PERFORMANCE OF A POLYMERIZATION REACTOR IN PERIODIC OPERATION R. L.
LAURENCE A N D G A N A P A T H Y WASUDEVAN
Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Md.
The performance of a polymerization reactor in periodic operation is analyzed. Analytical estimates of the deviations from the steady-state performance are obtained for both simple addition and combinationtermination polymerization mechanisms in the limit of very slow oscillation. The results using reported kinetic data indicate that a marked change in the distribution can b e obtained for these polymerization reaction mechanisms.
ECESTLY,
Douglas and Rippin (1966), Douglas (1967),
R and Horn and Lin (1967) considered the effect of periodic operation on the performance of a continuous stirred tank reactor. For a second-order, irreversible reaction it was shown that a periodic change i n the reactant feed concentration improves the time-averaged conversion over the steady-state conversion. Douglas (1967) argued that oscillations in reactor feeds already occurring in a plant should not be damped prior to introduction of the feed into the reactor. Improved conversion could be obtained if the oscillations are removed after the reactor. I n this paper, the e f i x t of periodic operation on the performance of a polymerization reactor is analyzed. Polymerization reactions constitute an infinite set of simultaneous chemical reactions which, generally, are nonlinear. Some deviatory behavior is to be expected when such a set of reactions is allowed to proceed under periodic conditions. The product of a polymerization reactor contains distribution of species and it is this distribution of molecular weights which determines the physical properties of the polymer product. Periodic operation of the reactor can modify the nature of the distribution and concomitantly the physical properties of the resultant polymer. I t is the intent of this work to establish the effect of periodic operation upon the molecular weight distribution of the polymer. Is it narrowed or does the polymer become more polydisperse? In contradistinction to Douglas' early work, the reactor would be subjected to intentional variations in feed composition to produce a material, perhaps, not possibly made in any other way.
material balances on the system, then, need be considered. Generally, an over-all material balance for species s in a CSTR is:
where 8 is the average residence time of material in the vessel, C, is the concentration of species s in the vessel, C,O is the concentration of species entering the vessel, and & is the volume rate of production of species s. Addition Polymerization
Miyake and Stockmayer (1965) proposed a fairly simple model for living end polymerization and studied the distribution that arose from a batch polymerization. This model is uscd because the steady-state solutions for the molecular weight distribution are particularly simple. I n this postulation, it is implicit that the reaction velocity constants are independent of polymer chain length.
+ M kl PI Pj + M Pj I
(initiation)
--t
(2)
122
(propagation)
+1
where k l is the initiation reaction velocity constant, kz is the propagation reaction velocity constant, I is the initiator concentration (moles per liter), Pi is the concentration of polymer with chain length j , and M is the monomer concentration. The material balances for this system are given by
The Models
The kinetics of the polymerization reactions used in this study are straightforward, but nonlinear, and provide, in spite of their simplicity, some insight into the dynamic behavior of the reactor without overly complicating the analytical study of the system. The two polymerization mechanisms treated are (1) addition polymerization without a termination step [the performance of a CSTR in which this reaction scheme proceeds was first studied by Denbigh (1947)], and (2) free radical polymerization with combination termination. The two mechanisms were selected for their applicability to real polymer systems and for the ready calculation of the properties of the steady-state distribution obtained in a CSTR. T h e model for the reactor is a n ideal one, a well mixed, isothermal continuous stirred tank reactor (CSTR). Only the
(4)
Abraham (1963) and Kilkson (1964) have used Z-transforms in the analysis of differential-difference equations of this type. Often the transform can be inverted to provide the entire species distribution. When it cannot, simple differentiation provides the moments of the distribution which, as can be shown, are related to measurable properties of the distribution. If the Z-transform of Pjis defined as
z-'P,
6(2) =
(7)
j
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427
the infinite set of differential Equations 4 , 5, and 6 can be collapsed into a set of four equations:
dI d0 d0
+ I - IO = -klIMS
+ M - Mo ds(l’e) d0
+ uo = im
(8)
- kzMb6(1,0)
-klIMb
=
duo d0
(9) where
+ s ( 1 , e ) = klIMb
a = kir91,
k&M,
(29)
7 = kibM,
(30)
=
where IOis the feed concentration of initiator, Mo is the feed concentration of monomer,!and 0 is a dimensionless time ( t / b ) . As reported by Saidel and Katz (1967), it is sometimes simpler to deal directly with the moments of the distribution, since measurable characteristics of a polymer are related to the moments. For example, the number average degree of polymerization is given by:
m n
nPn
= n=l
12
Pn
=
P~/PO
(12)
where PC(k is the kth moment of the distribution. The kth moment can be readily obtained from the 2-transform of P, by differentiating
I n the following sections, the responses of this set of equations subject to very slow oscillations in both monomer and initiator feed concentrations are evaluated to provide an estimate of the magnitude and direction of the deviation from steady-state performance. The Appendix shows that the set of equations linearized about the steady state has only real negative eigenvalues. For such a system, the response at low frequencies of oscillation is a good estimate of the maximum deviation from the steady state. If the energy equation is added to the set, there can be complex eigenvalues and resonant behavior is to be expected. The objective of this study was to establish the feasibility of periodic operation of polymerization reactors. For this reason the analysis was limited to isothermal reactors. Forced Oscillation of Monomer Feed Concentration
The corresponding set of moment equations for this system is obtained by differentiating Equation 11. The first three moment equations are
The first problem analyzed for this mechanism is the response of a CSTR to an oscillation of the monomer feed concentration about some steady value. In this case, Equations 23 through 27, in the limit of very slow oscillations when we can neglect the capacitance terms, assume the form: i m
- fio(1
- io
+ ax)
uz =
- fo
I, M,
- JTo
=
=
-kibImMm
- kzbMmpom
-kibM,Im
P O , = kibM,I, film
P Z = ~
+
(18 ) (19)
= kzbMmPom
kz19Mm(2~1,
(17)
=
I/I,;
Porn)
m = M/M,;
dm d0
--
428
+ i - io
+ m - mo
uk = p k / p k m
(22)
= -Tim
= -aim
aim
- aprnuo
(32) (33)
uom
(34)
=
+ 2P)-’(2Pul + uo)m
(35)
where 6 is the fractional amplitude of the input oscillation and x ( t ) is the input oscillatory function. Suppose the oscillation in the monomer concentration is expanded in a Taylor’s series about the steady value with x, the perturbation, as the independent variable. Since all quantities in the equations depend upon m and consequently upon x , any variable, y, can be expressed as
(21)
If Equations 8, 9 , 14, 15, and 16 are normalized, the concomitant dimensionless equations are di d0
-
(20)
The steady-state relations can be used now to formulate the problem for forced oscillation of the reactor. The concentrations and the moments can be normalized with respect to their steady-state values by
i
(1
=
(31)
u o = im u1
The solutions of Equations 8, 9 , 14, 15, and 16 a t steady state are given by
= --yzm
- apmuo
(24)
l&EC PROCESS DESIGN A N D DEVELOPMENT
The time-averaged shift in the value ofy is given by
G=;i
t +27/tu
b(X) - y(0)ldt’
(37)
If the Taylor’s series expansion of y in x ( t ) is used and if x ( t ) is periodic in t , then all terms containing odd powers in x vanish identically. For relatively small 6, only the term dzyldxz need be considered, since the contribution of the fourthorder term can be shown to be insignificant. Now if x = sin wt, we see from Equation 37 that
Applying Equation 38 and solving the set of Equations 31 through 35, the follclwing analytical relations are obtained for the time-averaged deviations from the steady-state values in the limit of slow oscillations.
Table 1.
Summary of Results for Addition Polymerization Mechanism
SteadyState Average Fractional Deviation QuanValues, from Steady-State Values tity MoleslLiter Monomer variation Initiator variation M 1.665 -19.75 X 10-6a2 0.346 PO 0.0985 -0.725 X 10-262 -0.606 X 10-26' -0.0702 62 P1 8.2 0.893 X 10-462 0.502 0.209 a2 !!L 1372 DP, 83.25 0.73 X -0.0642 a2 z 2.01 0 . 5 62 0.35 a2 (1 - O.14cS2)-1
in the performance of the reactor, the values of f i , 30,61,and 32 were calculated for a particular reaction system. The kinetic mechanism studied closely approximates that of the polymerization of caprolactam with water as the initiator (Kruissink et al., 1958). The reaction velocity constants for this system are actually size-dependent. An average value for the reaction velocity constants a t 254' C. was used. Data:
where
kl = 8 liters/mole-hr.
lo
k2 = 10 liters/mole-hr.
Mo = 10 moles/liter
= 0.1 mole/liter
Forced Oscillation of Initiator Feed Composition
The second problem is the system response to a low frequency oscillation in the initiator feed composition. For this case, Equations 31 and 32 'become
i - io(1 m
- fib0
+ 6 x ) = -Tim
=
-aim
- apmuo
c9 = 5 hours
With these constants and conditions, the dimensionless constants defined in Equations 28, 29, and 30 take on the values
(43) (44)
where x ( t ) is a sinusoidal disturbance. Equations 33, 34, and 35 remain the same. The time average deviations from the steady state (in the limit of very slow oscillations) are then obtained using Equation 38.
(Y
= kldZm = 0.0592
P
=
k2dM, = 83.25
66.6
= kibM,
Using the values for the constants, a, p, and y, the d.c. shifts for this system can be evaluated quantitatively subject to a variation of either the initiator or monomer feed concentration. Table I presents the results of these calculations along with the steady-state values of the moments of the distribution. Free Radical Polymerization with Combination Termination
Saidel and Katz (1967) studied a mechanism for free radical polymerization with combination termination using the continuum approximation of Zeman and Amundson (1963). I n this section we study the same kinetics, using a 2-transform analysis. The kinetics have the form:
I ~i
Discussion of Addition Polymerization Model
For forced oscillation of the initiator feed composition, Equations 45 and 46 show that monomer conversion is decreased and concomitantly the polymer production is decreased for all values of the kinetic constants. This is not the case for Equations 47 and 48, where the sign of the expression is not readily seen by inspection. However, since a is a measure of the initiator concentration and is generally much less than 1, 31 is usually negative, indicating a decrease in the number average degree of polymerization. The expression for 3 2 is even less tractable, but for most values of a, 8, and y, 3 2 is positive, indicating a broadening in the distribution. For an estimate of the magnitudes in the expected deviation
R,
ki
2R1 initiation
(49)
+ M 5 R ( + ~ propagation + R, ks
+ P,+j
(50)
termination
The material balances, equivalent to Equations 8 through 11 for the simple addition mechanism, using 2-transforms as before, can be written as:
dl d0
+ I - zo = --k1SI
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Forced Oscillation of Monomer Feed Composition
(55) The related moment equations are obtained by simple differentiation and yield:
If the monomer feed concentration is subjected to a periodic change, say rno = i i i O ( 1
+ 6 sin ut)
(72)
and if the solution for Equations 62 through 69 is expanded in terms of sin wt, in the limit of slow oscillations, we see that the shift in time-averaged values vanishes such that
i
(59)
= fi =
KO = XI
= Bo =
c1 = 0
(73)
There is no shift in the time-averaged steady-state value of initiator, the monomer, the zeroth and first moments of the polymer, and radical distribution functions. This is evident from the form of the equations describing the system, since the initiator concentration is unaffected by monomer variations and the total polymer and radical concentrations (the zeroth moments) depend only upon the initiator. There is a shift from the steady-state value for the second moment in the distributions. The relations, obtained as in the earlier section, are
Just as we did with the previous mechanism, the equations can be made dimensionless with respect to the steady-state solutions of equations given above,
(74)
Forced Oscillation of Initiator Feed Concentration
dm
- + rn d0
dho db’ dX 1 d0
+
+ A1
- rno
A0 =
(1
= -7mXo
+ s)i -
= -shah1
i - io(1
Tho2
+ (1 + T)mXo
This set of equations is certainly much more involved than the set provided by a simple addition polymerization mechanism. However, a striking difference in the performance of the two systems, because of the difference in the mechanism, is brought out by a study of the forced oscillation of the monomer feed. 430
The last aspect of this problem studied here is the case of periodic variation of the initiator feed concentration. Again only the system response to a very low frequency osciIIation is of interest. I n this instance, Equation 6 2 assumes the form
I L E C PROCESS D E S I G N A N D D E V E L O P M E N T
+ 6 sin w t ) =
- ti
(75)
I n the case of very slow oscillations, Equations 6 2 through 6 9 can be solved for the shift in the time-averaged steady-state value in the monomer concentration and the moment of the polymer and radical distribution. Since the initiation reaction is first order, the shift in the steady-state value of the initiator is invariant when averaged over one period. This is not necessarily the case for the other quantities describing the reaction system. Some of the results for the time-average shift in effluent compositions obtained from the tedious algebra are given below
The relations for the higher moments are extremely long and complex in terms of the parameters describing the system. I n any calculation of the performance of the reactor it is much easier to evaluate the shift in the higher moment for each particular numerical case-that is, given a set of parame.$, 7, NE, and Z R , we differentiate the equations for ters XI, u1, X 2 , and u2 and substitute the numerical values of the derivatives with respect to the variation x for the lower moments and monomer concentration. Below are reported values for the shift in the time-averaged values of these higher moments which were evaluated by the procedure outlined above.
r,
Discussion of Free Radical Mechanism with Combination Termination
The chain termination step in the free-radical mechanism sharply alters the steady-state performance of a polymerization reactor compared with the addition mechanism with no chain stopper (Denbigh, 1947; Kilkson, 1964). The polymer chain length distribution (PCLD) is narrower in the CSTR product than in the batch or plug flow reactor. Periodic operation of a polymerization reactor for the simple addition polymerization alters the PCLD appreciably, but there is a marked change when this mechanism is used. The striking difference between the mechanisms is emphasized by the fact that the system of equations describing the reactions becomes quasilinear when we oscillate the monomer feed concentration. Since the free-radical concentration depends only upon the initiator concentration which is invariant with time, the equation for the change in monomer concentration (Equation 63) and the equations for the first moment of both freeradical and dead polymer distributions (Equations 65 and 68) becorne a set of linear equations which by nature provides no variation in the time-averaged performance of the system. The higher moments are affected, however. For any free-radical mechanism, the dispersion of the freeradical distribution, Z R , is given by (79)
so that for long-chain radicals, Z R 2. The relation is independent of temperature or any other variable controlling the system. Therefore, the time-averaged deviations from the steady state in the case of long-chain radicals are always given by
The result of monomer feed variation is to broaden the distribution. A measure of the broadening is given by Equation 80. I n this case, the quantitative effect is appreciable. For the monomer feed half-wave amplitude of 20% of the steady value, a 2y0 increase in the weight-average degree of polymerization will occur. The best way to control a molecular weight distribution in a free-radical polymerization is to control the rate of initiator addition (Bamford et al., 1958). Variation of the initiator feed concentration does not lead to the gross simplicity of the effects of a periodic variation in the monomer feed concentration. The relations were very complex. Equations 76, 77, and 78 show that for any parameters of the system the monomer effluent concentration is increased, the free-radical concentration decreased, and the dead polymer concentration increased. Without numbers in the relations for the changes in the. higher moments of the polymer chain length distribution function, it is not obvious whether the distribution is sharper or broader. Reaction velocity constants from a particular reaction system will provide both the qualitative and quantitative measure of the change in performance of the reactor in periodic operation. Styrene polymerization conforms best to the mechanism studied and data were used as reported by Bamford et al. (1958) for styrene in benzene solution initiated by azobisbutyronitrile. At 5OoC, the reaction velocity constants are k1 =
1.8 X
(minute)-'
kz = 2.7 X lo3 liters/mole-minute k8 = 2.3 X 108 liters/mole-minute VOL. 7
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In a reactor achieving a monomer conversion of SO%’,, the required residence time is 624 minutes. The number-average degree of polymerization, NR, for the free radicals at steady state was 99. The dispersion of the radical distribution, Z,, was 2.01. The steady monomer and initiator feed concentrations were taken as 10 and 0.25 mole per liter, respectively. For a given monomer conversion, the system was studied a t 50°, 70°, and 100’ C. Table I1 gives the steady-state behavior of the system and the fractional deviation from the steady state a t all three temperatures. Although the distribution was altered markedly, the fractional shifts in the timeaveraged values were only weak functions of the temperature
where a = (it, m’, UO’, UI’, UZ’) and F is the matrix of coefficients. We can find a suitable nonsingular transformation, T, such that
b
=
T-la;
G = T-’FT
and
d -b=Gb de where 0 0 -1 1 (1 2P)-’
+
level. I t should not be inferred, however, that for a nonisothermal reactor no improvement over the isothermal performance could be obtained. Resonant behavior is to be expected and no a priori estimate of the magnitude or direction of the effect can be made. Conclusions
I t was the objective of this work to assess if any appreciable modification of the molecular weight distribution could be obtained by periodic operation of a polymerization reactor. Since the isothermal polymerization reactor has no natural frequency, the low frequency performance provides a good estimate of the magnitude of the modification possible due to periodic operation. For the systems studied, appreciable broadening was observed for variations in either monomer or initiator feed compositions. This provides ample justification for further study of the frequency response of the isothermal and nonisothermal reactor. Periodic operation of a polymerizer could result in a product not obtainable from an isothermal steady-state polymerization reactor. Appendix. Character of Eigenvalues of Polymerization Reactor Equations
For the addition polymerization mechanism, Equations 23 through 27 describe the performance of the CSTR. If there exists any natural frequency for this set of equations, it should be evident for the set of equations linearized about the steady state. These linearized equations are
di ‘ - = de
-(I
+ y)i’ - ym’
du, ‘ - - rn’ + U T ~ ’
dB
- u1’
the set of equations can be written as da - = Fa dB 432
I&EC P R O C E S S DESIGN A N D DEVELOPMENT
0 0
0 -1 2P(1 2p)-1
+
O 1 0
:I
-1
Since G and F are similar matrices, they have the same eigenvalues, 1,. I t is evident from the matrix G that the eigenvalues are
+ 4 1 + 2P) + Y(l + P)l + 4 1 + 2P)l 13=14=&= -1 11
12
= -11 = -[I
All eigenvalues are real and negative, confirming our intuitive notion that the isothermal reactor is stable and has no natural frequency. Acknowledgment
The authors thank the Petroleum Research Fund for support and J. Douglas for his helpful comments. Nomenclature a
= species vector (appendix) y)
-
A
= constant, a(P
b
= species vector (appendix) = constant, (1 y ) ( l cuP)
B
c, c,o -
+
+ + 4 1 + 0)
concentration of species s in vessel concentration of species s in feed stream DP, number average degree of polymerization D = constant, 1 a ( 1 6 ) F = matrix of coefficients (appendix) G = matrix of coefficients (appendix) I = initiator concentration, moles/liter Io = feed concentration of the initiator, moles/liter Io = steady-state feed concentration of initiator, moles/ liter I, = steady-state exit concentration of initiator, moles/liter i = dimensionless initiator concentration dimensionless steady-state initiator concentration in feed stream kl = initiation reaction velocity constant, (minute) -1 kz = propagation reaction velocity constant, liter (mole)-’ (minutes) ka = termination reaction velocity constant, liter (mole) -1 (minutes)-l l j = eigenvalues of linearized equations (appendix) M = monomer concentration, moles/liter go = feed concentration of the monomer, moles/liter 1Mo = steady-state feed concentration of the monomer, moles/liter M , = steady-state exit concentration of the monomer, moles/liter m = dimensionless monomer concentration mo = dimensionless monomer feed concentration Pi0 = dimensionless monomer feed concentration at steady state average fractional deviation steady-state values of monomer = = =
+
+
3
z
degree of polymerization of free radicals concentration of polymer of chain length j , moles/ liter 2-transform #ofthe polymer chain length distribution, moles/liter volume rate of production of species s, moles (liter)-l (minute) -:‘ concentration of free radical of length i, moles/liter Z-transform of the free radical chain length distribution, mole!r/liter species s time, minutes frequency of the oscillation, radiansjminute input oscillatory function (sinusoidal in this case) any dependent variable dispersion defined by. Equation 71 . 2-transform variable
GREEKLETTERS dimensionless constant, kldl, dimensionless constant, kz8M, dimensionless constant, k&4, fractional amplitude of input oscillation dimensionless constant dimensionless constant average residence time of material in vessel, minutes dimensionles!r time dimensionless kth moment of free radicals average fractional deviation from steady-state value of kth moment of distribution of free radical chain length kth moment of distribution of polymer chain length, moles/liter
pkrn
=
Yk
=
vkrn
= =
Ck
=
3k
=
’
=
kth moment of distribution of polymer chain length a t steady state, moles/liter kth moment of distribution of free radical chain length, moles/liter kth moment of distribution of free radical chain length a t steady state, moles/liter dimensionless constant, kld dimensionless kth moment of distribution of polymer chain length defined by Equation 22 average fractional deviation from steady-state value of kth moment of distribution of polymer chain length notation to designate perturbed variables
literature Cited
Abraham, W. H., Ind. Eng. Chem. Fundamentals 2, 221 (1963). Bamford, C., Barb, W., Jenkins, A., Onyon, P., “Kinetics of Vinyl pp. 71, 219, ButterPolymerization by Radical Mechanisms,” worths, London, 1958. Denbigh, K. G., Trans. Faradav Sod. 43. 648 11947). Douglas,’ J. M.; IND.ENG. CHEM.PROCESS DESIGNDEVELOP. 6, 43 (1967). Douglas, J. M., Rippin, D. W. T., Chem. Eng.Sci. 21, 305 (1966). Horn, F. J. M., Lin, R. C., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 6, 21 (1967). Kilkson, H., Ind. Eng. Chem. Fundamentals 3, 281 (1964). Kruissink, Ch. A., Van der Want, G. M., Staverman, A . J., J . f % l p ~ eSci. r 30,67 (1958). Miyake, A., Stockmayer, W. H., Makromol. Chem. 88, 90 (1965). Saidel, G. M., Katz, S., A.Z.CI2.E. J . 13, 319 (1967). Zeman, R. J., Amundson, N. R., A.Z.Ch.E. J . 9,297 (1963).
__
RECEIVED for review October 30, 1967 ACCEPTED March 27, 1968
CHEMICAL REACTORS Intuence of Packing on Effective Reactor Volume A L B E R T R. SHUKI,’ T H O M A S E. CORRIGAN,2 A N D M I C H A E L J. Ohio Staie University, Columbus, Ohio
DEAN2
When an ernpty vessel with a low ratio of length to diameter is used as a continuous flow chemical reactor, the flow pattern involves a considerable amount of backmixing. The empty vessel may behave as the perfect mixer or the mixer with d e a d space. Putting a tower packing in such a vessel will have a twofold effect: the effective volume will b e lessened, and the amount of backmixing will be reduced. The net effect depends upon packing characteristics, packing size, and void volume. The flow pattern will depend upon the packing characteristics and can b e evaluated in terms of the axial dispersion model or the series of tanks model. This paper evaluates the two effects of adding packing for various types of packings.
a packing material to a n empty unstirred vessel of low ratio of length to diameter will considerably reduce the backmixing when the vessel is used as a continuous flow reactor (Bauer and Corrigan, 1967). The increase in capacity due to the lessened backmixing is counterbalanced by the decrease in active volume due to the space occupied by the packing. Whether the net effect of adding packing is to increase or decrease the reactor capacity depends upon the nature of the ADDING
Present address, Standard Oil Co., Cleveland, Ohio. Present address, Mobil Chemical Co., Edison, N. J.
packing, the length to diameter ratio of the vessel, the degree of conversion, and the kinetics of the reaction. The purpose of this paper is to evaluate this effect for various types of packing a t low LID ratios and at various degrees of conversion, Since unstirred vessels with low length to diameter ratios can behave as perfect mixers with up to 18y0dead space, depending upon flow patterns (Sonawala, 1966), the perfect mixer was chosen as the criterion for the vessel with no packing. The vessel with packing was represented by the axial dispersion model. Values of (D/uL) were determined from the packing characteristics and the length to diameter ratio of the bed (Leva, 1951 ; Levenspiel, 1962). VOL 7
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433
