.0, Bo& = 0 ) = 0.4, Do& = 0 ) = 1 X IO-*,k , = 5 X k , = 1 X lo-*, k , = 0.5, k , = 1X Q g H ~ < , /=V s0.5.
03
00 0
20
40
60
80
1 00
CHAIN LENGTH, j
-.
0.008
Figure 3. Concentration of species A , B, and C as a function of chain length at various batch reaction times. Parameters: Aoe(t= 0) = 0.25, B,(t = 0 ) = 0.5, D,.(t = 0 ) = 1 X k, = 5 X k, = 1 X lo-*, k , = 0.5, k , = 1 X St,, = 1. QgHBo/Vs= 1 X
-
,.-
'5 z
il I< E
0.006-
2 0 z 0 0 W
50
0004
I
2 +
e
0.002
0 000 0
100
200
300
400
500
600
REACTION TIME, t
Figure 2. Total molar concentration of species D, E, and F as a function of batch time. Parameters: same as Figure 1.
chemical constants to obtain the simulation results presented here was based on model parameter sensitivity studies along with order-of-magnitude estimates. Precise identification of these constants using laboratory or pilot-scale polymerization reactor data by parameter estimation techniques and the application of engineering correlations lies outside the scope of this current work and remains a topic for future investigation. In Figure I, the total molar concentration or zeroth moments of mono- and dichloroformate oligomers of bisphenol A (species C, and B,) as well as dihydroxy oligomer (species A,) in the solvent phase are given as a function of elapsed batch time for two different values of the Stanton number for gas-liquid mass transfer. For this case, the organic solvent was assumed to be initially saturated with the gaseous reactant at a value defined by the
inlet concentration, i.e., B,(t = 0) = Bog,'/Hk Species A,, and B, are seen to decrease along the same path for t 5 50, indicating that the observed rate is unaffected by changes in the gas-liquid mass-transfer rate over the given range of Stanton number. When t > 50, the concentration vs. time curve for species B, at the smaller value of St,, decreases exponentially until t = 150 and then reaches a nearly constant value, while the concentration vs. time curve for species A , has a constant slope or rate over the indicated range of time. The curve for B, a t the larger value of S t , decreases slower than that for the smaller value until it reaches a minimum at an intermediate value of time at which point it increases to a value less than the initial equilibrium concentration. Even at the larger value of St,, the gas-liquid mass-transfer rate is not great enough to maintain the liquid phase saturated with the gas. The production of species C, increases in a fashion that is consistent with the consumption of species A, and B,, with the curves for the larger value of St,, being more pronounced as expected. In Figure 2, the total molar concentration or zeroth moments of various monocapped oligomers (species D, and E,) as well as the dicapped polycarbonate polymer (species F,) in the solvent phase are shown as a function of elapsed batch time. The total molar concentration of the dicapped polymer steadily increases as the monocapped oligomer D, and E , react with the mono- and dichloroformate oligomers (species C, and B,) and dihydroxy oligomer (species A,) to form species of increasing chain length. The magnitude of the Stanton number is seen to have a significant effect on the growth rate for all species which is analogous to the results given in Figure 1 for species A , B, and C. Figure 3 compares the concentration vs. chain-length dependence for uncapped polymer species A , B , and C at various increasing values of the batch reaction time, while
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 1.0
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