Ind. Eng. Chem. Res. 1992,31,861-868
86 1
Computation of Optimal Temperature Policy for Molecular Weight Control in a Batch Polymerization Reactor Jyh-Shyong Chang* and Jia-Lin Lai Department of Chemical Engineering, Tatung Institute of Technology, Taipei, Taiwan, ROC
This paper proposes a modified two-step method for estimating the reactor temperature in order to obtain a polymer which has a prescribed molecular weight distribution (MWD) in a free-radical polymerization batch reactor. First, profilea of instantaneous average chain length and polydispersity
to give the desired MWD are estimated. Next, the time profile of reactor temperature is obtained via efficient tracking of the profile of instantaneous average chain length only. In this way, the proposed algorithm can be adapted to a more realistic polymerization system. The possibility of achieving the time profile of reactor temperature is also examined.
Introduction The important mechanical properties of polymers have been shown to depend strongly on the molecular weight distribution (MWD). The typical goal in operating a polymerization reactor system is to minimize polydispersity and achieve a minimum number molecular weight around lo6to produce a polymer possessing superior mechanical and rheological properties (Nunes et al., 1982). Nunes et al. (1982) found that thermal properties, stress-strain properties, impact resistance, strength and hardness of films of poly(methy1methacrylate), and polystyrene were all improved by narrowing the MWD. Although the molding characteristics can be improved by blending short-chain polymer into long-chain polymer, it always causes loss of energy. Hence, the development of methodology for adjusting the MWD of a polymer during reaction is desired in many industries. Depending on the practical demand of the high-quality polymer product, polymers with various types of MWD are required (Takamatsu et d., 1988). Several works have reported that narrow MWD can be obtained in batch free-radical polymerization reactors by maintaining an average degree of polymerization (DP) or an instantaneous DP constant throughout the polymerization reaction (Hicks et al., 1969; Osakada and Fan, 1970; Thomas and Kiparissides, 1984; Louie and Soong, 1985; Ponnuswamy et al., 1987). The derived optimization problem, however, required a great amount of iterative calculation on nonlinear differential equations to solve the two-point boundary value problems. It is also necessary to change the calculation technique according to the problem. These calculation techniques usually require simplification of the kinetics involved. A simple and efficient numerical solution is expected to handle more complicated systems. Takamatsu et al. (1988) proposed a method for obtaining the optimal time profile of reactor temperature to obtain a final product with a prescribed MWD. In this report, a two-step calculation method, based on the instantaneous MWD, is proposed. This method can determine the time profiles of reactor temperature and initiator concentration in a general freeradical polymerization batch reactor. If temperature and initiator concentration are maintained along the calculated profiles, a polymer product with prescribed MWD will be obtained at the preestablished monomer conversion rate. The MWD control problem in a styrene suspension polymerization batch reactor initiated by benzoyl peroxide has been successfully demonstrated (Takamatsu et al., 1988). However, in comparison with bulk or near-bulk *To whom all correspondence should be addressed.
0888-5885J92J263l-O86l$O3.OOJO
Table I. Generalized Free-Radical Polymerization Mechanism k
I 4 2R' k. R' + M 2 P'l
P', + M P', P', P', P',
+M +S + P', + P',
k,
% !-
% !-
k!i
-!L
P*n+l
initiator decomposition initiation propagation
D, + P', chain transfer to monomer
D, + chain transfer to solvent D, + D, termination by disproportionation
D,,,
termination by combination
where I = initiator, R' = primary radical, M = monomer molecule, S = solvent molecule, Pom,,= macroradical of length m, n, and D,,, = dead polymer of length m, n
solution polymerization systems such as poly(methy1 methacrylate) polymerization (Chiu et al., 1983; Baillagou and Soong, 1985a,b), the polymerization system studied by Takamatau et al. (1988) was rather simple. More realistic kinetic models need to be used to represent polymerization systems at high conversions. Also, chain transfer, volume contraction, and gel effeots need to be properly incorporated (Louie and Soong, 1985). The gel effect model adopted by Baillagou and Soong (1985a,b) is not a simple explicit function of conversion and temperature as are those cited by Takamatau et al. (1988), Thomas and Kipariasides (1984),and Ponnuswamy et al. (1987). So if we apply the optimization algorithm proposed by Takamatau et al. (1988) for calculating the time profiles of reactor temperature and/or initiator concentration for MWD control of the polymerization based on the kinetic model proposed by Baillagou and Soong (1985a,b), we will be hampered by the difficult numerical solution of highly nonlinear algebraic equations. Hence, in this study, the main objective is to develop a modified two-step method to simplify the calculation procedure for estimating the time profile of temperature for MWD control. It is also necessary to cheese a criterion to judge whether the obtained optimal path is achievable. During the calculation, if the required heat removal exceeds the cooling resources that can be supplied by the existing batch reactor system, the temperature time profile will not be achievable. This problem will be studied in the following derivations.
Summary and Discussions of the Two-step Method for MWD Control The control problem considered here is to find an operating condition to produce a polymer with the prespecified cumulative average chain length and the polydispersity in a free-radical batch polymerization reactor at the final conversion rate. 1992 American Chemical Society
862 Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 Table 11. Objectives Used in M W D Control control objective
derived control objective
For*
X*(tf) PN*(tf) HI(tf)
plf* p*f*
''V(0) [M(0)1 X*(tf)/PN*(tf) V(0) [M(0)1 X*(tf) 'V ( 0 )[M(O)]X * ( t J PN*(tf) HI(tf)
The generalized free-radical polymerization mechanism is shown in Table I. The ith moment of the dead polymer pi is defined as m
pi
= Cni[Dn(t)] V(t) i = 0, 1, 2, n=l
...
(1)
where [Dn(t)] is the concentration of dead polymer of length n at time t and V(t) is the reactant volume at time t. The number-average degree of polymerization PN(tf) and polydispersity HI(tf) of the final polymer product can be represented by po(tf),pl(tf) and p2(tf)at final time tf as PN(tf) = Nl(tf)/CLO(tf) (2)
HI&) po(tf) ~ z ( t f ) / [ ~ i ( t f ) l ~ (3) The final conversion X(tf) is given by X(tf) (V(0) [M(0)1 - v(tf) [M(tf)I)/(v(o)[M(0)1) (4) where [M(O)] and [M(t)] are the concentration of the monomer at time 0 and t. When the quasi-steady-state assumption is applied, pl(tf) is given as pi(tf) = V(0) [M(0)1 - v(tf) tM(tf)l (5) Therefore, the final conversion becomes x(t3= pi(tf)/(V(O) [M(0)1) (6) The control objectives X*(tf),PN*(tf) and HI*(tf),as well as the derived control objectives po*, pl*,pz*, are shown in Table 11. The number-average degree of polymerization fin and polydispersity hi of instantaneous polymer are defined as fin (dpl/dt)/(dpo/dt) (7) (dpo/dt)(drz/iit)/(dpl/dt)2 From eqs 7 and 8, we obtain dp,/dpo = P n hi JL
Equations 13 and 14 give 2 degrees of freedom for designing the possible solution form. Based on this principle, the following three types of solutions of fin&) were studied by Takamatau et al. (1988): (a) rectangular type in terms of p0, (b) second-order polynomial in terms of po, and (c) a mixed type of zero- and firsborder polynomials in terms of Po. After the solution of fin(po)is calculated, T&) which gives the profile of &&) should be calculated on the basis of the mathematical model. The differential equations of the moments po, pl,and p 2 in a batch polymerization reactor are derived from the moment method: dpo/dt = Vl(ktmIM1 + kta[SI)Xo + (ha+ 0.5ktzJXo21 (15) dpl/dt = V[(ktm[Ml + k,[SI)Xl+ kth~hll (16) dpz/dt = V[(ktm[Ml + k,[SI)h + kth0h2 + k J 1 2 1 (17) The ith moment hi of polymer radical P'j is defined as m
h,(t) = Cn'[P*,(t)] n=l
i = 0, 1, 2, ...
(18)
where [P'Jt)] is the concentration of the polymer radical of length n at time t. If the assumption of quasi steady state is applied to the moment of &li4,1,2, the accumulation term of the following three equations can be set to zero. Then the moments of hili=0,1,2 can be derived as
b
= (2fkd[Il/(kk
+ ka))1'2
(19)
(8)
In a free-radical polymerization reaction scheme not including chain-transfer reaction to polymer, a dead polymer does not disappear and the final MWD of the polymer in the batch polymerization reaction becomes a cumulative MWD of instantaneous dead polymer during the operating period. When the derived control objectives hf*, plf*, and p 2 f* are attained, the following equations shall hold:
The original two-step method (Takamatsu et al., 1988) suggested that the MWD control proble-m can be reduced, for the first step, by fiding fin(&) and hi(&)which satisfy eqs 11 and 12 and for the second step, by finding the reactor temperature TR(t)and iniJiator concentration [Ut)] which guarantee the fin(&) and hi(&) obtained at the first step. It is possible that fin and hi can be operated independently by manipulating reactor temperature TR(t)and @tiator concentration [Ut)]. For simplicity of calculation, hi(po)is fixed at the constant value hi. Under this condition, the following two equations must be satisfied.
If eqs 15-21 are substituted into eqs 7 and 8 and the assumption of quasi steady state is used, then
hi=(,)(1+1+E 2-2
+ Z)
(23)
are derived where parameters E and Z are given by
From values of E and Z obtained above, reactor temperature T R and initiator concentration [I] as operating conditions for MWD control are solved by using eqs 24 and 25 and system equations of the solvent [SI and [MI. (26)
where conversion X is given as
Ind. Eng. Chem. Res., Vol. 31,No. 3, 1992 863
V(0)[M(0)1 - V(t) [M(t)l (28) V(0) M O I The method proposed above for calculating the time profiles of TR and [I] is called the two-step method. This twestep method does not require any iterative calculations of differential equations. It requires only iterative calculations on algebraic equations which are deduced from differential-algebraic equations, eqs 24-27. However, the detailed calculation procedure was not presented (Takamatau et al., 1988).
X=
Development of the Modified Two-step Method for MWD Control In order to reduce the mathematical derivation and programming required to obtain time profiles of manipulating variables for M W D control in which more realistic kinetic models are used to describe the free radical polymerization batch reactor system, the following modified two-step method for MWD control is now proposed as follows. Step 1. For simplicity of calculation, hi(po)is fixed at the constant value hi. Any of the following solution forms of Pn(po)can be adopted. type 1:
+
Pn(po) = Pn(0)
+ a+O2
(29)
type 2: when 0 Ipo Ia, Pn(0)
when a,
< po Ipof* Pn(po)
= Pn(0) + Q Z
(30)
type 3: when 0 Ipo Ia, P n ( d
when a,
= Pn(0)
< po Ipof* P n G o ) = Pn(0)
+ a ~ ( -~a,) o
(31) &(O) in the above equations can be calculated using the given initial condition. Given the proper control objectives, any solution form given above can be inserted into eqs 13 and 14 to obtain the undetermined parameters a, and a2 by solving the nonlinear algebraic equations. The resulting optimal path is defined as Pn*(po).If the parameters al and a2 cannot be obtained, it signifies that the given control objectives are too tight and further modification of the control objectives is required. Step 2. The MWD control objective hinges on the accurate tracking of Pn(po). If eq 13 is satisfied accurately along the whole domain [O,pOf*],then eq 14 will also be satisfied. The resulting problem will now be formulated as
lowing equation can be coupled with system equations in mapping the obtained T R ( p o )into TR(t). dt IV[(ktm[MI + k,[SI)X, + (ha + 0.5k,)X,211-' (34) dP0
---
Description of the Proposed Calculation Procedures in Obtaining T R ( t )for MWD Control On the basis of the proposed modified two-step method for MWD control, the algorithm is flexible depending on the user's choice. However, the following proposed calculation algorithm has proven to be acceptable. (a) Given the control objectives, the derived control objective, pof* can be calculated as shown in Table 11. Then discretization of the independent variable along [0 hf*] for about 100 points is performed. The representative optimal path fin*&) at these points can then be calculated depending on the chosen solution form, Le., eq 29,30,or 31. (b) Along each subsection of [0 pof*], three internal collocation points and two end points of each subsection are used (Villadsen and Michelsen, 1978). Nonlinear programming based on the reduced gradient method (Ladson et al., 1980)is executed on the problem defined in eq 32 in search of the optimal TR*(po)at the specified collocation points while the system equations were integrated numerically using the variable-step integration routines, DGEAR,provided in the package of IMSL.During the numerical integration, the required TR(h)at any point along this subsection can be easily calculated by interpolation. The obtained optimal end condition, TR* at one subsection, will be the initial condition for the next subsection. The whole calculation proceeds until pof* is reached. TR*(t) can be obtained by a mapping procedure using eq 34. Determination of Achievable Optimal Temperature Path for MWD Control Given the proper control objectives, X*,PN*, and HI, the optimal path Pn*(po)can be obtained. Execution of the proposed calculation procedure will assure that the optimal temperature policy, TR*(t),can be obtained. Whether the calculated temperature profile is achievable hinges on the capability of heat removal at each instantaneous time when the temperature of the reaction medium is driven along TR*(t).The heat balance equation is given as
Assuming that the temperature of the jacket-side cooling is kept constant and the overall heat-transfer medium, Tj, coefficient, U, is the manipulated variable, then an achievable path is given by (-hHp)kp(T~*)[M]Xo AU(T,*(t) - Tj)
-
Qgenerated
I O
Qremoved
subject to system equations for X, [I], and [SI Using eqs 9,15,and 16,one can obtain
for 0 It Itf (36) However, the value of U is a function of reactor design and polymerization system. So a minimum reasonable constant value, U-, is assumed here in judging whether the optimal path is achievable.
Quation 33 is the model solution which can be calculated once the states of the system, TR(fi0),X ( p o ) , [ 4 ( p o ) ,and [S](po),are known. In the mean time, considering the processing time, t, as a dependent variable, then the fol-
MWD Control of MMA Bulk and Near-Bulk Polymerization Batch Reactor The applicability of the proposed modified two-step method for MWD control is first examined for the MMA bulk and near-bulk polymerization batch reactor initiated by 2,2'-azobis(isobutyronitrile) (AIBN). The mathematical
864 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 Table 111. Parameters and Given Conditions for MMA Bulk and Near-Bulk Polymerization Batch Reactor paremeter casel case2 case3 case4 TR~, "C 50 70 70 70 0.58 0.58 0.58 0.58 f 0.093 0.093 0.093 0.093 U, kJ/ (m*.s.OC) 4.5 4.5 4.5 3.6 wh4, kg 0 0 0.9 WE .,.(solvent = 0 toluene), kg Wb kg 1.27 X 10"' 1.31 X 10"' 7.64 X 1.32 X lo4 Table IV. Desired Final Conversion, Number-Average Degree of Polymerization, Polydispersity, and Instantaneous Polydispersity for the Case Studies Considered in Table I11 casel case2 case3 case4 0.8 0.8 0.8 0.8 X*(td 7000 7000 7000 7000 PN' (tr) 3 3 3 Hiitj 3 1.99 1.99 1.99 hi 1.98
0
300
200
100
T h a (-1 Figure 2. Patterns of &(t) of type 1 (-1, type 2 (--), and type 3 (-.-). Symbls +, 0,and * represent the tracking points. Operating conditionsand control objectives are given in Tables III and IV (case 1). Bulk polymerization. -
._ , - - -
--
7 -
1
1
4
1.51
0 0
2
4
6
XlOd 0-Th W t ob PeLpa, & (d) Figure 1. Patterns of fi,(h)of type 1 (-), type 2 (--), and type 3 Symbols +, 0,and * represent the tracking points. Operating conditionsand control objectives are given in Tables 111and IV (caw 1). Bulk polymerization. (-e-).
model and reaction rate constants of this polymerization are listed in the Appendix. Related parameters and given conditions for MMA bulk and near-bulk polymerization for three case studies are given in Table 111. The desired final conversion, number-average degree of polymerization, and polydispersity are depicted in Table IV. Given the initial operating conditions, the instantaneous polydispersity hi for each case study is also calculated and included in Table IV. hi is assumed to be constant during the course of the reaction. Therefore, MWD control can be attained by manipulating only reador temperature T p Given the control pattern of fill(u,,) and &(t) for case 1( T R ~ = 50 O C , bulk polymerization) are plotted in Figures 1and 2. Using the proposed calculation procedures, the obtained manipulating variable, TR(t),and resulting profiles of X ( t ) for each pattern of D,(t) are shown in Figure 3. The symbols plotted in Figures 1and 2 show that tracking of each predetermined path is quite accurate. Hence the attainment of the control objectives given in Table IV is guaranteed. Because the proposed modified two-step method for M W D control results in a one-parameter tracking problem, i.e., P,(t), the searching routines work efficiently even when facing the complicated and stiff mathematical model required to describe the bulk MMA polymerization behavior. From the time profiles of TR(t) in Figure 3, paths 1and 3 seem to be attained more easily than path 2. This is because the step change of P,(fio) or
Time (rnin)
Figure 3. Time profiles of the reactor temperature and conversion for trackingP,(t) of type 1 (-), type 2 (--I, and type 3 (-*-). Operating conditions and control objectives are given in Tables I11 and IV (case 1). Bulk polymerization. =r--
I at
-2s
'
,
1 7
'
I
XI
_
1M
1so
203
250
303
Time (min)
Figure 4. Time profiles of the instantaneous rate of net heat accumulation in the batch reador for tracking fi&) of type 1 (-1, type 2 (- -1, and type 3 Operating conditions and control objectives are given in Tables I11 and IV (case 1). Bulk polymerization. (-e-).
in path 2 induces a stiff drop in TR(t)at around 190 min. However, heat removal limitations prevent the attainment of these patterns. In Figure 4, the left-hand side of eq 36 is plotted against time. This figure shows that the heat removal criterion given in eq 36 cannot be met along the full run. To demonstrate the applicability of the
PJt)
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 865
' I
--x
0 8L 0 6 1
0 aC
1
'I
I
0 6-
/,
0 4b
0 2 0
o
2m
sm
4.33
ma
-,
ixa
~ u p
ma
Time (min)
Time (minj
Figure 5. Time profiles of the reactor temperature and conversion for tracking P J t ) of type 1 (-), type 2 (--), and type 3 (---). Operating conditione and control objectivesare given in Tables I11 and IV (case 2). Bulk polymerization. 350
em
nm
1803
. ,
Figure 7. Time profiles of the reactor temperature and conversion for tracking P,(t) of type 1 (-), type 2 (--), and type 3 (---). Operating conditions and control objectivesare given in Tables I11 and IV (case 4). 20 wt % solution polymerization. 0
1
.................................................
--.__ --.*
,
-
:I
fl - -y" ____ ___----------
0
o
w
~
m
m
a
m
m
x
a
~
~
~
o
~ xa e
YXI
Time (min)
Figure 6. Time profiles of the reactor temperature and conversion for tracking P,,(t) of type 1 (-1, type 2 (--I, and type 3 (-*-). Operating conditione and control objectivesare given in Tables I11 and IV (case 3). Bulk polymerization.
proposed modified two-step method for MWD Control further, the initial reactor temperature is raised from 50 to 70 OC, while all other operating conditions are kept constant. The time and conversion profiles for an initial reactor temperature of 70 O C are depicted in Figure 5. The control objectives given in Table IV are all met successfully. If the amount of initiator initially fed is reduced kg for an initial reactor from 1.31 X lo4 to 7.64 X temperature of 70 "C, the time and conversion profiles depicted in Figure 6 can be obtained. The required time profiles of TR(t)for path 2 depicted in Figures 5 and 6 eIongate much longer than those for paths 1 and 3. This is mainly attributed to a low rate of initiator decomposition induced by lower TR(t)for path 2 which makes the terms in the right-hand side of eq 34 much larger for path 2. Unfortunately, cases 2 and 3 are not achievable. The plots about this point are omitted for brevity. Further, Figure 7 shows the time profile of TR(t) and X ( t ) for the 20% solution polymerization case study. Figure 8 depicts that all of these three paths are achievable. Hence the actual control implementation can be assured beforehand.
MWD Control of MMA Solution Polymerization Batch Reactor To examine the flexibility and applicability of the proposed modified two-step method for MWD control, the mathematical model and related kinetic parameters
a
e a
803
im
tzm
1100
~ w o IBM
-
l
o m
Time (min)
Figure 8. Time profiles of the instantaneous rate of net heat accumulation in batch reactor for tracking P&) of type 1 (-), type 2 (--), and type 3 Operating conditions and control objectives are given in Tables I11 and IV (case 4). 20 wt % solution polymerization. (-e-).
(Ponnuswamy et al., 1987) describing the behavior of MMA solution polymerization were adopted to calculate the time profiie of TR in achieving the required objectives. The kinetic parameters for MMA solution polymerization, operating conditions, and MWD control objectives are shown in Table V. The phenomena of gel effect and volume contraction are not included in this solution polymerization model. Hence the developed program can be easily adapted by neglecting these two effects. The results for tracking the path pattern of type 1are shown in Figurea 9 and 10 for two different initiator loadings in 60 w t % solution polymerization. The time profiles of TR are achievable. The experimental demonstrations in the work of Ponnuswamy et al. (1987)based on the same mathematical model can partially support this judgment.
Comparison of the Two-step Method and the Proposed Modified Two-step Method in Calculating the Optimal Temperature Policy Takamatsu et al. (1988)demonstrated the applicability of their two-step method for a MWD control problem in a styrene suspension polymerization batch reactor initiated by benzoyl peroxide. At any point po between 0 and hf*, E and Z can be determined using eqs 22-23. Then, solution of the four nonlinear algebraic equations, i.e., eqs 24-27,can give the staka TR,X,[SI,and [I]at this specific
866 Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 Table VI. Physical Properties of the Polymerizing Mixture (Reprinted with Permission from Baillagou and Soong (1985a) Copyright 1985 Pergamon Press PLC) &/(kg/m3) = 966.5 - 1.1(T/"C) ps/(kg/m3) = 883.0 - 0.9(T/OC) C,M/(kJ/(kg'C)) = 1.675 C, s/ (kJ/ (ke"C)) = 2.240 WM'/(kg/kmol) = 100.12 W,'/(kg/kmol) = 92.14 -AH/(kJ/kmol) = 5.65 X lo4 assuming PPMMA (kg/m3) = 1200
Table V. Kinetic Parameters and Given Conditions for MMA Solution Polymerization Batch Reactor Kinetic Parameters kd/@ = 3.75 X 10l6exp[(-1.390 X lo6 (kJ/mol))/R(T/K)] k,/(m3/(s.kmol)) = 1.20 X log exp[(-4.025 x lo4 (kJ/kmol))/R(T/K)] k,/ (m3/ (~.kmoi))= 2.113 X lo8 exp[(-4.242 X lo3 (kJ/kmol))/R(T/K)]
kt = k d + k, k,, = 3.88 X 10" exp[(-1.154 X lo5 (kJ/kmol))/R(T/K)] k, = 4.41 X loB exp[(-1.419 X lo6 (kJ/kmol))/R(T/K)] exp[(1.712 X lo4 (kJ/kmol))/R(T/K)] k,/kd = 3.956 X par ameter case 1 case 2 Operating Conditions 70 70 0.58 0.58 0.093 U,k J / ( m 2 d C ) 0.093 1.71 WM, kg 1.71 Ws (solvent = toluene), kg 2.565 2.565 WI, kg 3-78 x 10-4 2.5 x 10-4
Control Objectives 0.5 500 2 1.989
X*(t3 PN*(tf) Hut3 hi
cpPMMA
t
=
cpM
= -0.1946 - 0.916 X 10-3(T/"C)
where AH = enthalpy of PMMA polymerization, c = volume expansion factor, WM', WS' = molecular weights of monomer and solvent, pM, ps, p p = densities ~ of monomer, solvent, and polymer, and C, M, C, s, C, PMMA = heat capacities of monomer, solvent, and polymer
Table VII. Composition Variables and Mixture Physical Properties in Terms of Conversion, Temperature, and Solvent Fraction (Reprinted with Permission from Baillagou and Soong (1985~).Copyright 1985 Society of Plastic Engineers)
0.5 500 2 1.989
y
4M = 1 - x
o 2 r
-5
,/'
4s = s'(PM'PS) - volume fraction Y
us =
I
S' -
weight fraction 1+s' where p = density of the polymerizing mixture, C, = heat capacity of the polymerizing mixture, s' = s/(1 - s) (s is the weight fraction of solvent in the feed), and y = 1 + cX + S'(PM/PS)
Table VIII. Intrinsic Kinetic Rate Constant Used in the Model (Reprinted with Permission from Baillagou and Soong (1985a). Cowright 1985 Peraamon Press PLC)O f = 0.58 kdo/s-l = 1.053 X 10l6exp[(-1.283 X lo6 (kJ/kmol))/R(T/K)] k,/(m3/(s.kmol)) = 4.917 x lo5 exp[(-1.821 X lo4 (kJ/kmol))/R(T/K)] k,/(m3/(s.kmol)) = 9.800 X lo7 exp[(-2.934 x lo3 (kJ/kmol))/R(T/K)] k,/k, = 9.480 X lo3 exp[(-5.601 X lo4 (kJ/kmol))/R(T/K)] k,/k,, = 1.010 X lo3 exp[(-4.772 X lo4 (kJ/kmol))/R(T/K)] k, = k,, + k , k,,/kdo = 3.956 X exp[(1.712 X lo' (kJ/kmol))/R(T/K)]
t
kd = kdo in the text of this work.
-39
io0
Za,
300
400
xa
5m
Time (min)
Figure 10. Time profiles of the instantaneous rate of net heat accumulation in the batch reactor tracking Pn(t)of type 1: (-) [I] = 0.075 kmol/m3; (- - - - -) [I] = 0.05 kmol/m3. Operatir4 cenrlitions and control objectives are given in Table V (cases 1 and 2). 60 w t % solution polymerization.
w0. It is well-known that the Newton-Raphson method ( V i d s e n and Michelsen, 1978) is usually used for solving
nonlinear algebraic equations. Because the gel-effect induced termination rate constant cited by Takamatsu et al. (1988) is expressed as a simple function of TR and X explicitly 12, = k,,[exp(-BX - C P - DX3)I2 (37) and B, C, and D are functions of TR only, hence the functional Jacobian matrix can be formulated and calculated easily by applying the Newton-Raphson method. This is not the case in the bulk or near-bulk solution polymerization of MMA considered in this work. Although the physical properties of the polymerizing mixture and the volume contraction of reaction medium are functions of TR and X explicitly (Tables VI and VII), the gel-effect induced rate constants, k, and k, (Tables VI11 and IX and
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 867 Table IX. Gel-Effect Constitutive Equations (Reprinted with Permission from Chiu et al. (1983). Copyright 1983 American Chemical Society)
k, =
k, 1 + k,~,Xo/D
with 0.168 - 8.21 X 104(T - T,)’
A
B = 0.03
e,
=
1.1626 X lo5 (kJ/kmol)
R ( ( T / O C ) + 273.15) where Tm = the glass transition temperature of PMMA (114 “C), dJ, = the volume fraction of polymer, k, = the intrinsic rate constant k , in Table VIII, and k, = the intrinsic rate constant kt in Table VI11
eq 19) are rather complicated functions of TR,X, [I], [SI, and the physical properties of the polymerizing mixture. Hence, the functional Jacobian matrix cannot be formulated and calculated easily. Thus the applicability of the original two-step method (Takamatsu et al., 1988) in the polymerization system studied in this work possesses numerical difficulties. However, the proposed modified two-step method can be applied to both simple and complicated polymerization systems, as demonstrated in the above two sections. The proposed method is motivated by the fact that the solution of complicated nonlinear algebraic equations can also be obtained by integration of ita original dynamic system equations. Conclusions In this work, a modified two-step calculation procedure for estimating the reactor temperature to give the desired average chain length and polydispersity of polymer at the desired final conversion in a polymerization batch reactor has been proposed. The proposed method formulates the M W D control problem into a one-parameter searching problem. Hence the complicated numerical computations required to solve the nonlinear algebraic equations of the original two-step method for MMA bulk polymerization batch reactors can be avoided. The proposed method is accurate and effective. The adaptation of this calculation procedure to different mathematical models shows this procedure to be very flexible. The proposed criterion to judge whether or not the optimal time profile of the reactor temperature is reliable plays an important role in the actual implementation of the obtained optimal path in achieving the predefined control objectives. Acknowledgment We gratefully acknowledge the support provided by the as well National Science Council (NSC 79-0405-E-036-06) as Tatung Institute of Technology, Taipei, Taiwan, ROC.
Nomenclature A = surface area available for heat transfer, m2 C, = heat capacity of the polymerizing mixture, kJ/(kg.K) [D,]= concentration of dead polymer of length n,kmol/m3 f = initiator efficiency HI&) = polydisperity of final product hi = polydispersity of instantaneous polymer -AHp = enthalpy of PMMA polymerization, kJ/kmol [I] = initiator concentration, kmol/m3 kd = rate constant of initiator decomposition, (l/s) ki = rate constant of initiation, m3/(s.kmol) k,, = intrinsic rate constant of propagation, m3/(s.kmol) k, = rate constant of propagation, m3/(s.kmol) k , = intrinsic rate constant of overall termination, m3/(s.kmol) k , = rate constant of overall termination, m3/(s.kmol) k,, = intrinsic rate constant of termination by combination, m3/ (snkmol) k, = rate constant of termination by combination,m3/(~kmol) kd,, = intrinsic rate constant of termination by disproportionation, m3/(~kmol) ka = rate constant of termination by disproportionation, m3/(s.kmo1) k,, = ratio constant of transfer to monomer, m3/(s.kmol) k , = rate constant of transfer to solvent, m3/(s.kmol) [MI = monomer concentration, kmol/m3 PN(tf)= number-average degree of polymerization of final product p, = number-average degree of polymerization of instantaneous polymer [SI = solvent concentration, kmol/m3 T.= jacket temperature, O C or K = reactor temperature, oc or K t = time, s U = overall heat-transfer coefficient, kJ/ (m2.s.K) V = total volume of reaction medium, m3 W .= total mass of fluid in jacket, kg = total weight of monomer in feed, kg WM’= molecular weight of monomer, kg/kmol Ws’= molecular weight of solvent, kg/kmol X = monomer conversion Greek Symbols pi = ith moment of the concentration of dead polymer X i = ith moment of the concentration of polymer radical p = density, kg/m3 Superscripts * = desired value Subscripts b = batch-side f = final i = intial j = jacket-side M = monomer min = minimal P = polymer S = solvent
TR
dM
Appendix: Mathematical Description of MMA Bulk and Near-Bulk Polymerization Batch Reactor The Material Balances. [I] dV -411 - -kd[I] - dt V dt
-
-dX = (kp + k d l - X)X, (A21 dt The other balance equations of the moments po, pl,and pz are given in eqs 15-18, while the moments of )b, AI, and X2 are given in the eqs 19-21 in the text of this study.
Ind. Eng. Chem. Res. 1992,31,868-876
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The volume contraction of the reaction medium is induced by the density difference between monomer and polymer and is also influenced by the conversion, the temperature of the reaction medium, and the amount of solvent added. The volume of reaction medium, V, is given by
where WM is the weight of monomer added initially; t is the volume expansion factor determined by t = (pM pp)/pM; s is the weight fraction of the solvent in the feed; pM, ps, and pp are densities of the monomer, solvent, and polymer, respectively. The physical properties of the polymerization mixture are shown in Table VI. Table VI1 gives the composition variables and mixture physical properties in terms of conversion, temperature, and solvent fraction. The intrinsic kinetic rate constant as well as gel-effect constitutive equations used in the model are shown in Tables VIII and E.The gel model proposed by Chiu et al. (1983) is suitable for describing the polymerization behaviors of bulk or near-bulk (20% solvent in weight) MMA polymerization batch reactors. The Heat Balance. The heat balance equation is given in eq 35 of this study. Registry No. MMA, 80-62-6; PMMA, 9011-14-7.
Literature Cited Baillagou, P. E.; Soong, D. S. Major Factors Contributing to the Nonlinear Kinetics of Free-Radical Polymerization. Chem. Eng. Sci. 1985a, 40, 75-86.
Baillagou, P. E.; Soong, D. S. Molecular Weight Distribution of Products of Free Radical Nonisothermal Polymerization with Gel Effect. Simulation for Polymerization of Poly(Methylmethacrylate. Chem. Eng. Sci. 1985b, 40,87-104. Baillagou, P. E.; Soong, D. S. Free-Radical Polymerization of Methyl Methacrylate in Tubular Reactors. Polym. Eng. Sci. 1985c, 25, 2 12-231. Chiu, W. Y.; Carratt, G. M.; Soong, D. S. A Computer Model for the Gel Effect in Free-RadicalPolymerization. Macromolecules 1983, 16, 348-357. Hicks, J.; Mohan, A.; Ray, W. H. The Optimal Control of Polymerization Reactors. Can. J. Chem. Eng. 1969, 47, 590-597. Ladson, L. S.; Waren, A. D.; Ratner, M. W. GRG2 User’s Guide; Department of General Business, University of Texas: Austin, TX, 1980. Louie, B. R.; Soong, D. S. Optimization of Batch Polymerization Processes-Narrowingthe MWD. I. Model Simulation. J. Appl. Polym. Sci. 1985, 30, 3707-3749. Nunes, R. W.; Martin, J. R.; Johson, J. F. Influence of Molecular Weight and Molecular Weight Distribution on Mechanical Properties of Polymers. Polym. Eng. Sci. 1982, 22, 205-228. Osakada, K.; Fan, L. T. Computation of Near-Optimal Control Policies for Free-Radical Polymerization Reactors. J.Appl. Polym. Sci. 1970,14, 3065-3082. Ponnuswamy,S. R.; Shah,S. L. Computer Optimal Control of Batch Polymerization Reactors. Znd. Eng. Chem. Res. 1987, 26, 2229-2236. Takamatsu, T.; Shioya, S.; Okada, Y. Molecular Weight Distribution Control in a Batch Polymerization. Znd. Eng. Chem. Res. 1988, 27,93-99. Thomas, I. M.; Kiparissides, C. Computation of the Near-Optimal Temperature and Initiator Policies for a Batch Polymerization Reactor. Can. J. Chem. Eng. 1984,62, 284-291. Villadsen, J.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice-Hall, NJ, 1978. Received for review May 22, 1991 Accepted November 21, 1991
Dynamic Simulation of Catalytic Reactors Using the UCKRON-I Test Problem as the Kinetic Model Ferenc Z.TBtrai*J Department of Chemical Technology, Technical University Budapest, H-1521 Budapest, Hungary
Ern6 Varga and P a u l Benedek Department of Chemistry, Eotvos University, B P 112, P.O. Box 32, H-1518 Budapest, Hungary
Dynamic models were developed to investigate the behavior of the perfectly mixed methanol synthesis reactor. The temperature of the catalyst and that of the gas were always regarded as variables, and the heat capacities of both phases were taken into account as parameters. The continuous stirred tank reactor (CSTR) model is able to predict a temperature jump of the reactor from a low-temperature steady state to a high-temperature one. We have studied the following problems: the existence of multiple steady states and the phenomenon of hysteresis; the dynamic response of the reactor to step changes of the inlet parameters, with special emphasis on the dynamics of crossing the border of the multiple-steady-state domain; and detecting and forecasting temperature jump situations, leading from a low-temperature steady state to a high-temperature one. Introduction The UCKRON-I test problem was offered by Berty et ale (1989) for the mathematical of a reaction system on a computer. In this work uexperimentalw were gained by computer of the system in a continuous stirred tank reactor (CSTR), reaction mechanism and adding normally using a ‘Present address: Department of Chemical Engineering, The University of Queensland, Queensland 4072, Australia.
distributed random errors to the simulated results. The explicit rate equation for the true model was developed without assuming a rate-controlling step. Different research groups tried to identify the true mechanism on the basis of the best agreement between the published experimental resulta and the resulta of different, fitted kinetic expressions. The true kinetics was also published at the Denver AIChE meeting in August 1983 (Berty, 1989). According to our present knowledge, most of the published modeling work has been concentrated on the steady-state behavior of the system. Berty (1989) also
0888-5885/92/2631-0868$03.00/00 1992 American Chemical Society
