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Ind. Eng. Chem. Res. 2006, 45, 206-211
Control of an Emulsion Polymerization Reactor Nida Sheibat-Othman* and Sami Othman LAGEP-UniVersite´ Lyon I, Baˆ t 308, 43 BouleVard du 11 NoV. 1918, 69622 Villeurbanne Cedex, France
On-line control of emulsion polymerization is important in order to optimize process productivity. In this work, multivariable nonlinear geometric control is applied in order to maximize the reaction rate in a semicontinuous emulsion polymerization reactor. Two process variables are manipulated, the jacket temperature and the monomer flow rate, in order to control both the reactor temperature and the number of moles of monomer in the reactor. The controller performance is validated experimentally on a laboratory scale reactor during the emulsion polymerization of styrene. 1. Introduction The modeling, monitoring, and control of polymerization processes are of high importance in order to optimize the process performance. One of the objectives of controlling emulsion polymerization is to maximize the process productivity, that is, to minimize the process time while ensuring the security of the process and good product quality. The process variables that affect the process reactivity are the reaction temperature and the concentrations of monomer and initiator. Using multiple criteria, optimal initial concentrations of surfactants and initiator were calculated by Fonteix et al.1 to minimize the process time while ensuring the desired number of particles and polymer molecular weight. By input/ output linearization and optimization of the reactor temperature, Gentric et al.2,3 could control the polymer molecular weight and minimize the process time in a batch polymerization reactor. Other papers treated the on-line control of the process productivity and polymer properties (polymer molecular weight distribution and/or composition) in semicontinuous emulsion polymerization by manipulating the feed rates of monomers and, sometimes, that of the chain transfer agent.4-10 In this case, the reaction temperature is maintained at a constant value. The process productivity is optimized by maximizing the concentration of monomer in the polymer particles. The reaction temperature is one of the process operating conditions that affects the decomposition rate of the initiator and the propagation rate of monomer. Optimizing the process productivity can therefore be done by controlling both the reactor temperature and the concentration of monomer. A similar technique was proposed by Araujo and Giudici.11 The authors used an iterative dynamic programming technique to minimize the reaction time and control the polymer composition by manipulating both the reaction temperature and the monomer flow rate. Using two control variables requires the employment of a multivariable control technique that is adapted to the nonlinear model at hand. Differential geometry is used in this work in order to control the process (Isidori12). The nonlinear geometric control has been applied to some free radical polymerization processes. In most cases, a PI controller was used in the external loop.13,14 Sampath et al.15 used input/output linearizing control with robust control as an external loop to control the monomer conversion and the polymer molecular weight in batch solution methyl methacrylate polymerization. * To whom correspondence should be addressed. E-mail:
[email protected].
The objective of this work is to maximize the process productivity in a semicontinuous emulsion polymerization reactor. The jacket temperature and the flow rate of monomer are manipulated in order to control the reaction temperature and the concentration of monomer in the polymer particles that affect the reaction rate. In the first part of this work, the process model is presented. Calorimetry is used to estimate the heat produced by the reaction, and that is then used to estimate the reaction rate of monomer. With this measurement, the process is reconstructed; the number of moles of free monomer and radicals in the polymer particles are estimated. At this step, the controller is applied to control the heat produced by the reaction. The controller is validated experimentally in a 3 L reactor during the polymerization of styrene.
2. Process Model The material balance of monomer in a semicontinuous reactor is given by
N˙ ) F - RP
(1)
where N is the number of moles of free monomer, F is the monomer flow rate, and RP is the reaction rate given by the following equation:
njNP V[MP] RP ) kP(T) NA µ)
(2)
njNP V NA
where µ is the number of moles of radicals, [MP] is the concentration of monomer in the polymer particles, and kP is the propagation rate coefficient that is related to the reaction temperature (T) by the following relationship:
kP(T) ) kP0 exp
( ) -EA RT
(3)
During interval II, where monomer droplets exist, the polymer particles are saturated in monomer; therefore, the concentration of monomer in the particles ([MP]) is constant. During interval III, all the monomer is assumed to be in the polymer particles, as given by the following equation:
10.1021/ie0502483 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/10/2005
{
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[MP] ) (1 - φPP)Fm MWmN 1 - φPP MWm T if (N - N) g 0 MWm Fm Fm φPP (4) N else (NT - N) N MWm + FP Fm
(
(
)
)
NT is the total amount of monomer introduced into the reactor:
NT )
∫0tF dt
(5)
A heat balance around the reactor gives the following equation:
MCPT˙ ) QR + UA(Tj - T) + FfeedCPfeed(Tfeed - T) - Qloss (6) where Qfeed ) FfeedCPfeed(Tfeed - T) Qloss is the heat loss to the environment due to the unsurrounded regions of the reactor and to the condenser. The heat produced by the reaction (QR) is related to the reaction rate by the following equation:
QR ) (-∆H)RP
(7)
For more details on the process model, see ref 16. 3. Process Monitoring Due to the exothermic nature of the reaction, the process is usually monitored by calorimetry. The temperature of the reactor and that of the inlet and the outlet of the jacket are measured on-line. The unknown terms in the heat balance (Qloss and U) can be estimated by correlation to the solid contents.6,17 This requires measuring the solid contents at discrete intervals during the reaction. The heat produced by the reaction can therefore be estimated from the heat balance. The reaction rate can then be estimated using eqs 6 and 7. This can be done by an estimator in order to filter measurement noise. For instance, Fan and Alpay18 proposed a Kalman filter to do that. Here, a high gain observer is constructed (see ref 19). The observer uses the following system:
{
1 ((-∆H)RP + UA(Tj - T) + Qfeed - Qloss) MCP R˙ P ) (t)
T˙ )
(8)
in eq 9 above (ref 10). All this information will be necessary for the on-line control of the process. 4. Process Control The process model presented in eqs 1 and 6 is nonlinear. In eq 1, the nonlinearity comes from the fact that the concentration of radicals in the polymer particles is a result of radical decomposition and termination and depends on the particles size. Moreover, the reaction parameters kp and the decomposition coefficient depend on the reaction temperature. Several states that are interacting in a nonlinear manner are, therefore, involved in this equation. An important nonlinearity comes also from µ, which can vary slowly or increase dramatically during nucleation or if a gel effect takes place. In eq 6, the heat produced by the reaction is nonlinear, since it is proportional to the reaction rate. For this reason, the process has to be controlled by a nonlinear controller. Equation 2 shows that the reaction rate depends on the concentrations of monomer and radicals in the polymer particles and on the reaction temperature. To control the concentration of monomer in the polymer particles, the flow rate of monomer is manipulated and controlling the reaction temperature is done by manipulating the jacket temperature. No attempt is done in this work to manipulate the concentration of radicals. The controllability of the system can be verified by calculating the characteristic matrix that must be nonsingular.12 Input/Output Linearization. The input/output (I/O) linearization12 is calculated by the following state feedback transformation: 1
υ)
1 ((-∆H)R ˆ P + UA(Tj - T) + Qfeed - Qloss) Tˆ˙ ) MCP 2θ(Tˆ - T) (9) θ2 (Tˆ - T) Rˆ˙ P ) (-∆H) The estimated value of RP can then be injected in eq 1 to estimate the number of moles of monomer in the reactor (N). With these measurements, the parameter µ can be estimated as
(10)
where 〈 , 〉 denotes the dual product of the elements in the brackets separated by a comma, adfn(g) represents the Lie brackets of order n, and the parameters βk are to be adjusted in a way that ensures the stability of the states of the model. Applying input/output linearization to the process gives the following system:
[]
[
υ1 N˙ ) - UA T + UA υ T˙ MCP MCP 2
]
(11)
Nonlinear Controller. With the I/O linearization, the controller takes the following form:
[
υ1 + R P F MCP 1 Tj ) υ (Q + Qfeed - Qloss) + T UA 2 UA R
[]
y)T where (t) represents the unknown dynamic of Rp. The observer is easily tuned by the parameter θ and takes the following form:
βkLkf h + (-1)0β1〈dh, adf0(g)〉u ∑ k)0
]
(12)
Linearizing the process allows us to decouple the coupled variables in order to apply a linear controller to the process. The transformation υ can be replaced by a PID controller to control the linear part of the model. Using a proportional controller gives the following control system:
[]
F Tj )
[
]
κP1(N - Nsetpoint) + RP MCP 1 (κ (T - Tsetpoint)) (Q + Qfeed - Qloss) + T UA P2 UA R (13)
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where κP1 is the proportional constant of the controller of N and κP2 is that of the controller of Tj. Equation 13 shows that the control variables take into account the nonlinearity of the model, Rp and QR that are obtained from the high gain observer developed in eqs 7 and 9. Controlling the monomer flow consists of replacing the consumption of monomer (RP), and then, the proportional part of the controller allows consideration for the difference between the set point and the process. The set point of T is calculated as a function of the heat exchanged with the jacket and the heat produced by the reaction. The set points of N and T are given by the following equations:
{
(
)
Figure 1. Heat produced by the reaction by manipulating F and Tj, manipulating F alone, and without control.
(NT - N) N + FP Fm T setpoint N QR MWm Nsetpoint ) min 1 1 setpoint FP -∆HµkP - QR MWm Fm FP [MP]satMWm
(
Tsetpoint )
(
))
(14)
-EA R log
( ) ksetpoint P kP0
Figure 2. Concentration of monomer in the polymer particles obtained while manipulating F and Tj, manipulating F alone, or without control.
with
ksetpoint ) P
Qsetpoint R (-∆H)µ[MP]
Table 1. Data Used in the Simulation
(15)
The required value of N is defined in two ways: the value necessary to bring QR to the set point and that necessary to bring [MP] to the saturation value. The minimum between these two equations is then used as the set point of N. This ensures that the constraints on the state N are respected. This means that, if the value of N required to control QR would exceed the saturation of the polymer particles, then it would not be introduced into the reactor, since this will generate monomer droplets that do not participate in the reaction but might destabilize the latex. 5. Discussion Experimental Setup. A 3 L jacketed reactor was used for the validation of the controller. The reactor is supposed to be well-stirred (stirring rate of 200 rpm). A condenser is introduced in the cover of the reactor to cool the evaporating components. Temperatures are measured in the reactor at the inlet and at the outlet of the jacket using PT100 probes. These data are acquired every 10 s. A controlled pump allows the introduction of monomer. To have the exact mass introduced into the reactor, monomer is put on a balance where the mass is also registered every 10 s. The introduced charge is heated by a jacketed tube where the fluid of the reactor jacket circulates. This tube measures 20 cm, which ensures that the components will be heated at the jacket temperature even at the highest flow rate. The heat exchanged with the introduced charge is therefore minimized and can be calculated by the following equation:
Qfeed ) FfeedCPfeed(Tin j - T) where Tin j is the temperature measured at the inlet of the jacket.
parameter
value
parameter
value
Tjmin Tjmax Tj0 kp0
40 °C 90 °C 60 °C 1.05 × 1010 cm3/(mol s)
EA κP1 (T) κP2 (N)
2.9544 × 104 J/mol 1 0.01
As required by the calorimetric technique, samples are withdrawn from the reactor every 20 min and analyzed by gravimetry using a thermobalance. This value is then introduced in an optimization algorithm in order to estimate the unknown parameters of the heat balance (the heat loss and the heat transfer coefficient between the reactor and the jacket). The calorimetric technique used has been validated on laboratory scale reactors and on a 250 L reactor.20 The heat capacity of the reactor inserts is usually neglected in the calorimetric techniques proposed in the literature, since it has a negligible effect when working under stationary conditions. However, in this work, the reactor temperature is controlled and would not be constant during the reaction. During transitions in the reactor temperature, the heat capacity of the reactor inserts must not be neglected. The capacity of the area of exchange with the jacket, the stirrer, and the baffle was estimated and introduced in the heat balance. Simulation Results. The data used in the simulations of the controller are presented in Table 1. Styrene was used as monomer. Only a proportional (P) controller was employed. A first-order reference model was added to the set point (T) with a time response that is related to the response of the reactor temperature to changes in the jacket temperature. This depends on the reaction volume and on the jacket capacity of heating and cooling. The simulation results are presented by Figures 1-4. Figure 1 shows the evolution of QR obtained while controlling F and Tj, while controlling F alone, and without control. In the case of no control, constant temperature (60 °C) and monomer flow
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Figure 3. Monomer flow rate obtained while manipulating F and Tj, manipulating F alone, or without control.
Figure 5. Experimental validation of the multivariable control of the heat produced by the reaction.
Figure 4. Reactor temperature obtained while manipulating F and Tj, manipulating F alone, or without control.
Figure 6. Controlled monomer flow rate and reaction rate.
rate (2.5 × 10-4 mol/s) were applied. In this case, the number of moles of monomer consumed to produce polymer during the simulation time was 2.96 mol. In the case of controlling QR by manipulating only the monomer flow rate, 4.13 mol of polymer were produced. In the case of controlling QR by manipulating both the jacket temperature and the monomer flow rate, 4.5 mol of polymer were produced in the same lapse of time. It is obvious that the time of convergence depends on the jacket capacity that was limited at 1 °C/min in order to be compared to big reactors. This means that if no control is applied, a longer time is required in order to consume the same amount of monomer, which shows that the controller allows minimizing the process time. Figure 2 shows the concentration of monomer in the polymer particles. In the controlled process, [MP] does not exceed the saturation value during the reaction, respecting, therefore, the constraints in eq 14. However, in the case where the monomer flow rate is not constrained in a way to respect the saturation value of [MP], nothing ensures that this condition is respected and more monomer than desired can be introduced. The employed monomer flow rate and the reaction temperature are shown in Figures 3 and 4, respectively. When the monomer flow rate is manipulated, it is null during interval II, where the polymer particles are saturated in monomer, but is maintained constant in the open-loop simulation. The jacket temperature was maintained at 60 °C in the case of manipulating F alone and when no control was employed. In the controlled process, the convergence time of T to the set point is limited by the jacket capacity. Experimental Validation. Styrene was used for the experimental validation of the controller. The used recipe is given in Table 2. The desired heat produced by the reaction was set to 60 W. Figure 5 shows the evolution of QR during this experiment. No attempt was made to accelerate the convergence time to the set
Figure 7. Concentration of monomer in the polymer particles ([MP]). Table 2. Experimental Validation of the Controller (C12)
styrene dodecyl sulfate, sodium salt potassium persulfate H2O
initial charge (g)
feed (g)
145 4 4 1000
1000 9 500
point by increasing the reaction temperature. The jacket temperature was fixed at 60 °C until QR attained the set point (about 15 min). This was done to anticipate the heat generated by the nucleation taking place at the beginning of the reaction. During the first 20 min, the monomer flow rate (Figure 6) was equal to zero, since the polymer particles were saturated in monomer as shown in Figure 7. Afterward, the monomer flow rate was controlled in a way that maintained the number of moles of monomer close to the set point calculated by eq 14, as given by Figure 8. The difference at the beginning is due to the initial values chosen for the nucleation of new particles. The second control variable, the reactor temperature (Figure 9), was calculated in a way that maintained kP close to the set point calculated by eq 3. The desired value of T was not really applied during the first 15 min in order to anticipate the heat produced by the nucleation, as mentioned above.
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Figure 8. Desired and real number of moles of free monomer.
Figure 11. Number of moles of radicals in the polymer particles (µ) as a function of [MP].
Figure 9. Desired and real reaction temperature. Figure 12. Average number of radicals per particle (nj) obtained off-line by introducing the particle size.
Figure 10. Number of moles of radicals in the polymer particles (µ).
While the system converged to the set point, two peaks of QR could be seen: The first one is due to the heat produced by the nucleation, which is a rapid phenomenon that is difficult to control. The second peak (at about 50 min) corresponds to a rapid change in the number of moles of radicals in the polymer particles, the gel effect. Figure 10 shows the evolution of µ as a function of time. At 50 min, µ increased from 1 × 10-6 to 6 × 10-6 in 20 min. This is due to the decrease in [MP]. Figure 11 shows that, at low [MP] values, µ increases importantly. It should be mentioned, however, that µ depends also on the reactor temperature. This is why a multiplicity can be seen in Figure 11. It is worthy to mention that a reference model was added on Tsetpoint in order to avoid oscillations which slow the convergence of Tj to the set point. It can be seen that, even under these conditions, the controller could bring QR back to the set point. It is worthy to say that using the flow rate as a unique control variable would not suffice to control this process in such a situation, since decreasing the flow rate would perhaps increase µ for a fraction of time, after which µ would, of course, decrease. This was avoided by controlling the temperature. Figure 5 shows that the reaction temperature decreased by 2 °C at 50 min. After these two peaks, QR was maintained at the set point with an error of 5%. Figure 5 shows that the reactor
Figure 13. Number of particles.
Figure 14. Experimental and real monomer conversion.
temperature increases by about 8 °C over the course of the reaction in order to maintain QR close to the set point. This increase was a result of a decreasing [MP] and an increasing µ. By introducing the number of particles (Figure 13), obtained from the particle size off-line, in eq 2, the average number of radicals per particle can be calculated off-line (Figure 12). As
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usual, nj is close to 0.5 during interval II. Afterward, nj increases due to a slight gel effect taking place during interval III. Finally, Figure 14 shows the evolution of the monomer conversion compared to the experimental data obtained gravimetricaly. It can be seen that the calorimetric strategy gives accurate results. 6. Conclusion A multivariable nonlinear controller was developed to control the reaction rate in emulsion polymerizations. Input/output linearization with a proportional controller was applied to the material and heat balances of a semicontinuous reactor. This allows for control of the monomer flow rate and the jacket temperature to maintain QR close to the set point. It was found that a reference model on the set point was necessary in order to avoid oscillations in the controlled variables. The control scheme can be ameliorated by adding constraints on the reaction temperature that can be related to the polymer properties, such as the polymer molecular weight. Actually, the polymer molecular weight depends on the reaction temperature and the concentrations of monomer, radicals, and the chain transfer agent. Since the controller was found to have different degrees of freedom, the control objective can be realized at different levels of the reaction temperature. A constraint on the reaction temperature would limit these levels. Nomenclature A ) heat transfer area (m2) CP ) heat capacity of the reaction medium (J/(kg °C)) CPfeed ) heat capacity of the feed (J/(kg °C)) EA ) activation energy (J/mol) F ) monomer flow rate (mol/s) Ffeed ) input flow rate (kg/s) kP ) propagation rate coefficient (cm3/(mol s)) kP0 ) pre-exponential factor (cm3/(mol s)) M ) reaction medium mass (kg) [MP] ) concentration of monomer in the polymer particles (mol/ cm3) MWm ) monomer molecular weight (g/mol) N ) number of moles of free monomer (mol) NT ) total amount of monomer introduced to the reactor (mol) NA ) Avogadro’s number NP ) particle number Qloss ) heat loss to the environment (W) QR ) heat produced by the reaction (W) RP ) reaction rate (mol/s) R ) universal gas constant (J/(mol K)) T ) reaction temperature (°C) Tj ) jacket temperature (°C) Tfeed ) input temperature (°C) U ) heat transfer coefficient (W/(°C m2)) V ) reaction medium volume (cm3) µ ) number of moles of radicals in the polymer particles (mol) φpP ) volume fraction of polymer in the particles under saturation nj ) average number of radicals per particle Fm ) monomer density (g/cm3) Fp ) polymer density (g/cm3) ∆H ) reaction enthalpy (J/mol)
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ReceiVed for reView February 25, 2005 ReVised manuscript receiVed October 4, 2005 Accepted October 10, 2005 IE0502483
