Ind. Eng. Chem. Res. 2007, 46, 7677-7683
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Continuous Emulsion Copolymerization of Acrylonitrile and Butadiene: Simulation Study for Reducing Transients during Changes of Grade Roque J. Minari,† Luis M. Gugliotta,† Jorge R. Vega,†,‡ and Gregorio R. Meira*,† INTEC (UniVersidad Nacional del Litoral-CONICET), Gu¨emes 3450, Santa Fe 3000, Argentina, and UniVersidad Tecnolo´ gica Nacional (Facultad Regional Santa Fe), LaVaisse 610, Santa Fe 3000, Argentina
This article theoretically investigates the reduction of off-specs generated during changes of grade in an emulsion copolymerization of acrylonitrile (A) and butadiene (B) performed in an industrial train of eight continuous stirred tank reactors (CSTRs). The investigated nitrile rubber (NBR) grades exhibit differences in the copolymer composition, molecular weights, and levels of branching. In the Normal plant configuration, all the reagents are fed into the first reactor. In the Improved plant configuration, intermediate additions of A, B, and chain-transfer agent (CTA) are admitted along the train for increasing production and reducing chain branching, polydispersity, and compositional drift. Optimal transitions between the Normal steady states and the Improved steady states were calculated. They involved changes between two typical copolymer grades and the application of “bang-bang” feed profiles onto A, B, and CTA. According to the model, the bangbang strategies reduce the off-specs 35%-60%, with respect to simultaneous step changes in all of the reagents. Introduction Nitrile rubber (NBR) is a random copolymer produced by copolymerizing acrylonitrile (A) and butadiene (B) in a “cold” emulsion process. The reaction can be conducted in batch, semibatch, or continuous reaction systems.1 The quality of an NBR grade is determined by the molecular weights, the copolymer composition, and the level of long-chain branching. The chemical composition is possibly the most important quality variable,2 because of the sharp (and almost linear) increase of the glass-transition temperature (Tg) with the content of A in the copolymer.3 The content of A also increases the rubber resistance to fuels and hydrocarbons, to abrasion, and to gas permeation.1,2 The Mooney viscosity (MV) is an important process variable, from the point of view of the rubber processability,1 and it is a complex function of the molecular weights, degrees of branching, and chemical composition.4 Typical NBR grades are classified according to the following values of Tg and MV: (a) AJLT, -56 °C and 40-50 MS; (b) BJLT, -37 °C and 40-50 MS; (c) BJJLT, -37 °C and 20-30 MS; and (d) CJLT, -27 °C, and 40-50 MS. Ideally, all of these materials should exhibit narrow composition distributions. In grade CJLT, the mass fraction of A is close to the azeotropic composition (which is 38%),5 and, therefore, batch reactors are perfectly adequate. In grades BJLT and BJJLT, the batch production generates a copolymer with a moderately decreasing A content. Grade AJLT requires semibatch control of the composition, to avoid the large composition drifts that would be otherwise produced in batch polymerizations. A batch reactor is equivalent to a train of many continuous stirred tank reactors (CSTRs) operating in the steady state (SS). In addition, a semibatch operation is equivalent to intermediate SS injections of some of the reagents along the continuous train. The NBR emulsion process has been modeled by Dube´ et al.,6 Vega et al.,7 and Rodrı´guez et al.8 Vega et al.7 investigated the batch production of grade BJLT with tert-dodecyl mercaptan as CTA; observing that, while the number-average molecular * To whom correspondence should be addressed. Tel.: 0054-342451-0348.Fax: 0054-342-451-1079.E-mailaddress:
[email protected]. † INTEC (Universidad Nacional del Litoral-CONICET). ‡ Universidad Tecnolo ´ gica Nacional (Facultad Regional Santa Fe).
weight (M h n) remained essentially constant along the reaction, both the weight-average molecular weight (M h w) and the average number of trifunctional branches per molecule (B h n3) steadily increased. In addition, an almost-constant copolymer composition was produced by the semibatch addition of discrete pulses of the faster reacting comonomer (A). In the work of Gugliotta et al.9 and Vega et al.,10 the model by Vega et al.7 was combined with calorimetric measurements to estimate the monomer conversion (x), average molecular weights, and average degree of branching. Vega et al.10 also developed a model-based semibatch strategy for controlling the main molecular characteristics. Several publications have investigated the SS optimization of continuous emulsion processes. Poehlein and Dougherty11 adjusted the mean residence time of a single CSTR to maximize the number of generated polymer particles (Np) and, therefore, polymer production (G). Following an original suggestion by Hamielec and MacGregor,12 Vega et al.13 and Minari et al.14 investigated a styrene-butadiene rubber (SBR) process conducted in a train of CSTRs (a) to maximize the SS production by reducing the total “inert” monomer droplets phase in the first reactors of the train through intermediate addition of the comonomers mixture into such reactors, and (b) to improve the final rubber quality by intermediate feeds of CTA along the train. In Minari et al.,15 the semibatch NBR model by Vega et al.7 was extended to optimize the SS production of grades BJLT and AJLT in a train of eight CSTRs. In the Normal plant configuration, all the reagents were fed into the first reactor, and the final molecular properties were somewhat deteriorated, with respect to equivalent batch operations. In the Improved plant configuration, intermediate additions of A, B, and CTA along the train were admitted to improve rubber quality and increase polymer production.15 In many continuous processes, large amounts of off-spec product are often generated between SSs, because of the frequent changes of grade and/or in the level of production. For changes in production for the same grade, the off-spec product can be almost eliminated by appropriate manipulation of the total feed flow (qT).14,16 For changes of grade at a fixed level of production, more-complex strategies are required. For a continuous SBR process, Vega et al.16 and Minari et al.14 minimized the off-
10.1021/ie070392j CCC: $37.00 © 2007 American Chemical Society Published on Web 07/21/2007
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specs during changes of grade where the SS specifications involved different molecular weights and degrees of branching but a common copolymer composition. The operation required the entire recipe into the first reactor to be changed at t ) 0, and it required the introduction of an independent transient feed of CTA into the last reactor of the train.16 While rapid transitions were observed for the final M h n, the other molecular specifications (M h w and B h n3) remained out of specification for longer periods.16 Vega et al.16 also presented a control policy that changed the entire SBR recipe into the first reactor at t ) 0, except for the CTA, which exhibited a “bang-bang” feed profile with an intermediate (constant) value. The intermediate constant values were maximum for changes of grade that required a decrease in the final molecular weights, or were minimum for increased final molecular weights.16 For the continuous production of NBR in a reactor train under the Normal plant configuration, a brief introduction into the problem of reducing the off-spec product generated during changes of grade was considered in Minari et al.17 To this effect, suboptimal bangbang profiles for the comonomers and CTA feeds into the first reactor were calculated, without application of an explicit optimization algorithm. Bang-bang controls are optimal when the manipulated variables operate at their limiting values along the transient.18 The present article is a continuation of Minari et al.,15 and its objective is to minimize the off-spec product generated during changes of grade between Normal or Improved SSs. The changes of grade involve not only different final molecular weights and degrees of branching, but also different final copolymer compositions. Investigated Plant and Optimal Steady States (after Minari et al.15) Reconsider the industrial NBR process simulated in Minari et al.15 It consisted of a train of eight identical CSTRs operated at 10 °C, with each reactor exhibiting a reaction volume of 17 473 L. The final product was periodically sampled and the following was measured: monomer conversion, average copolymer composition, and MV. The final conversion was limited to ∼80%, to control branching, crosslinking, and compositional drift. Taken from Minari et al.,15 Table 1 reproduces the SS recipes and final conditions of grades AJLT and BJLT. In the Normal SS configuration, all the reagents were fed into the first reactor. In the Improved SS configuration, intermediate feeds of A, B, and CTA were admitted (see Table 2). Compared with grade BJLT, grade AJLT requires higher initiator and emulsifier concentrations. The final rubber quality is represented by the molecular weights, average branching (B h n3), weight-average mass fraction of A in the copolymer (pjA), and the ratio pjA/pjAn (where pjAn is the number-average mass fraction of A in the copolymer; and the ratio pjA/pjAn represents the polydispersity of the copolymer composition distribution, given as the copolymer mass versus its mass fraction of A). With respect to the Normal SSs, the Improved SSs exhibit higher polymer productions (G), and reduced values of B h n3, M h w/M h n, and pjA/pjAn.15 General Considerations The industrial specifications of pjA and MV are given at the bottom of Table 1c. The SS grades exhibit a common specification for the MV but differences in pjA and in the molecular characteristics M h w, M h w/M h n, and B h n3 (see Tables 1b and 1c). The MV range of grade AJLT coincides with that of grade BJLT,
Table 1. Normal and Improved Steady States (SSs) of Nitrile Rubber (NBR) Grades BJLT and AJLT (after Minari et al.15): Recipes and Global Simulation Results normal SSs parameter
BJLT
AJLT (a) Recipe 321.7 85.0 15.0 170.2 4.18 0.0156 0.300
improved SSs BJLT
AJLT
362.7b 67.1c 32.9c 166.4 3.41 0.0093 0.427c
362.3b 83.3c 16.7c 166.9 4.10 0.0191 0.316c
x [%] G [kg/min] M h n [g/mol] M h w [g/mol] M h w/M hn B h n3 [molecule-1] pjA [%] pjA/pjA,n
(b) Final Product Characteristics 72.7 74.1 71.8 72.2 72.7 81.7 61000 74700 59300 210000 349000 199000 3.44 4.67 3.36 0.467 0.643 0.436 34.8 18.7 34.6 1.04 1.20 1.00
74.5 83.3 71600 311000 4.34 0.573 18.9 1.00
MV [MS] pjA [%]
(c) Process Quality Specifications 45.0 ( 5.0 45.0 ( 5.0 45.0 ( 5.0 34.6 ( 2.5 18.9 ( 2.0 34.6 ( 2.5
45.0 ( 5.0 18.9 ( 2.0
qT [L/min] B [pphm]a A [pphm]a water [pphm]a emulsifier [pphm]a initiator [pphm]a CTA [pphm]a
321.7 68.6 31.4 170.2 3.49 0.0096 0.409
a Parts per hundred monomer. b Outlet flow rate. c Total feed concentration (see Table 2 for feeds into each individual reactor).
because the larger average molecular weights and degree of branching of the former are compensated by its reduced contents of A. Thus, the MV is of little use for analyzing transitions between grades AJLT and BJLT; instead, variations of (10% h w/M h n, and B h n3 are considered (in around the SS values of M h w, M addition to the quality specifications for pjA given in Table 1c). Unfortunately, no quantitative correlations between MV and the molecular characteristics are available and the molecular specifications are generally not routinely measured in industry. However, it was known that the proposed common (10% variations in the molecular variables were sufficiently adequate to maintain each of the grades within specifications. The investigated process was simulated with the same model and model parameters applied by Minari et al.15 The model is an extension of the batch model proposed by Vega et al.,7 and it is based on a kinetic mechanism that considers reactions in both the aqueous and polymer phases. In the aqueous phase, it includes redox initiation, homopropagation of A, termination of polyacrylonitrile radicals, radical deactivation by O2, and oxidation of the ferrous activator by O2. In the polymer phase, it includes propagation, termination, chain transfer to the comonomers, CTA and polymer, reactions with internal double bonds, and reaction with impurities such as O2. The model assumptions include the following: (1) The polymer particles are generated by the combined mechanisms of micellar and homogeneous nucleations; (2) The latex particles are monodisperse in size; (3) The low-molecular-weight species (i.e., A, B, initiator, and CTA) are distributed between three phases (aqueous, monomer, and polymer), according to constant partition coefficients; (4) Primary A radicals can desorb from the polymer particles; (5) The instantaneously produced polymer exhibits an average copolymer composition calculated from the consumption of the reacted comonomers in polymer and aqueous phases, and the cumulative copolymer composition distribution is obtained by accumulation of the corresponding instantaneous distribution; (6) The moments of the MWD are calculated by assuming a pseudo-bulk homopolymerization, and the molecular weights are determined by chain-transfer reactions;
Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7679 Table 2. Improved Steady States (SSs): Stationary Feeds of A, B, and CTA (≡ X) along the Reactor Train (after Minari et al.15) grade BJLT reactor, r 1 2 3 4 5 6 7 8
F(r) A
[kg/min] 4.88 3.39 3.23 3.16 1.78 0.90 0.77 0.65
F(r) B
[kg/min] 27.34 19.91 19.40 19.94 7.33 0 0 0
grade AJLT F(r) X
[kg/min]
0.1287 0.0857 0.0643 0.0456 0.0189 0.0032 0.0039 0.0034
(7) Trifunctional branches are produced by chain transfer to the polymer (which are, in turn, caused by hydrogen abstraction from B repetitive units); (8) Tetrafunctional branches are produced by reaction between growing free radicals and internal double bonds of B units; and (9) There is a negligible generation of gel. The mathematical model was first adjusted to the following measurements obtained in an industrial batch reactor: average particle diameter, x, pjA, M h n, and M h w.7 These values were later validated with measurements from the same reaction, but conducted in the described continuous plant under Normal SS configuration (not included here, for proprietary reasons). Consider some general ideas on the transient control between the Normal and Improved SSs (Table 1). The changes of grade BJLT f AJLT involve increasing the molecular weights and decreasing the average copolymer composition pjA. This requires an increase in the inlet concentrations of B, the emulsifier, and the initiator, and it requires a simultaneous decrease in the inlet concentrations of A and CTA (see Table 1). For changes of grade between Normal SSs, bang-bang profiles for the feed flows of A, B, and CTA into the first reactor were proposed. They involved intermediate constant feed flows of A, B, and CTA between their initial and final SS values, applied during a (1) common period starting at t(1) i and finished at tf , where the superscript “1” indicates the first reactor. For changes of grade between Improved SSs, delayed bang-bang profiles for A, B, and CTA were proposed, successively applied along the reactor train. In this case, the intermediate constant flows were applied (r) during common periods (t(r) i , tf ), with r indicating the reactor number (r ) 1, 2, ..., 8). Independently of the feeds and plant configuration, the off-spec product was collected along a period (t1, t2), with switching times that were defined as follows: t1 is the time when the first quality variable (pj(8) h (8) h (8) h (8) A , M w , M w /M n , or (8) B h n3 ) fell out of specification of the first grade, and t2 is the time when all the quality variables entered into specifications of the final grade. In all cases, the feed flows of A and B were combined in the sense that, when the mass flow of A was increased by a certain quantity, then the mass flow of B was decreased by the same quantity, and vice versa. This procedure not only accelerated the copolymer composition transitions, but also reduced the oscillations in the polymer production and monomer conversion (by imposing a constant feed flow for the total comonomers). Changes of Grade between Normal Steady States Consider minimizing the off-specs generated between Normal SSs (Table 1), through the following control strategy: (a) At t ) 0, all the reagent feeds were changed to their final (1) values, except for the feed flows of A, B, and CTA (F(1) A , FB , (1) and FX , respectively), which were maintained at their initial (1) (1) (1) SS values (FA,SS1 , FB,SS1 , and FX,SS1 ) until t ) t(1) i (g 0);
F(r) A
[kg/min]
F(r) B [kg/min]
F(r) X [kg/min]
22.21 17.54 16.94 16.11 4.52 0 0 0
0.1642 0.1204 0.0958 0.0749 0.0280 0 0.0072 0.0015
11.17 7.58 7.36 6.94 2.65 0.87 0.74 0.62
(b) At t ) t(1) i , the manipulated variables were changed to (1) (1) their intermediate bang-bang values (F(1) A*, FX*, and FB* ) (1) (1) (1) (1) (1) (FA,SS2 +FB,SS2) - FA*), where FA,SS2 and FB,SS2 were the final (1) SS feeds of F(1) A and FB , respectively; and (1) (1) (1) (c) At t ) tf , FA , F(1) B , and FX are adjusted to their final SS (1) (1) (1) values FA,SS2, FB,SS2, and FX,SS2, respectively. The optimization problem aimed at minimizing the mass of polymer produced between the previously defined switching times t1 and t2. The four adjustable parameters were the intermediate flows F(1) A* (1) (1) and F(1) X*, and the time (ti and tf ). In addition, each intermediate flow was limited between zero and twice the highest SS (1) value, and F(1) B* was calculated from FA* to produce a constant comonomers flow. The numerically algorithm is formalized as follows:
min
(1) (1) (1) t (1) i ,t f ,F A*,F X*
∫t t
{
2
G(8)(t) dt}
(1a)
1
with (1) (1) 0 e F (1) i* e 2 max{F i,SS1,F i,SS2} (i ) A, B, X)
(1b)
(1) (1) (1) F (1) A* + F B* ) F A,SS2 + F B,SS2
(1c)
(8) (8) e pj(8) jA,SS1,max ∧ t1 ) max{t:pjA,SS1,min A (t) e p (8) (8) M h w,SS1,min eM h (8) h w,SS1,max ∧ w (t) e M (8) (8) h n)SS1,min e (M h w/M h n)(8)(t) e (M h w/M h n)SS1,max ∧ (M h w/M (8) (8) B h n3,SS1,min eB h (8) h n3,SS1,max } (1d) n3 (t) e B
(8) (8) t2 ) min{t:pjA,SS2,min e pj(8) jA,SS2,max ∧ A (t) e p (8) (8) eM h (8) h w,SS2,max ∧ M h w,SS2,min w (t) e M (8) (8) (M h w/M h n)SS2,min e (M h w/M h n)(8)(t) e (M h w/M h n)SS2,max ∧ (8) (8) eB h (8) h n3,SS2,max } (1e) B h n3,SS2,min n3 (t) e B
In eq 1a, G(8)(t) is the instantaneous polymer production; therefore, ∫tt12 G(8)(t) dt is the total mass of off-specs. The optimization problem was numerically solved with a sequential quadratic programming (SQP) algorithm.19 The final results are depicted in Figures 1 and 2, and the data are given in Table 3. For the change BJLT f AJLT, the required bangbang profiles are shown in Figure 1a and b). In this case, the intermediate bang-bang period starts immediately at t(1) i ) 0, and coincides with the main change of recipe (which includes modifications in the feed flows of the initiator, emulsifier, and (1) water). Although the intermediate values of F(1) X and FA are below their final SS values, the intermediate values of F(1) B are above its final SS value. In Figure 1a, note that the increased
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Figure 1. Optimal “bang-bang” control for the change of grade BJLT f AJLT, between Normal steady states (SSs): (a, b) feed profiles of A, B, and CTA into the first reactor; (c-h) final evolutions of x(8), G(8), pj(8) A , (8) (8) (8) M h (8) , M h /M h , and B h (solid lines). The off-spec product is accumulated w w n n3 between t1 and t2. Also shown are the evolutions due to a direct step change in all the reagent feeds at t ) 0 (dashed lines).
Figure 2. Optimal “bang-bang” control for the change of grade AJLT f BJLT, between Normal SSs: (a, b) feed profiles of A, B, and CTA into the first reactor; (c-f) final evolutions of x(8), G(8), pj(8) h (8) h (8) h (8) A ,M w ,M w /M n , and (8) B h n3 (solid lines). The off-spec product is accumulated between t1 and t2. Also shown are the evolutions due to a direct step change in all the reagent feeds at t ) 0 (dashed lines).
mass flow of B is exactly compensated by a reduction in the mass flow of A. However, the density of A (819.1 g/L) is different from that of B (621.1 g/L), therefore generating step changes in the total volumetric feed flow. The off-specs were accumulated between t1 ) 260 min (when pj(8) A was the first variable to come out of specification of the initial grade; see Figure 1e), and t2 ) 450 min (when M h (8) h (8) w /M n was the last variable to enter into specification of the final grade; see Figure 1g). The total off-spec product (13 610 kg; see Table 3) corresponds to the shaded area under G(8)(t) between t1 and t2 (see Figure 1d). In this case, the manipulated variables were not maintained at their initial SS values after t ) 0, to produce an early and fast change of grade and to collect the off-specs while x(t) and G(t) were both at relatively low values. The algorithm immediately reduced the intermediate feed flow of the most reactive comonomer (A) to almost zero, to compensate for the increase in production induced by the increased initiator and emulsifier feeds at t ) 0. Figures 1c-h present (in continuous trace) the predicted evolutions of x(8), G(8), pj(8) A , (8) (8) (8) M h (8) , M h /M h , and B h . w w n n3 Consider the change AJLT f BJLT (see Figure 2 and Table 3). A fast but delayed transition was produced in this case, with an increased mass of total off-spec, with respect to the opposite change (see Table 3). As stated previously, pj(8) A was the first variable to come out of specification, and M h (8) h (8) w /M n was the last variable to enter into specification. In this case, the feeds of A, B, and CTA were maintained at their original SS values until t(1) i ) 290 min (see Figure 2a and b) and the off-specs could not
be collected during the lower values of G(t) (see Figure 2d). The reduction of the initiator and emulsifier feeds at t ) 0 induced a reduction in x(t) and G(t). However, in this case, the off-specs could not be collected at the lower values of G(t), because of the necessary increase of the most reactive comonomer feed (A), to produce the required increase in the final copolymer composition (see Figure 2e). In both changes of grade, the fast transitions of the quality variables also generated undershoots and overshoots that remained, however, within specification bands. This is particularly the case of the undershoots and overshoots of pj(8) A (t) observed in Figures 1e and 2e, which almost reach the specification limits of the final grade. In Figures 1d and 2d, discontinuities in G(t) are observed. These were generated by the step changes in the volumetric feed flows of water at t ) 0, and of the comonomers at ti and tf. For comparison, Figures 1c-h and Figures 2c-h also present (in dashed trace) the changes of grade produced by a direct step change in all of the feeds into their final SS values at t ) 0. All of the resulting switching times (t1, t2) and total off-spec masses are presented in Table 3. With respect to the direct step changes, the bang-bang strategies reduced the off-specs by 52% for BJLT f AJLT and 35% for AJLT f BJLT. Changes of Grade between Improved Steady States Consider minimizing the off-specs generated during changes of grade between Improved SSs (see Tables 1 and 2). In this case, delayed bang-bang controls were applied along the train,
Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7681 Table 3. Changes of Grade between Normal and Improved Steady States (SSs): Total Off-Spec Masses and Switching Times for Off-Spec Accumulation Periodsa BJLT f AJLT strategy
a
off-spec [kg]
AJLT f BJLT
t1 [min]
t2 [min]
“bang-bang” control direct step change
13610 28520
(a) Transitions between Normal SSs 260 (pjA) 450 (M h w/M h n) 320 (pjA) 710 (M h w)
“bang-bang” control direct step changes delayed-step changes
14905 32700 28160
(b) Transitions between Improved SSs 250 (pjA) 440 (pjA) 250 (pjA) 660 (M h w) 320 (pjA) 660 (M h w)
off-spec [kg]
t1 [min]
t2 [min]
20850 32060
480 (pjA) 270 (pjA)
750 (M h w/M h n) 690 (M h w)
19018 46500 24650
480 (pjA) 200 (pjA) 405 (pjA)
710 (pjA) 720 (M h w/M h n) 720 (pjA)
Items shown in parentheses after the switching times are the quality variables that determined the off-specs accumulation period.
with simultaneous changes in the feeds of A, B, and CTA into each of the reactors. The optimization involved four unknowns per reactor: the starting and ending times of the intermediate (r) bang-bang period for A, B, and CTA (t(r) i and tf , respec(r) (r) tively), and the intermediate values FA* and FX* (r ) 1, ..., 8), because the values of F(r) B* were automatically adjusted, based on F(r) , to produce a constant total feed of comonomers into A* each of the reactors. As stated previously, lower and upper limits were imposed onto the feeds flows of A, B, and CTA. In summary, the optimization procedure involved the simultaneous adjustment of 32 unknowns (4 unknowns per reactor), and it was formalized as follows:
min
(r) (r) (r) t(r) i ,tf ,FA*,FX*
{
∫t t
2
G(8)(t) dt}
(for r ) 1, ..., 8)
reduced the off-specs by >55% (see Table 3). As in the case of transitions between the Normal SSs, the change BJLT f AJLT required modification of the manipulated variables at t(1) i ) 0, whereas the change AJLT f BJLT the feed flows of A, B, and CTA to be maintained at their original SS values (until t(1) i ) 228 min). Finally, an intermediate control strategy was calculated, which replaced the delayed bang-bang profiles of A, B, and CTA for simple delayed step changes of the manipulated variables (r) (r) F(r) A , FB , and FX into their final SS values. Thus, the objective of the optimization problem was to determine the delay times along the train t(r) (r ) 1, ..., 8) of the three manipulated i variables into their final SS values, through
(2a)
min {
1
t (r) i
∫t t
2
G(8) dt}
(r ) 1, ..., 8)
(3a)
1
with with (r) (r) (r) e 2 max{F i,SS1 ,F i,SS2 } 0 e F i* (for r ) 1, ..., 8; i ) A, B, X) (2b)
F (r) A*
+
F (r) B*
)
(r) F A,SS2
+
(r) F B,SS2
(for r ) 1, ..., 8)
(8) (8) e pj(8) jA,SS1,max ∧ t1 ) max {t:pjA,SS1,min A (t) e p (8) (8) eM h (8) h w,SS1,max ∧ M h w,SS1,min w (t) e M
(2c)
(8) (8) t1 ) max {t:pjA,SS1,min e pj(8) jA,SS1,max ∧ A (t) e p
(8) (8) (M h w/M h n)SS1,min e (M h w/M h n)(8)(t) e (M h w/M h n)SS1,max ∧ (8) (8) eB h (8) h n3,SS1,max } (3b) B h n3,SS1,min n3 (t) e B
(8) (8) eM h (8) h w,SS1,max ∧ M h w,SS1,min w (t) e M (8) (8) (M h w/M h n)SS1,min e (M h w/M h n)(8)(t) e (M h w/M h n)SS1,max ∧ (8) (8) eB h (8) h n3,SS1,max } (2d) B h n3,SS1,min n3 (t) e B
(8) (8) t2 ) min {t:pjA,SS2,min e pj(8) jA,SS2,max ∧ A (t) e p (8) (8) eM h (8) h w,SS2,max ∧ M h w,SS2,min w (t) e M
(8) (8) t2 ) min {t:pjA,SS2,min e pj(8) jA,SS2,max ∧ A (t) e p
(8) (8) (M h w/M h n)SS2,min e (M h w/M h n)(8)(t) e (M h w/M h n)SS2,max ∧ (8) (8) eB h (8) h n3,SS2,max } (3c) B h n3,SS2,min n3 (t) e B
(8) (8) M h w,SS2,min eM h (8) h w,SS2,max ∧ w (t) e M (8) (8) h n)SS2,min (M h w/M e (M h w/M h n)(8)(t) e (M h w/M h n)SS2,max ∧ (8) (8) eB h (8) h n3,SS2,max } (2e) B h n3,SS2,min n3 (t) e B
The results are depicted in Figures 3 and 4, and the data are given in Table 3. The required bang-bang profiles for A and CTA are shown in Figures 3a-d and 4a-d. Because of space reasons, the feed flows profiles of B along the train were not represented in such figures, but were determined using eqs 2b and 2c. For the change AJLT f BJLT, the feeds of B into the last three reactors were all equal to zero, because of the nonnegativity requirement of eq 2b. The evolution of the final quality variables are represented in Figures 3e-h and 4e-h. The results of the delayed bang-bang strategy (represented by the solid lines), were compared with the evolutions of a direct step change in the all feeds at t ) 0 (represented by the dashed lines). Compared to the direct step change, the proposed strategy
For space reasons, the resulting feed profiles are not shown in Figures 3 and 4; however, the delayed step changes were (2) (3) (4) applied at t(1) i ) 29 min, ti ) 55 min, ti ) 62 min, ti ) 194 (5) (6) (7) min, ti ) 204 min, ti ) 222 min, ti ) 233 min, and t(8) i ) 236 (1) min for the change BJLT f AJLT; and at ti ) 103 min, t(2) i ) (4) (5) (6) 229 min, t(3) i ) 326 min, ti ) 364 min, ti ) 415 min, ti ) 450 (8) min, t(7) i ) 495 min, and ti ) 512 min for the change AJLT f BJLT. The final quality variables are shown in Figures 3e, 3f, 4e, and 4f (represented by the dash-dotted lines). The resulting off-spec masses were intermediate between those of the optimal bang-bang and direct step changes (see Table 3). Conclusions Grades BJLT and AJLT exhibit important differences in their chemical composition specifications, but moderate differences
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Figure 3. Optimal “bang-bang” control for the change of grade BJLT f AJLT, between Improved SSs: (a-d) feed profiles A and CTA into the successive reactors of the train; (e-h) final evolutions of pj(8) h (8) h (8) A , M w , M w / (8) (8) M h n , and B h n3 (solid lines). Also shown in panels e-j are the evolutions due to a direct step change in all the reagent feeds at t ) 0 (dashed lines), and due to delayed-step changes (dash-dotted lines).
in their average molar masses and degrees of branching, and a common Mooney viscosity (MV). For that reason, the chemical composition and the molecular structure variables were selected for analysis of the transients during changes of grade. Unfortunately, however, the average molar masses and degree of branching are generally not measured in industrial nitrile rubber (NBR) plants. The optimal control calculations involved a highly multivariate system. However, while the copolymer composition is mainly affected by the A/B ratio (where A denotes acrylonitrile and B denotes butadiene), the molar masses and degree of branching are mainly affected by the chain-transfer agent (CTA) feed. In addition, the monomer conversion and polymer production are directly affected by the feeds of A, B, initiator, and emulsifier. In the developed “bang-bang” strategies, fast transitions in the molar masses and degree of branching were obtained by overreacting with the intermediate feed flow of CTA, as follows: (a) To increase the final molar masses, the intermediate CTA feed was lower than the corresponding final steady-state (SS) feed (see Figures 1b, 3c, and 3d); and
Figure 4. Optimal “bang-bang” control for the change of grade AJLTfBJLT, between Improved SSs: (a-d) feed profiles of A and CTA into the successive reactors of the train; (e-h) final evolutions of pj(8) h (8) A ,M w , (8) (8) M h (8) /M h , and B h (solid lines). Also shown in panels e-j are the w n n3 evolutions due to a direct step change in all the reagent feeds at t ) 0 (dashed lines), and due to delayed-step changes (dash-dotted lines).
(b) To decrease the final molar masses, the intermediate CTA feed was higher than the corresponding final SS feed (see Figures 2b, 4c, and 4d). Similarly, for changes of grade that involved a reduction in the final copolymer composition, the intermediate A/B feed ratios were less than the corresponding final SS feed ratios (see Figures 1a, 3a, and 3b), whereas, for the changes of grade that involved an increase in the final copolymer composition, the intermediate A/B feed ratios were higher than the corresponding final SS feed ratios (see Figures 2a, 4a, and 4b). As expected, the fast final quality transitions also generate undershoots or overshoots at the limits of the final specification ranges. For both plant configurations, changes BJLT f AJLT produced lower off-specs and earlier transitions than changes AJLT f BJLT. This was due to a more favorable combination of feed flows changes for BJLT f AJLT. For changes between the Normal SSs, the total off-specs masses were similar to those between the Improved SS. However, the Improved SSs exhibit increased polymer productions (by ∼13%), and, therefore, the corresponding optimal changes exhibit lower values of the ratio between the total off-spec mass and the mass of polymer
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generated during the transients. For transitions between the Improved SSs, the simpler policy of introducing delayed step changes produced intermediate off-specs between the optimal bang-bang control and the simultaneous step change in all the reagents at t ) 0. The developed open-loop strategies are totally dependent on the accuracy of the process model that is used. One of the model limitations is, for example, that it neglects the emulsification times of A, B, and CTA. Another way of calculating the required feed profiles is the use of “virtual” controllers.14 These algorithms produce continuous (rather than bang-bang) solutions, and they are based on feeding-back the model outputs through the use of numerical proportional-integral (P+I) controllers. Although not shown, the off-specs of “virtual” controller solutions were similar to those obtained with the simpler bang-bang strategies. Acknowledgment To CONICET, SECyT, and Universidad Nacional del Litoral for the financial support. To Petrobras Energı´a S.A., for providing us with the industrial data. Nomenclature B h (r) n3 ) number-average number of trifunctional branches per molecule in reactor r (dimensionless). ) mass flow of reagent i (i ) A, B, X) in reactor r F h (r) i (kg/min) G(r) ) polymer production in reactor r (kg/min) M h (r) n ) number-average molar mass in reactor r (g/mol) M h (r) w ) weight-average molar mass in reactor r (g/mol) MV ) Mooney viscosity (MS) Np) total number of polymer particles pj(r) An ) number-average mass fraction of polymerized S in the copolymer in reactor r pj(r) A ) weight-average mass fraction of polymerized S in the copolymer in reactor r qT ) total volume flow in the train (L/min) SS ) steady state t ) time (min) t(r) f ) ending of the bang-bang period (min) t(r) i ) beginning of the bang-bang period (min) t1 ) time when the first quality variable fell out of specification of the first grade (min) t2 ) time when the last quality variable entered into specification of the final grade (min) x ) gravimetric monomer conversion X ) chain transfer agent, CTA Superscripts r ) reactor number SS1 ) initial ss SS2 ) final ss min ) lower limit of specification max ) upper limit of specification
Literature Cited (1) Kirk, R. E.; Othmer, D. F. Encyclopedia of Chemical Technology, 4th Edition; Wiley: New York, 1998. (2) Blackley, D. C. Diene-based Synthetic Rubbers. In Emulsion Polymerization and Emulsion Polymers, Lovell, P. A., El-Aasser, M. S., Eds.; Wiley: New York, 1997. (3) Hofman, W. Nitrile Rubber. Rubb. Chem. Technol. J. 1964, 37, 1. (4) Kensuke, O.; Tsutomu, H.; Mitsuru, M.; Tooru, T.; Mitsutoshi, F.; Nobuyuki, B. Estimation Method of Mooney Viscosity. Jpn. Patent No. JP52145080, 1977. (5) Ambler, M. E. Studies on the Nature of Multiple Glass Transitions on Low Acrylonitrile, Butadiene-Acrylonitrile Rubbers. J. Polym. Sci. Polym. Chem. Ed. 1973, 11, 1505. (6) Dube´, M. A.; Penlidis, A.; Mutha, R. K.; Cluett, W. R. Mathematical Modeling of Emulsion Copolymerization of Acrylonitrile/Butadiene. Ind. Eng. Chem. Res. 1996, 46, 3203. (7) Vega, J. R.; Gugliotta, L. M.; Bielsa, R. O.; Brandolini, M. C.; Meira, G. R. Emulsion Copolymerization of Acrylonitrile and Butadiene. Mathematical Model of an Industrial Reactor. Ind. Eng. Chem. Res. 1997, 36, 1238. (8) Rodrı´guez, V. I.; Estenoz, D. A.; Gugliotta, L. M.; Meira, G. R. Emulsion Copolymerization of Acrylonitrile and Butadiene. Calculation of the Detailed Macromolecular Structure. Int. J. Polym. Mater. 2002, 51, 511. (9) Gugliotta, L. M.; Vega, J. R.; Antonione, C. E.; Meira, G. R. Emulsion Copolymerization of Acrylonitrile and Butadiene in an Industrial Batch Reactor. Estimation of Conversion and Polymer Quality from Online Energy Measurements. Polym. React. Eng. 1999, 7 (4), 531. (10) Vega, J. R.; Gugliotta, L. M.; Meira, G. R. Emulsion Copolymerization of Acrylonitrile and Butadiene. Semi-Batch Strategies for Controlling Molecular Structure on the Basis of Calorimetric Measurements. Polym. React. Eng. 2002, 10 (1&2), 59. (11) Poehlein, G. W.; Dougherty, D. J. Continuous Emulsion Polymerization. Rubber Chem. Technol. 1977, 50, 601. (12) Hamielec, A. E.; MacGregor, J. F. Modeling Copolymerizationss Control of Composition, Chain Microstructure, Molecular Weight Distribution, Long Chain Branching and Crosslinking. In Polymer Reaction Engineering: Influence of Reaction Engineering on Polymer Properties; Reichert, K. H., Geiseler, W., Eds.; VCH Publishers: Berlin, 1983. (13) Vega, J. R.; Gugliotta, L. M.; Brandolini, M. C.; Meira, G. R. Steady-State Optimization in a Continuous Emulsion Copolymerization of Styrene and Butadiene. Lat. Am. Appl. Res. 1995, 25, 207. (14) Minari, R. J.; Vega, J. R.; Gugliotta, L. M.; Meira, G. R. Continuous Emulsion Styrene-Butadiene Rubber (SBR) Process: Computer Simulation Study for Increasing Production and for Reducing Transients between Steady States. Ind. Eng. Chem. Res. 2006, 45, 245. (15) Minari, R. J.; Vega, J. R.; Gugliotta, L. M.; Meira, G. R. Continuous Emulsion Copolymerization of Acrylonitrile and Butadiene: Computer Simulation Study for Inproving the Rubber Quality and Increasing Production. Comput. Chem. Eng. 2007, 31, 1073-1080. (16) Vega, J. R.; Gugliotta, L. M.; Meira, G. R. Continuous Emulsion Polymerization of Styrene and Butadiene. Reduction of Off-Spec Product between Steady-States. Lat. Am. Appl. Res. 1995, 25, 77. (17) Minari, R. J.; Vega, J. R.; Gugliotta, L. M.; Meira, G. R. Emulsion Copolymerization of Acrylonitrile and Butadiene in a Train of CSTRs. Intermediate Addition Policies for Improving the Product Quality. Lat. Am. Appl. Res. 2006, 36, 301. (18) Kirk, D. E. Optimal Control Theory: An Introduction, 1st Edition; Prentice Hall: Englewood Cliffs, NJ, 1970. (19) Powell, M. J. D. A Fast Algorithm for Nonlineary Constrained Optimization Calculations. In Numerical Analysis; Watson, G. A., Ed.; Lecture Notes in Mathematics 630; Springer-Verlag: Berlin, 1978.
ReceiVed for reView March 15, 2007 ReVised manuscript receiVed May 18, 2007 Accepted May 24, 2007 IE070392J
