Polymer Composition Control in Emulsion Co- and Terpolymerizations

Ind. Eng. Chem. Res. 2002, 41, 1261-1275

1261

Polymer Composition Control in Emulsion Co- and Terpolymerizations Nida Othman,† Gilles Fe´ votte,‡ and Timothy F. McKenna*,† LCPP-CNRS/ESCPE-Lyon and LAGEP, Universite´ de Lyon I, Baˆ t 308, 43 Boulevard du 11 Nov. 1918, 69616 Villeurbanne Cedex, France

A new strategy for composition control in emulsion copolymerization and terpolymerization processes, based on the use of nonlinear estimation and control techniques, is presented in the current work. The strategy uses the estimation of the concentration of monomers in the reactor, and the manipulated variables used for the composition control are the flow rates of the most reactive monomers, calculated in such a way as to account for the reaction rate, thereby allowing us to anticipate changes in the polymer composition. The controllers are experimentally validated for several composition set points and systems. It was found that the controllers maintain the composition at the desired value even for monomers with a wide difference in their reactivity ratios. 1. Introduction It is a well-known fact now that the economical production of polymers with the desired properties, under safe conditions requires that the polymerization process be controlled online. Process control for emulsion polymerization remains a particularly difficult task because of the lack of online measurements of most of the polymer properties of interest: high reaction rates, the sensitivity of the reaction to small amounts of additives, and the highly complex kinetic mechanisms that result in a complex, nonlinear model with a number of unknown variables. One of the main objectives in the control of an emulsion polymerization process is to maintain the polymer composition at some predetermined level because this variable plays an important role in determining a number of end-use properties of the final polymer, e.g., glass transition temperature, particle morphology, mechanical and chemical resistances. Intelligent control of the polymer composition becomes particularly important when several monomers with different reactivity ratios are involved in the reaction at a nonazeotropic monomer composition, which is the case of most industrial systems. If no control action is taken in such cases, the polymer composition will drift during the reaction, and this will lead to a heterogeneity in the polymer properties (unless the polymer is made in a continuous stirred tank reactor). To control the polymer composition, one requires above all else an online measurement of this quantity. Online sensors such as gas chromatography (GC) or densimetry require a sampling device or an auxiliary recirculation loop. This can present certain difficulties, such as coagulation in the circulating loop. Others, such as ultrasonic or infrared spectroscopy, can be used in situ but require an extensive calibration before use or are very sensitive to polymer viscosity and coagulation of polymer particles in the device.1-3 Thus, despite * To whom correspondence should be addressed. E-mail: [email protected]. Fax: (+33) 4 72 43 17 68. † LCPP-CNRS/ESCPE-Lyon. ‡ LAGEP.

recent progress in this area, it is still difficult to obtain direct online measurements of the process states of interest for composition control. This difficulty of performing online measurements has encouraged the use of estimation techniques to infer estimates of certain variables that are not available online from auxiliary measurements. A high-gain nonlinear estimator will be used in this work to estimate the polymer composition online in both co- and terpolymerization processes.4,5,21 The overall conversion is obtained by calorimetry5,6 and used as the input to the high-gain observer that returns values of the number of moles of residual monomer in the reactor, as well as an estimate of the total number of moles of radicals in the particles in the reactor. The polymer composition is then calculated based on the estimated individual reaction rates. To control the polymer composition, some authors7,8 envisaged using the reaction temperature to modify the reactivity ratios of monomers in order to pilot the composition in batch reactions. Temperature control might be applicable to the production of small amounts of polymer in small batch reactors, where agitation is not an issue, or in viscous systems, where converting to semicontinuous operations is difficult, but cannot realistically be applied to large reactors. Moreover, it is limited to pairs of monomers with very different activation energies and on a limited range of composition. A second, more efficient means of controlling the composition consists of manipulating the monomer feed flow rates in semicontinuous reactors. The polymer composition is very sensitive to the monomer ratio in the reactor, which can be directly controlled by employing appropriate monomer feed flow rates. Several monomer addition policies have been reported in the literature.9-11 The most widely used policy consists of adding a mixture of monomers at the desired composition. The most efficient monomer addition policy consists of applying a variable monomer feed flow rate of the more reactive monomer(s).10,12 The monomer flow rate should be a function of the residual amount of monomer remaining in the reactor and their reactivity ratios to maintain the monomer ratios (fi) in the reactor at the desired value. This means that the concentration

10.1021/ie0101994 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/30/2002

1262

Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002

of the residual monomer must be known at every moment in order to calculate the flow rates. In other studies, the composition control was based on the direct measurement of the polymer composition (e.g., by mass spectroscopy, NMR, or GC), which provides a continuous online correction of the model, particularly of the monomer reactivity ratios, and offers therefore the possibility of accurately estimating and controlling the process.11-16 Other authors used calorimetry to measure the heat generated by the reaction17,18 or online GC19 to calculate the feed flow rate of the more reactive monomer to control the copolymer composition. This last paper presented an interesting discussion on the control of composition and molecular weight. The authors used the same three monomers as those used in this paper and developed a model-based “adaptive” controller to adjust feed flow rates based on GC measurements. Their system depended on online adjustment of the constants of a polynomial that was used to fit the value of n j NP from online data. They found that the addition of a PI term to the controller was not necessarily advantageous. However, despite the highly nonlinear behavior of the polymerization process, the polymer composition was usually controlled by linear control techniques [proportional-integral derivative (PID) and adaptive] applied to the linearized nonlinear system. This was normally attributed to difficulties in dealing with nonlinear systems. However, linear estimation and control techniques are inadequate for highly nonlinear processes. Nowadays, recent developments in nonlinear theory allow us to implement nonlinear control techniques (e.g., geometric control20) to several classes of systems without extensive calculations. In the work presented in this paper, the co- and terpolymer compositions are controlled by adding the monomers separately to the reactor. The control output is the concentration of monomers in the reactor. The flow rate of the most reactive monomer is used to control the monomer ratio and is calculated by using nonlinear geometric control. This requires the online estimation of the concentration of monomers in the reactor that is obtained by the calorimetric estimation of the heat produced by the reaction and the monomer instantaneous conversion combined by the kinetic estimators developed for copolymerization4,5 and terpolymerization processes for the estimation of the number of moles of residual monomer and the number of moles of radicals in the polymer particles.21 In addition to the differences in how the parameters are estimated online in a number of literature studies such as the one presented by Urretazbizkaia et al.19 (the number of moles of monomer and of radicals in the organic phase are estimated online using calorimetry), the controller proposed is a geometric, nonlinear controller with input-output linearization and includes PI. It will be shown that that the proportional part is indispensable if we are to correct any eventual errors online or to overcome errors in the initial values. The controllers developed below will be used to input these previously developed estimators and will be validated using methyl methacrylate (MMA), butyl acrylate (BuA), and vinyl acetate (VAc) monomers. 2. Mathematical Model 2.1. Emulsion Copolymerization. The instantaneous polymer composition, or the molar fraction of monomer i being added to the polymer (Fi), is related

to the reaction rates (Rpi) by the following equation:

Rpi

Fi )

n

(1)

Rpj ∑ j)1 where n is the total number of monomers involved in the reaction and Rpi is the reaction rate of monomer i in moles per liter. Ideal composition control would consist of maintaining the instantaneous composition, defined in eq 1, at a predefined trajectory throughout the reaction. Typically, this would be at a constant set point, but one could also envisage a profile of composition that evolves during the reaction. In the rest of this work, we will only treat the constant composition. If we assume that the polymer particles are the only reaction loci (it has been shown that aqueous phase polymerization has very little influence on the composition in most cases21), the monomer material balance is simplified as follows for i ) 1 and 2:

N˙ i ) Qi - Rpi ) Qi - µ([Mip][Kp,1iP1p + Kp,2i(1 - P1p))]

(2)

The reaction rate that appears in the material balance (2) is a function of various parameters related to the monomer concentration in the polymer particles [Mip], Ρip is the probability that the ultimate unit of an active chain in the polymer particles be of type I, and µ is the number of moles of radicals in the polymer particles:

µ)n j NPT/NA

(3)

where n j is the average number of radicals per particle, NPT is the total number of particles in the reactor, and NA is Avogadro’s number. The concentration of monomers in the polymer particles is calculated assuming that the monomers are not soluble in the aqueous phase, and therefore they are partitioned only between the monomer droplets and the polymer particles. The compositions of monomers in the different parts of the organic phase, i.e., monomer droplets and polymer particles, are regarded as equal.22 Therefore, the concentration of monomers in the polymer particles can be written as a function of the solubility of monomer in the polymer particles. The expressions used to define monomer concentrations and the conditions for the existence of monomer droplets are given in Fe´votte et al.4 Based on the material balance, an estimate of the number of moles of residual monomer can be obtained using the observers mentioned above.4,5 The observer is based on the overall monomer conversion obtained by calorimetry that is used to calculate the total number of moles of monomers in the reactor (y), the real observer output. To obtain y, the measurement of the total amount of monomers added to the reactor (initial amounts plus feed flow rates) must be known online. The model output (y) is calculated by the following equation:

∑i NiTMWi

y ) (1 - X) )

∑i NiMWi

(4)

Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1263

where the monomer molar conversion is defined by

X)

∑i MWi(NiT - Ni)

The composition is well controlled if the ratio F1/F2 remains at the desired value during the reaction. This ratio is given by

F1 RP1 ) F2 RP2

(5)

∑j MWjNj

T

where i and j are the numbers of monomers. 2.2. Emulsion Terpolymerization. The material balance is written as

N˙ i ) Qi - RPi, i ) 1-3

(6)

In this work, we will be interested in the production of a constant composition during the reaction. Therefore, the objective of our control strategy will be to maintain the molar ratio of polymers at a desired constant value. When the assumptions about monomer partitioning discussed above are invoked, the composition ratio becomes

where p

p

p

p

RPi ) µ[Mi ](Kp,1iP1 + Kp,2iP2 + Kp,3iP3 )

3. Control Law The controllers used in this work are constructed using nonlinear geometric control theory.20 The control output is the concentration of residual monomer in the reactor. As we discussed above, it will be assumed that controlling the monomer ratio in the reactor allows us to control the polymer composition (which is true if the reactivity ratios of monomers are well-known). To do this, the flow rates of the most reactive monomers are used as the control variables, or the controller inputs. The controller is constructed using an input/output linearizing technique. Once the input/output relationship is linear, we can apply a linear controller such as a traditional PID controller. 3.1. Emulsion Copolymerization. The overall polymer composition is determined by integrating the instantaneous polymer composition produced per unit time. The ideal way to control the copolymer composition is to control the instantaneous composition, i.e., to maintain Fi, in eq 1, at some desired value Fid. In a copolymerization process, the molar fraction of homopolymer i in the copolymer is given by the following equation:

Fi )

RPi RP1 + RP2

F1 ) F2

(7)

Because it has been shown that the monomer ratio in the polymer particles is not particularly sensitive to the monomer solubility in the aqueous phase for the processes that interest us here, it is supposed that the monomers are water-insoluble and partitioned only between the polymer particles and the monomer droplets.21 While this is strictly speaking not true, we will see below that this assumption turns out to be quite reasonable. Under the assumptions that the ratio of monomer concentrations are the same in the two parts of the organic phase,23 the concentration of monomer in the polymer particles is only a function of the saturation of polymer particles and the residual number of moles of each monomer. Details on the model used to calculate monomer concentrations and the existence of droplets for the terpolymerization observer can be found in Othman et al.21 This approach has been experimentally validated; we can assume that the instantaneous copolymer and terpolymer compositions are available online.

(8)

(9)

(

)

N1 N1 Kp,11 + Kp,12 N2 Kp,22 N1 N2 Kp,12 + Kp,12 N2 Kp,21

(

)

(10)

This equation shows clearly that it is sufficient to control the ratio of monomers N1/N2 in order to maintain the instantaneous composition at a constant set point (if the Kp,ij values are well-known). If we assume that monomer 1 is more reactive than monomer 2, the ratio N1/N2 decreases as a function of time in a closed system. Hence, the number of moles of monomer 1 must be manipulated to control the ratio N1/N2. Q2 can be set to be constant, but its value must be known at every moment in order to be able to calculate N1/N2. The following equation can be used to calculate the monomer ratio N1/N2 that gives us the desired composition:

( ) x ( )

Kp,12 F1 N1 ) -1 + N2 2Kp,11 F2 1 2Kp,11

Kp,122 1 -

F1 F2

2

Kp,22 F1 + 4Kp,11Kp,12 (11) Kp,21 F2

To do this, one must start by defining the desired composition and therefore (F1/F2)d; then eq 11 can directly be used to calculate the desired monomer ratio (N1/N2)d that must be tracked in order to achieve the control purpose. The controller objective now becomes controlling the monomer ratio in the reactor and not the polymer composition. If we suppose that N1, N2, and µ are available online from the observer, we can write the mass balance of copolymerization with the output y ) N1:

The system given by eq 12 is a nonlinear single input single output (SISO) system with the states x. u is the manipulated input and y the model output. Q2 is not considered as a manipulated variable but as a known input. To test the controllability of the system and whether u can be used to control N1, we first calculate the relative order (r). For r ) 1, we calculate

1264


〈dh,adfr-1(g)〉 as given by Isidori:20

Lfh )

∂h f ∂x

[

-RP1 ) [1 0 ] Q RP2 2

]

) -RP1 * 0

(13)

Lfh is the Lie derivative of the function h in the direction of the vector f. The relative order of the system r is equal to 1, and we can therefore calculate a nonlinear input/ output linearizing controller. To do so, we define the following state feedback transformation:8 1

υ ) Ω(x,u) ) ) β0h + β1

βkLkf h + (-1)0β1〈dh,adf0(g)〉 u ∑ k)0 ∂h ∂h f + β1 g ∂x ∂x

) β0N1 + β1(-RP1) + β1u

(14)

and we can therefore calculate the input u:

u) )

υ - β0N1 + β1RP1 β1 β0 υ - N + RP1 β1 β1 1

the reaction, if the control error ) 0 (i.e., the value of N1 equals the desired value), then the resulting flow rate of monomer 1 will not be zero but equal to the reaction rate of this monomer. This limits the oscillations of the controller around the set point, which would occur if a PI is applied alone. The implementation of the PI controller becomes necessary when the monomer ratio is not well initialized or if any perturbation in the monomer flow rates occurs. The proportional (P) action is indispensable to account for the set point, and the use of the integrator (I) part allows us to eliminate possible steady-state offsets, due to modeling uncertainties. The integrator gain must, however, be chosen wisely, because it augments the order of the linear model and the integral gain might add an imaginary pole to the linear system, which causes oscillations in the response. As a result, the combination of the PI controller and the nonlinear controller avoids oscillations around the set point, accounts for modeling errors, and compensates for perturbation in the process. 3.2. Emulsion Terpolymerization. As in the case of copolymerization, the composition produced during a batch terpolymerization reaction involving monomers with different reactivities will undoubtedly vary if the composition is not azeotropic. When three monomers are involved, a constant composition is obtained if the following two ratios are maintained at the desired values throughout the reaction:

F2 RP2 F1 RP1 ) and ) F3 RP3 F3 RP3

(15)

where u is in moles per second. The external input υ can be used to add a linear PI loop, as follows:

Consider the model for hydrophobic monomers (i ) 1-3). Assuming that the monomers are not water soluble and that they have the same solubility in both parts of the organic phase (polymer and monomer), the composition ratio becomes

F1 N1(a1Kp,11 + a2Kp,21 + a3Kp,31) ) F3 N3(a1Kp,13 + a2Kp,23 + a3Kp,33)

Hence, the complete control variable becomes

u ) Q1 )

(

∫0t dt

1 1 κ + β1 P τI

f

)

+ RP1

(19)

F2 N2(a1Kp,12 + a2Kp,22 + a3Kp,32) ) F3 N3(a1Kp,13 + a2Kp,23 + a3Kp,33)

(17)

The parameters β0 and β1 must be chosen in a way that ensures the stability of the states of the model. In eq 17, we can take β1 ) 1 without any loss of generality. In the case where υ is taken to be a PI linear controller, the stability of the system is governed by κP and τI. Note that the other states of the model (N2) are stable for all values of Q1. N2 decreases if all of monomer 2 is added to the reactor at t ) 0, or it can depend on Q2, where Q2 must be set at some reasonable rate. The PI gains must be chosen in a way that guarantees stable and rapid convergence to the desired composition. The influence of the values of these gains will be studied experimentally. Equation 17 is therefore translated to

Equation 18 shows that the nonlinear controller accounts for the model nonlinearity by accounting for the reaction rate of the monomers. Therefore, during

(20)

where

( (

a1 )

N1 N1 N2 K K + Kp,21Kp,32 + Kp,31Kp,23 N3 p,31 p,21N3 N3

a2 )

N2 N1 N2 Kp,12Kp,31 + Kp,12Kp,32 + Kp,32Kp,13 N3 N3 N3

(

N1 N2 a3 ) Kp,13Kp,21 + Kp,23Kp,12 + Kp,13Kp,23 N3 N3

)

) )

(21)

Equation 20 shows that it is sufficient to manipulate the ratios N1/N3 and N2/N3 in order to keep the composition at some desired fractions, F1/F3 and F2/F3. If we assume that monomers 1 and 2 are more reactive than monomer 3, these ratios must be controlled by manipulating N1 and N2. The desired ratios (N1/N3)d and (N2/N3)d can be calculated from F1/F3 and F2/F3 from eq 20. Both ratios must be controlled simultaneously in order to guarantee the production of a polymer with the


desired composition. This can be done by manipulating the flow rates of monomers 1 and 2 together. To do so, a controller of dimension two must be constructed using the terpolymerization model. The following system can be used to represent the terpolymerization model for hydrophobic monomers:

We obtain the following inputs:

u1 ) )

υ1 - β00N1 + RP1u2 β10 υ2 - β01N2 + RP2 β11

(27)

The external input υ1 can be replaced directly by the set point of N1, or υ1 - β00N1 can be used to add a proportional loop, as was done in the case of copolymerization, and the same argument can be applied to υ2. We obtain the following inputs:

u1 ) Q1 ) ) Q2 ) where Q1 and Q2 are the two manipulated inputs of the system and N1 and N2 are assumed to be the model outputs and are supposed to be known at every moment using the observer mentioned before. To guarantee the controllability of the systems 22, the characteristic matrix must be nonsingular and the relative orders equal to 1. The characteristic matrix corresponding to the system 22 is

[

] [ ]

Lg h1 Lg h1 1 0 C(x) ) L 1h L 2h ) 0 1 g1 2 g2 2

(23)

and is therefore nonsingular, and for r1 ) 1 and r2 ) 1, we can calculate 〈dhi,adrf i-1(g)〉, i ) 1 and 2:

Lfh1 )

[

]

-RP1 ∂h1 f ) [0 1 0 ] -RP2 ) -RP1 * 0 ∂x Q3 - RP3 L fh 2 )

∂h2 f ) -RP2 * 0 ∂x

(24)

Therefore, the relative orders r1 ) 1 and r2 ) 1. We can, therefore, perform two input/output linearizing transformations, correlating Q1 with N1 and Q2 with N2: 1

υ1 ) Ω1(x,u) )

βk0Lkf h1 + (-1)0β10〈dh1,adf0(g1)〉u ∑ k)0

) β00h1 + β10

∂h1 ∂h1 f + β10 g ∂x ∂x 1

) β00N1 + β10(-RP1) + β10u1

(25)

and the same transformation for input u2: 1

υ2 ) Ω2(x,u) )

βk1Lkf h2 + (-1)0β11〈dh2,adf0(g2)〉u ∑ k)0

) β01N2 + β11(-RP2) + β11u2

(26)

κP1 + RP1u2 β10 1 κP2 + RP2 β11 2

(28)

where ui is in moles per second. The controller parameters (κPi, β1j) must be chosen in a way that guarantees a stable composition profile. First of all, we can set β10 ) 1 and β11 ) 1 without any loss of generality. As we discussed above, the proportional constant is indispensable in order to account for possible errors in the initial values and any perturbations, caused by the sensors or actuators. The integral action is important in order to account for model uncertainties. 4. Validation 4.1. Copolymerization. The controller was initially tested in simulation. The system MMA and BuA is chosen to study the controller robustness because it is known to introduce a large composition drift in batch operations and it does not have an azeotropic composition. In this system, MMA is the more reactive monomer, and its flow rate is, therefore, the manipulated variable, calculated by eq 18. The gains of the controller are adjusted by simulation, and the optimal ones were found to be κP ) 0.1 and τI ) ∞ because no modeling error is assumed in the simulation test. The composition set point is 30% MMA and 70% BuA, by mole. The initial number of moles of BuA is 3.12 mol, and its flow rate is supposed to be zero (Figure 1). The initial number of moles of MMA is calculated by eq 24 and is equal to 0.525 mol. An error of 10% of the initial value of MMA was voluntarily introduced in order to test the rapidity of the convergence of the controller to the set point. The value of µ was supposed to be constant during the simulation (the models of n j and NPT are not used). Rate constant data are presented in Table 1. Figure 1 shows that the flow rate of MMA is, as expected, a function of the reaction rate of BuA, which is constant during interval II. Because the initial value of MMA was lower than the desired value, the flow rate of MMA was higher at the beginning of the reaction. The maximum admissible flow rate was voluntarily fixed at 5 × 10-4 mol/s to simulate pump saturation. This caused a short delay in the convergence of the monomer ratio to the set point. During this time, the obtained instantaneous composition does not exactly match the desired value but converges quickly to the exact set point. During interval III, the reaction rate was a function of the concentration of monomer in the

1266


Figure 1. Simulation of the copolymerization of MMA and BuA: (left) the flow rate of MMA calculated by the controller; (right) the instantaneous composition, with an error in the initial mass to test the robustness of the controller. Table 1. Reaction Rate Constants and Reactivity Ratios of Monomers BuA MMA VAc

Kp (L/mol/s)

T (°C)

ref

33700 843 9460

60 60 60

25 25 25

monomer

r1

r2

ref

BuA-MMA VAc-BuA VAc-MMA

0.4 0.0262 0.0261

2.15 5.939 24.025

25 26 26

polymer particles, which decreases with time, and therefore Q1 decreases linearly in interval III (because µ is assumed to be constant). To validate the controller experimentally, a set of experiments was realized using the pair of monomers MMA and BuA because of the strong composition drift that occurs if inadequate control is applied. The polymerizations were carried out at 60 °C in the 3 L reactor. The heating, charging, and degassing of water and monomers is performed as explained in work by Othman et al.21 To avoid errors in the composition of the initial monomer charge due to evaporation of the reactive species, the monomers are deoxygenated for only 5-10 min. The temperatures, the monomer mass added to the reactor, and the pump voltage are measured every 10 s, and the data are saved in a file. These data are continuously fed to a Matlab Simulink program to solve the energy balance.5,21 The estimators developed in these same works allow us to obtain a real online estimate of the polymer composition. At the beginning of the reaction, Q1 is calculated from the reaction rate, which, in turn, is calculated from the energy balance using the values of UA and Qloss obtained by calibration until samples are withdrawn to correct these estimates. In the semicontinuous control strategy, the reactor is initially charged with all of the desired amount of the less reactive monomer and with the amount of the more reactive monomer required to correctly initialize the ratio N1/N2. Thereafter, Q1 is calculated by the control law given by eq 18. Q2 can also be set to a known positive value if it is not possible to add all of the desired amount of BuA in the initial charge, especially if this might provoke a large increase in the rate of heat produced. An initial set of experiments was performed to determine the optimal controller gains κP and τI, which

Table 2. Control Experiments of Copolymerization for the Evaluation of the Controller Gains (Desired Molar Composition: 30% MMA-70% BuA) C 11 initial charge (g) H2 O MMA BuA Triton KPS κP τI

1502 53 400 4.09 3.01 0.01 1e4

feed (g) 64 150

C 12

C 13

C 14

initial initial initial charge feed charge feed charge (g) (g) (g) (g) (g)

1501 53 400 5.14 5.55 3.0455 0.001 0.1 8 1e ∞

feed (g)

67 1500 70.8 1504.7 70 150 53 150 53 150 400 400.8 5.1 7.9 5.4 8.0311 8.353 3.0429 3.0506 0.01 ∞

provided rapid, nonoscillating convergence of the composition. The recipes and the trial controller gains used for these experiments are shown in Table 2. The composition set point was 30% MMA-70% BuA for all of the experiments. The initial amounts of monomers were precisely charged to minimize the composition drift at the beginning of the reaction. The flow rate of MMA was calculated from the controller 18 as a function of the chosen controller gains. The “best” controller gains will be those that give us a smooth flow rate Q1 that brings the composition to the desired trajectory quickly (if it is not well-initialized or if it deviates during the control experiment due to some perturbation) and maintains it on this trajectory with minimum oscillations. Simulations showed that Q1 is primarily a function of the reaction rate of MMA (RP1), which, in turn, must follow the rate of reaction of BuA (RP2), especially if the initial amounts are correctly chosen. The PI part is indispensable if there is an error in the initial values of N1 and N2 or if a perturbation occurs in the ratio N1/ N2, which gives a positive error . In this case, both the linear (PI) and nonlinear parts of the controller contribute to the action. Parts a-d of Figure 2 show the values of Q1 and the reaction rates of MMA and BuA, obtained for the different experiments. In this set of experiments, the obtained polymer composition is validated by GC. In experiment C11, a strong combined PI action was used (κP ) 0.01 and τI ) 1 × 104). Figure 2a shows that the calculated value of Q1 was, therefore, either zero or equal to the maximum admissible flow rate of the pump, which means that a bang-bang controller is obtained. This kind of control is to be avoided because it exhausts the pumps and causes oscillations in the resulting instantaneous composition (Figure 3a,b). These oscilla-


Figure 2. (a) Experiment C11: κP ) 0.01 and τI ) 1 × 104. (b) Experiment C12: κP ) 0.001 and τI ) 1 × 108. (c) Experiment C13: κP ) 0.1 and τI ) ∞. (d) Experiment C14: κP ) 0.01 and τI ) ∞.

Figure 3. Instantanoues (a) and cumulative (b) polymer composition for experiment C11 for controller parameters κP ) 0.01 and τI ) 1 × 104.

tions become critical when there is a small amount of monomer in the reactor, because at this stage the monomer ratios and thus the polymer composition become very sensitive to the monomer flow rate. In this experiment, the oscillations in the instantaneous composition increase at the end of the reaction, but fortunately the cumulative composition remained unaffected

by these oscillations. In other cases, for example, for the experiments that remain a long time under starved conditions, the cumulative composition can be critically influenced. In experiment C12, a different set of gains was used to adjust the controller (κP ) 0.001 and τI ) 1 × 108). Figure 2b shows that the obtained curve of Q1 has the

1268


Figure 4. Instantanoues (a) and cumulative (b) polymer composition for experiment C12 for controller parameters κP ) 0.001 and τI ) 1 × 108.

Figure 5. Instantanoues (a) and cumulative (b) polymer composition for experiment C13 for controller parameters κP ) 0.1 and τI ) ∞.

shape of RP2 but with a significant delay. The composition of the polymer produced in this case (Figures 4a and 5b) also drifts from the set point, which means that the controller gains are not high enough. In experiment C13 the proportional gain is set to a higher value, κP ) 0.1. The integral action was eliminated (τI ) ∞) because it was found that it augments the calculation time of the controller and seems to be sensitive to the response time of the pump. Moreover, as we mentioned earlier, the I action is used to eliminate the static error that usually comes from a modeling error. In fact, we do not need this in our case because the closed-loop observer of Ni and µ accounts for modeling uncertainties. Figure 2c shows that the resulting flow rate of MMA reasonably followed the evolution of the reaction rate of BuA (RP2) at the beginning of the reaction, but at the end of the reaction, RP2 was decreasing even though RP1 was increasing. This causes a drift in the ratio RP1/RP2 and, therefore, in the polymer composition. This perturbation is due to an overbearing proportional action. The resulting polymer composition (Figure 5a,b) begins to drift from the set point at low conversion and finishes by a noticeable drift at the end of the reaction that also influences the cumulative composition. In experiment C14, the controller parameters were κP ) 0.01 and τI ) ∞. The obtained curve of Q1 is smooth

and follows exactly the shape of RP1 and RP2, except during the rapid part of the reaction where small oscillations are observed in the curve of Q1. In this experiment, the perturbations were not due to the controller but to the optimization procedure. The controller was able to precisely manipulate Q1 at high conversions. The estimated copolymer composition is presented in Figure 6a,b. The drift in the polymer composition in this experiment occurs at very high conversion where the monomer ratio (Figure 6c) is very sensitive to small changes in the monomer flow rates. In theory, controlling N1/N2 should allow us to control F1/F2. The polymer ratio F1/F2 is thereafter calculated by eq 10. It was noticed during this set of experiments that a small error was integrated in the flow rate of the pumps. This caused a steady-state offset between the real mass (measured by the balance) and the mass calculated from the pump frequency. To track exactly the desired masses, a local model-based controller was associated with the pumps. The controller was then tested during the following set of experiments with different composition ratios of MMA and BuA, using the optimal gains obtained by the first test. Table 3 shows the recipes used for these experiments. In experiment C15 the composition set point is 30% MMA-70% BuA by mole. Figure 7a shows the reaction


Figure 6. Experiment C14. (a) Estimations given by the continuous line; experimental GC validation given by the points. (b) Cumulative polymer composition validated by experimental GC obtained offline. (c) Real monomer ratio N1/N2 compared to the desired one. Table 3. Semicontinuous Experiments for the Validation of the Control Strategy C15 (30:70 MMA-BuA)

C16 (50:50 MMA-BuA)

initial initial charge feed charge (g) (g) (g) H2O MMA BuA Triton SDS KPS final solids content (%) final particle diameter (nm)

1498 53 400 8 3.01 26

71.5 1500 150 132.3 400 8.7 8.35 0.35 3 31 33

266

295

feed (g)

C17 (70:30 MMA-BuA) initial charge (g)

100.2 1500 217.5 257 300 8.54 8.04 0.58 3.01

feed (g) 100.2 320.8 8.61 0.602

315

rates of MMA and BuA and the controlled flow rate of MMA. It can be seen that the curve of Q1 follows the shape of RP1 and RP2. Because the value of Q1 is the result of both the composition controller and the local controller of the pump, some oscillations are observed in the curve of Q1. Figure 7c shows that this flow rate gives a constant cumulative composition during the entire experiment. The composition was validated by offline GC measurements. In experiment C16, the composition set point is 50% MMA-50% BuA by mole. Parts a-e of Figure 8 show the results of this experiment. The rate of heat

produced by the reaction (QR) is shown in Figure 8a. A peak of heat is produced at the beginning of the reaction, during the particle nucleation. QR then increases slowly during the semicontinuous addition of monomer, perhaps because of an increase in the gel effect. Because the desired composition is 50% MMA50% BuA, the reaction rate of both monomers should be equal. Figure 8b shows that RP1 is maintained close to RP2 during most of the reaction time. At the end of the experiment, the sensitivity of the reaction rate to the monomer flow rate yields a small difference between RP1 and RP2. This difference influences the instantaneous composition that is shown in Figure 8c. The cumulative composition was found to be less sensitive to these errors, because the reaction rate at the end of the experiment is low and the amount of monomer produced with this drift of composition is very small compared to the amount of polymer produced at the desired composition. The polymer composition and individual conversions were validated by off line GC measurements. Figure 8e shows the number of moles of free monomer versus time validated by the GC measurements. In experiment C17, the desired composition is 70% MMA-30% BuA. Figure 9a shows the overall and individual conversions estimated during this experiment and validated by GC. Figure 9b shows that the cumulative composition was maintained constant throughout the reaction.

1270


Figure 7. Experiment C15. (a) Reaction rates of MMA and BuA and flow rate of MMA (mol/s). (b) Number of moles of free monomer. (c) Desired composition ) 30% MMA-70% BuA.

4.2. Terpolymerization. The system MMA, BuA, and VAc is chosen to validate the terpolymerization controller. In this system, the monomers MMA and BuA are more reactive than VAc. The manipulated parameters are, therefore, the flow rates of MMA and BuA. The first test of the controller was done in simulation. The flow rate of VAc was set to zero during the simulation, the physical constants were those used above, and the set point was chosen at 45:45:10 MMABuA-VAc (mol). The simulation results are shown in Figure 10a,b. Figure 10b shows that the controller quickly brings the composition to the desired value even though the initial values contained an intentional error in the monomer ratios. In Figure 10a, we can see that, even though Q1 and Q2 were calculated by two different controllers, the ratio Q1/Q2 was constant during the addition. Furthermore, it was found that

Q1/Q2 ) (N1/N2)initial

(29)

To analyze this situation, let us rewrite the control objective, that is to maintain the monomer ratios at predefined values:

( )

N1 N1 ) N3 N3

d

and

( )

N2 N2 ) N3 N3

d

(30)

If these ratios are always constant, then the following ratio is also constant:

N1 (N1/N3)d ) N2 (N /N )d 2

(31)

3

Therefore, it is sufficient to maintain this ratio N1/N2 at the desired value in the preemulsion and to employ the single control law in eq 28 that maintains one of the monomer ratios at the desired value. The second ratio will be directly obtained if the monomer ratio N1/ N2 in the reactor is at the desired level. The possibility of introducing the monomers together has some advantages. First of all, this simplifies the experimental setup because a single pump is required in this case. Second, this guarantees that the monomer ratio N1/N2 in the feed remains at the desired value, even if a fault is produced in one of the pumps. This will ensure that the monomer ratio N1/N2 in the reactor be at the desired value if and only if the initial monomer ratio in the reactor is correct. Finally, this technique can be applied to any triplet of monomers. However, it should be clear that if a large perturbation causes a big drift in the monomer ratio N1/N2 in the reactor, the controller will not be able, in this case, to rectify this error. Therefore, for monomers that have a wide difference in their reactivity ratios, where a drift might rapidly occur, it is recommended to employ two pumps.


Figure 8. Experiment C16. (a) Rate of heat produced by the reaction (QR). (b) Reaction rates of MMA and BuA. (c) Estimated instantaneous composition. (d) Estimated cumulative composition. (e) Estimated number of moles of free monomer.

In the case of MMA and BuA, it was found that these monomers react at the same ratio when involved in a terpolymerization with VAc because it is impossible that MMA reacts and not the BuA. Therefore, a single pump was used to introduce the monomers MMA and BuA. The flow rate of VAc is set to zero in these experiments because all of VAc was completely charged in the reactor at the beginning of the experiment. The recipes used to carry out these experiments are shown in Table 4. Two different compositions were tested: 50:35:15 and 45:45: 10 MMA-BuA-VAc by mole.

To obtain a composition of 45:45:10 by mole, in experiment C23, the initial monomer ratios were found by eq 20 to be N1/N3 ) 0.4648 and N2/N3 ) 0.2269, which gives N1/N2 ) 2.0485. This ratio is, therefore, set in the preemulsion. Figure 11a shows the evolution of the monomer ratios during this experiment. It can be seen that the monomer ratios oscillate around the desired values, especially at the end of the reaction. This is due to the fact that when the amount of monomer in the reactor is very small, the monomer ratio becomes very sensitive to the monomer feed rates. However,

1272


Figure 9. Experiment C17. (a) Overall and individual conversions validated by GC. (b) Estimated cumulative composition validation by GC.

Figure 10. Simulation of the controller. (a) Molar flow rates of MMA and BuA. (b) Instantaneous molar composition. Table 4. Experiments of Terpolymerization of MMA-BuA-VAc for the Validation of the Composition Controller

H2O MMa BuA VAc Triton DSS KPS final solids content (%) final particle diameter (nm)

C23 (45:45:10 MMA-BuA-VAc)

C26 (50:35:15 MMA-BuA-VAc)

initial charge (g)

initial charge (g)

1200 23 60 86 6.87

feed (g) 300 450 544 7.8 3.06

1200 33.6 54.3 172 7.42

3.097 242 36

feed (g) 300 640 550 8.91 3.01

3.096 44 308

because the reaction rate decreases at the end of the reaction, even if the instantaneous composition oscillates slightly around the set point (Figure 11b), the cumulative composition (Figure 11c) remained stable and close to the desired value. The instantaneous terpolymer composition is calculated from the monomer reaction rates, as given by the following equation:

Fi )

RPi RP1 + RP2 + RP3

(32)

The cumulative composition is the integrated molar ratio of the amount of polymer i produced at every moment, with respect to the whole amount of monomer. The composition was validated by offline NMR measurements. The NMR measurements show that a constant composition was maintained throughout the reaction. However, the NMR measurements present a difference with the estimated composition, at the beginning of the reaction. In experiment C26, the composition set point was 50: 35:15 MMA-BuA-VAc. The overall and individual conversions obtained for this experiment are shown in Figure 12a. The individual conversions were validated by measuring, offline, the amount of residual monomer by means of GC. The obtained experimental measurements seem to agree with the estimated values. The estimated and experimental number of moles of residual monomer are shown in Figure 12b. The corresponding polymer composition produced at every moment is shown in Figure 12c. Oscillations were observed in the instantaneous composition, especially at high conversions. The evolution of the cumulative composition, however, is stable and very close to the desired value but does not exactly fit the experimental values obtained by NMR analysis of the terpolymer. However, if we look at parts a and b of


Figure 11. Experiment C23. (a) Monomer ratios N1/N3 and N2/N3, compared to the desired values. Desired composition ) 45:45:10 MMA-BuA-VAc by mole. (b) SP ) 45:45:10 MMA-BuA-VAc. (c) SP ) 45:45:10 MMA-BuA-VAc. Validation by NMR given by the points.

Figure 12, it can be seen that the GC measurements agree with the estimated values of the monomer individual conversions and the residual number of moles of monomer in the reactor. If we assume that the value of the polymer composition obtained by analyzing the polymer by NMR measurements is more precise than the value obtained by analyzing the residual monomer by GC, then the difference between the desired composition and the real one is due to a modeling error in the reaction rates of the monomers: Kp,ij and rij. The controller depends on these kinetic data, and therefore the control results are sensitive to the use of different reaction rate constants. At this level, the controller works in an open-loop fashion, which means that the kinetic values are assumed to be well-known. The constants used in the control strategy are given in Table 4. They were chosen for use in the control strategy because they were found to give a realistic representation of the evolution of the process.24 5. Conclusion The control strategy developed in this work is adapted for experimental practice. First of all, the composition control robustness is independent of the desired polymer composition. It was possible to maintain the composition

of the polymer produced at a constant value for several compositions using the copolymerization system MMABuA and the terpolymerization system MMA-BuAVAc. The technique does not require that we start the polymerization with a seed. A constant composition can be obtained even during the nucleation period. Moreover, the technique is robust to modeling errors related to the evolution of the number of radicals in polymer particles and of the monomer solubility in the aqueous phase. Finally, the control strategy is dependent on the estimated overall conversion, which is obtained by calorimetry in this work, but any other online method can be used. If an accurate estimation of the overall conversion is obtained, the controller is then able to track the desired composition trajectory. However, the composition control necessitates good knowledge of the reaction kinetics, Kp,ij and rij. These parameters are usually well-known or can be identified in laboratoryscale experiments. It was observed that the estimation of the overall conversion is very sensitive to the flow rate variations at high conversion which leads to a composition drift, or oscillation. However, the amount of polymer produced at this time is not very important. We can, therefore, stop the control and terminate the reaction in batch if

1274


Figure 12. Experiment C26. (a) Experimental global conversion obtained by gravimetry; individual conversions validated by GC. (b) Number of moles of free monomer validated by GC. (c) Desired composition ) 50:35:15 MMA-BuA-VAc. Validation by NMR measurements.

the residual amount of monomer is not important and does not effect the polymer final composition. While no attempt was made to control anything other than the polymer composition, it should be pointed out that the scheme used here can easily be adapted to control both composition and monomer concentration in the particles (thus, the molecular weight) for both coand terpolymerization. The advantage of using estimators that provide both the number of moles of residual monomer and the concentration of radicals in the organic phase (µ)4,5 would allow us to perform modelbased control of the molecular weight without the need to resort to semiempirical models for µ. Notation A ) frequency factor (L/mol/s) Ea ) activation energy (kJ/mol) Fi, Fid ) instantaneous and desired instantaneous polymer compositions Kp,ij ) reaction rate constant between the active chain i and monomer j Lfh ) Lie derivative of the scalar field h with respect to the vector field f (Chapter 2) [Mip] ) concentration of monomer i in the polymer particles n j ) average number of radicals per particle Ni ) number of moles of free monomer i

NPT ) total number of particles in the reactor PiP ) time-averaged probability that the ultimate unit of an active chain is of type i Qi ) molar flow rate of monomer i R ) universal gas constant (kJ/mol/K) Rpi ) rate of reaction of monomer i [mol/s] u ) input V ) voltage x ) state variables Greek Letters ) error κp ) proportional gain of the P controller τI ) integral gain of the PI controller τ ) system time constant (s) µ ) number of moles of radicals in the polymer particles υ ) linearizing input-output transformation

Literature Cited (1) Guinot, P.; Othman, N.; Fe´votte, G.; McKenna, T. F. The On-line Monitoring of Emulsion Polymerization using calorimetry and other hardware sensors. Polym. React. Eng. 2000, 8 (2), 115. (2) Kiparissides, C.; Morris, J. Intelligent Manufacturing of Polymers. Comput. Chem. Eng. 1996, 20, S1113. (3) Kammona, O.; Chatzi, E. G.; Kiparissides, C. On-Line Monitoring of Emulsion Copolymerization Using a MIR probe in

Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1275 Combination with Factor Analysis. In DECHEMA Monographs; Moritz, H., Reichert, K.-H., Eds.; VCH Publishers Inc.: Berlin, 1998. (4) Fe´votte, G.; McKenna, T. F.; Othman, S.; Hammouri, H. Nonlinear tracking of glass transition temperature for free radical emulsion copolymers. Chem. Eng. Sci. 1998, 53, 773. (5) Othman, N.; Fe´votte, G.; McKenna, T. F. On-line evaluation of calorimetric state estimators for copolymerisation in emulsion. Can. J. Chem. Eng. 2000, accepted for publication. (6) McKenna, T. F.; Othman, S.; Fe´votte, G.; Santos, A. M.; Hammouri, H. An integrated approach to polymer reaction engineering: a Review of Calorimetry and State Estimation. Polym. React. Eng. 2000, 8, 1. (7) Tirrell, M.; Gromley, K. Composition control of batch copolymerization reactors. Chem. Eng. Sci. 1981, 36, 367. (8) Kravaris, C.; Wright, R. A.; Carrier, J. F. Nonlinear controllers for trajectory tracking in batch processes. Comput. Chem. Eng. 1989, 13, 73. (9) Hamielec, A. E.; MacGregor, J. F.; Penlidis, A. Multicomponent free-radical polymerization in batch, semi-batch and continuous reactors. Makromol. Chem., Macromol. Symp. 1987, 10/11, 521. (10) Arzamendi, G.; Asua, J. M. Monomer addition policies for copolymer composition control in semi-continuous emulsion copolymerization. J. Appl. Polym. Sci. 1989, 38, 2019. (11) Canu, P.; Canegallo, S.; Morbidelli, M.; Storti, G. Composition control in emulsion copolymerization. I. Optimal monomer feed policies. J. Appl. Polym. Sci. 1994, 54, 1899. (12) Arzamendi, G.; Asua, J. M. Copolymer composition control during the seeded emulsion copolymerization of vinyl acetate and methyl acrylate. Makromol. Chem., Macromol. Symp. 1990, 35/ 36, 249. (13) Guyot, A.; Guillot, J.; Graillat, C.; Llauro, M. F. Controlled composition in emulsion copolymerization application to butadieneacrylonitrile copolymers. J. Macromol. Sci. Chem. 1984, A21 (6 & 7), 683. (14) van Doremaele, G. H. J.; Schoonbrood, H. A. S.; Kurka, J.; German, A. L. Copolymer composition control by means of semicontinuous emulsion copolymerization. J. Appl. Polym. Sci. 1992, 45, 957. (15) Leiza, J. R.; de la Cal, J. C.; Meira, G. R.; Asua, J. M. Online copolymer composition control in the semicontinuous emulsion

copolymerization of ethyl acrylate and methyl methacrylate. Polym. React. Eng. 1993, 1, 461. (16) Canegallo, S.; Storti, G.; Morbidelli, M.; Carra, S. Densimetry for on-line conversion monitoring in emulsion homo- and copolymerization. J. Appl. Polym. Sci. 1993, 47, 961. (17) Buruaga, I. S.; Echevarria, A.; Armitage, P. D.; de la Cal, J. C.; Leiza, J. R.; Asua, J. M. On-line control of a semibatch emulsion polymerization reactor based on calorimetry. AIChE J. 1997, 43, 1069. (18) Buruaga, I. S.; Armitage, Ph. D.; Leiza, J. R.; Asua, J. M. Nonlinear control for maximum production rate latexes of welldefined polymer composition. Ind. Eng. Chem. Res. 1997, 36, 4243. (19) Urretabizkaia, A.; Leiza, J. R.; Asua, J. M. On-line Terpolymer Composition Control in Semicontinuous Emulsion Polymerization. AIChE J. 1994, 40 (11), 1850. (20) Isidori, A. Nonlinear control systems. An introduction, 2nd ed.; Springer-Verlag: Berlin, 1989. (21) Othman, N.; Fe´votte, G.; McKenna, T. F. On-line monitoring of emulsion terpolymerization processes. Polym. React. Eng. 2000, accepted for publication. (22) Nomura, M.; Fujita, K. On the prediction of the rate of emulsion copolymerization and copolymer composition. Makromol. Chem. Suppl. 1985, 10/11, 25. (23) Poehlein, G. W. Emulsion polymerization with water soluble comonomers in batch and continuous reactors. Macromol. Symp. 1995, 92, 179. (24) Othman, N. Advanced strategies for composition control in semi-continuous emulsion polymerization. Ph.D., Universite´ Claude Bernard Lyon I, Villeurbanne, France, 2000. (25) van Herk, A. M. Pulsed initiation polymerization as a means of obtaining propagation rate coefficients in free-radical polymerizations. J.M.S.-Rev. Macromol. Chem. Phys. 1997, C37, 633. (26) Beauchemin, R.-C.; Dube, M. A. Bulk terpolymer composition prediction from copolymer reactivity ratios. Polym. React. Eng. 1999, 7, 485.

Received for review March 1, 2001 Revised manuscript received August 1, 2001 Accepted November 20, 2001 IE0101994

Polymer Composition Control in Emulsion Co- and Terpolymerizations

Recommend Documents