Ind. Eng. Chem. Res. 2002, 41, 4233-4241
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Online Reaction Calorimetry. Applications to the Monitoring of Emulsion Polymerization without Samples or Models of the Heat-Transfer Coefficient S. BenAmor,† D. Colombie´ ,†,‡ and T. McKenna*,§ ATOFINA, CERDATO, BP 19, 27470 Serquigny, France, and LCPP-CNRS/CPE-Lyon, 43 Bd du 11 Novembre, Baˆ t 308F, BP 2077, 69100 Villeurbanne Cedex, France
A high-gain nonlinear cascade state estimator is presented in this work and used to monitor conversion and polymer composition in an emulsion copolymerization process. The advantage of this cascade observer is that it requires neither occasional online samples nor prior knowledge of the dependence of the overall heat-transfer coefficient with the solids content of latex. A simple test is used to ensure that the results are robust to process noise and to rapid changes in conditions during the semibatch phase. Applications of this technique are presented in both laboratory-scale and industrial pilot-scale reactors. 1. Introduction Much work has been done on the design of systems for the monitoring and control of polymer reactors, for the simple reason that being able to control the polymer/ latex properties opens the door to new products with tighter specifications. The complex nature of the polymerization reaction, combined with a lack of online sensors for the determination of polymer properties, requires us to rely on software sensors or state estimation techniques to follow what is occurring in the reactor. Recently, combinations of hardware sensors with state estimation techniques have been discussed in the literature (e.g., refs 1-22). The software sensors discussed in these papers are based on a model of the process and use online measurements reflective of the polymerization reaction to provide knowledge on properties such as the polymer composition. Such measurements are often referred to as “indirect” because while they are related to the polymerization reaction, they do not give a direct indication of the polymer properties. The use of reaction calorimetry in polymerization reactors has been widely explored in recent years.3,5-21,23,24 Because modern reactors are typically well instrumented in terms of measuring temperature and (usually) material flow rates, and because vinyl polymerizations are usually exothermic, the measurement of the amount of heat generated by the reaction can be used to monitor the rate of polymerization if we can clearly identify the heat-transfer coefficient and heat loss from the reactor. The noninvasive nature and low cost of this technology make it quite attractive.11,13,15 The reliability of calorimetric techniques resides in the relation between the heat flow, kinetics, and thermodynamics that are involved in the reacting medium.19,25 However, one of the most difficult problems to overcome in heat flow calorimetry is the need to know the global heat-transfer coefficient (UA) at all times during the reaction. Fouling * Corresponding author. E-mail:
[email protected]. Telephone: (+33) 4 72 43 17 75. Fax: (+33) 4 72 43 17 68. † ATOFINA, CERDATO. ‡ Current address: Alcatel Fibres Optiques, Atelier Fibrage/ Usine de Conflans, 53 rue Jean Broutin, 78703 Conflans Ste Honorine Cedex, France. § LCPP-CNRS/CPE-Lyon.
problems and semibatch operation make this difficult, although in some cases it is reasonable to make certain approximations (e.g., assume that fouling is not important and that UA varies in a known manner as a function of the solid content for a given reactor).5,6,9,10,16,17 If it is either not possible or not desirable to use a model for UA, an additional sensor can be used to provide input such as conversion or solids content measurements to help calculate UA online. This approach, referred to as adaptive calorimetry, has also been discussed in the literature.3,8,14,22,23 Briefly, in this method, we forego the use of predefined parameters in a model for UA and use occasional measurements of the overall conversion to optimize them online. The advantage is that this method works when there is fouling in the reactor; the disadvantage is that it requires an additional measurement. In this paper, we will look at a method for the online monitoring of emulsion terpolymerization by the simultaneous estimation of the heat of polymerization and the overall heat-transfer coefficient using only the measurements of temperatures and feed flow rates and state estimation techniques. The method we propose provides a measurement of the heat of reaction, Qr, and of UA without the need for an extra measurement of the overall conversion. From this result, estimates of the conversion and polymer composition using the approach developed by Urretabizkaia et al.19,24 are calculated. The focus of the work is on the use of state estimators for calorimetry rather than for estimation of the composition. The technique developed below is based on the use of a “cascade” of two state observers. It will be experimentally tested online for the emulsion terpolymerization of styrene (STY)-butyl acrylate (BuA)-methyl methacrylate (MMA) in both laboratory- and pilot-scale reactors. The use of the estimation strategy for the monitoring of the scaling up of recipes is discussed. The pertinence of the choice of the initial value of the heat global coefficient is also illustrated. In addition, some comments regarding the online diagnosis are given and supported by examples. 2. Motivation and Problem Setup The general energy balance includes the heat balance of the jacket and the reactor heat flow balance:
10.1021/ie010948h CCC: $22.00 © 2002 American Chemical Society Published on Web 07/19/2002
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Qr ) UA(Tr - Tj) + (mCp)r
∑i
dTr
+ dt FiCpi(Tr - Tfeed) + Qcond + Qloss (1)
The enthalpy balance on the reactor in eq 1 involves the contribution of the rate of heat generation due to reaction Qr, the heat flow through the reactor walls, the accumulation of enthalpy in the reactor, the sensible heat of the feed, the heat removal through condensation, and a heat loss term, respectively. A similar balance can be developed for the cooling jacket, but if we wish to use the jacket heat balance, we will need information on the flow rate of the cooling fluid and the inlet and outlet jacket temperatures (see, e.g., refs 3 and 13). In the current study, the only measurements available were the inlet jacket temperature for the laboratoryscale reactor and the inlet and outlet temperatures for the pilot- and industrial-scale reactors. In both cases, the flow rate of the cooling fluid was not measured. For these reasons, our approach was based on the reactor energy balance in eq 1. For the determination of the heat of reaction, Qr, the global heat-transfer coefficient UA as well as the temperatures, Qcond and Qloss, must be known. In this study, Qcond was lumped in with the heat loss term Qloss, which in turn was considered to be known (or measurable). Available sensors gave the temperature measurements and the mass of monomers fed to the reactor. The initial value of the heat-transfer coefficient U as well as that of the effective heat-exchange area is also assumed to be known (see refs 13 and 22). Both U and A may change during the course of the reaction because of increasing viscosity, addition of feed streams, changing density, and reactor wall fouling so that the product of these two values will be calculated throughout the reaction. The laboratory-scale reactor is a 3-L commercial reaction calorimeter (RC1; Mettler-Toledo) that is optimized as a so-called heat flow calorimeter (Tr mode). The operating principle of the RC1 is presented by Moritz and co-workers in ref 3, where IR spectroscopy and additional sensors are used to control the chemical composition of a free-radical copolymerization. The RC1 software calculates the value of U based on a linear (or other) extrapolation between the initial and final values. This behavior of U is perfectly adequate for polymerization reactions having a low viscosity of the latex, which is not the case for the polymerization reactions that we consider because the final solids content is about 50%. For example, measurements of the heat-exchange coefficient in the RC1 were performed at different solids contents, and Figure 1 shows the behavior of U versus the mass fraction of polymer in the latex (based on calibration runs where a heating element is used to generate a known quantity of heat). The estimation of the heat-exchange coefficient is crucial to the determination of the heat of polymerization even when the reaction calorimeter RC1 is used. In addition, it is necessary to complete the reaction in order to have the pre- and postpolymerization values of U to do the extrapolation if we use this type of commercial system. In previous studies, certain authors (e.g., refs 16 and 17) developed correlations linking UA to conversion or solids content. This is probably a valid approach in a laboratory-scale glass reactor but not in industrial reactors at larger scales where fouling (e.g., due to
Figure 1. Behavior of the heat-transfer coefficient versus the solids contents based on calibration runs. Different symbols indicate replicate runs. The value of U0 (zero solids) is not shown as an experimental point. However, the extrapolation of this curve to zero solids content is in very close agreement with the value calibrated using the Mettler software.
splashing of the walls during semibatch operation or foaming) of the heat-exchange surface can be a problem. Other authors proposed “adaptive calorimetry”, where occasional measurements of the overall conversion were obtained by gravimetry (or some other method) and used to correct the terms UA and Qloss in the energy balance.8,14,22,23 This method presented the advantage of dispensing with the need for a model of UA but required online samples of the overall conversion to recalibrate the estimate of UA in real time. While being fine for research studies, the need for online measurement of the conversion can make this approach problematic if probes such as ultrasonic or IR sensors are not available in an industrial reactor. 3. Estimation Strategy The objective of the current paper is to simultaneously determine the rate of heat generation caused by the polymerization reaction and the overall heat-transfer coefficient, using heat flow calorimetry described by eq 1 and state estimation techniques. The results of state estimation will serve as a substitute measurement for the conversion and polymer composition estimation. If enough information were available, we could simultaneously estimate the heat of polymerization Qr and the global heat-transfer coefficient UA using the only equation of heat flow from the following system:
{
dTr
(
∑)
dt
)
Qr
-
(mCp)r
∑i FiCp (mCp)r
UA
(Tr - Tj) + (mCp)r
i
(Tfeed - Tr) -
Qloss (mCp)r
(2)
dQr
) 1(t) dt dUA ) 2(t) dt y ) Tr
where y is the system output. Unfortunately, system (2) is not observable if both UA and Qr are unknown.26 To overcome this drawback, recall that, at t ) 0, the initial values of both Qr and UA are known. Because there is no reaction at t ) 0, Qr,0 ) 0. The value of the initial
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heat-transfer coefficient, U0, can easily be obtained, for instance, by using the technique given in ref 13. Another technique estimating U0 is given in ref 22. The initial effective exchange area should also be known because both the initial charge and the reactor shape are known. Now, if UA is known, we can estimate Qr from the following system:
{
dTr
(
∑1)
1
)
dt
(mCp)r
Qr +
1 (mCp)r
i
dQr
(3)
{
step 7
1
)
dt
(mCp)r
Q ˆr +
1 (mCp)r
[-v1(T ˆ r - Tj) +
∑i FiCp (Tfeed - Tˆ r) - Qloss] - 2θ1(Tˆ r - y)
(O1)
i
ˆ r - y) ) -θ12(mCp)r(T
dt
(4)
is an observer for system (Σ1), where θ1 is a tuning parameter. The symbol ∧ indicates that the variables are estimated values. The observer synthesis is given in Appendix A1. To estimate UA in the case where Qr is known, consider the system
{
dTr
(
∑2)
)-
dt
UA(Tr - Tj)
+
(mCp)r
1 (mCp)r
[v2 +
∑i FiCp (Tfeed - Tr) - Qloss] i
(5)
dUA
) 2(t) dt y ) Tr
{
)-
dt
U ˆ A(T ˆ r - Tj) (mCp)r
+
1 (mCp)r
[v2 +
∑i FiCp (Tfeed - Tˆ r) - Qloss] - 2θ2(Tˆ r - y)
(O2)
[
i
]
(mCp)r U ˆA + θ22 (T ˆ r - y) ) 2θ2 dt T ˆ r - Tj T ˆ r - Tj
dU ˆA
using Qr ) 0 at time t ) 0 and ν1 ) U0A0 for t ∈ [0, δt], estimate Qr by integrating observer (O1) from t ) 0 to δt estimate UA using U0A0 as the initial state variable and take ν2 ) last estimated value of Qr for t ∈ [0, δt], by integrating observer (O2) from t ) 0 to δt calculate conversion and polymer composition using the last estimated value of Qr estimate Qr with ν1 ) last estimated value of UA from t ) δt to 2δt estimate UA with ν2 ) last estimated value of Qr from t ) δt to 2δt calculate conversion and polymer composition using the last estimated value of Qr repeat steps 4-6 for every time stage [(k - 1)δt, kδt], k g 2, until the final reaction time
Using the estimated heat of polymerization, information related to the overall conversion is directly available. For instance, Urretabizkaia et al.19 developed an approach to estimate the evolution of the copolymer composition in batch emulsion copolymerization systems using calorimetric data. This method is based on the knowledge of the reactivity ratios and the monomer partitioning between the different phases.27 This approach can be easily extended to semibatch co- and terpolymerization reactions and is used in the present work. When the profiles of the jacket and reactor temperatures never cross (i.e., when one is always greater than the other), system (Σ2) is always uniformly observable. In this case, the estimation procedure at each time stage of duration δt (i.e., the time between two consecutive temperature and feed flow-rate measurements) would be as outlined in Table 1. Note that, in practice, even if the measurements are available every 2 s, the time stage is taken between 10 and 20 s. It is worth pointing out that this procedure gives good enough results, as will be shown in the next section, because the behavior of the dynamics of the global heat-transfer coefficient is smooth enough. If during a time interval the dynamics of UA were to change significantly, the loss of information might adversely influence the estimation results, and this strategy could fail. 4. Experiments
where ν2 is a signal input. Consider U as the set of inputs such that Tr * Tj. The system (Σ2) is observable for every input belonging to U. Hence, the system
dT ˆr
step 5 step 6
) 1(t)
where y is the system output and ν1 is a signal input. System (Σ1) is observable13 because (mCp)r is always nonzero and ν1 can be any signal input. Moreover, the system (Σ1) is uniformly observable.12 Hence, the system
dQ ˆr
step 2
step 4
dt y ) Tr
dT ˆr
step 1
step 3
[-v1(Tr - Tj) +
∑i FiCp (Tfeed - Tr) - Qloss]
Table 1. Algorithm Used for Cascade Estimation of UA and Qr
(6)
is an observer for system (Σ2) for every input in U and θ2 is a tuning parameter. The proof of observability and the observer design for (Σ2) are given in Appendix A2.
Semicontinuous, unseeded terpolymerizations of STYBuA-MMA were carried out at 80 °C in 3- and 250-L reactors. The reactor temperature is maintained at the set point by circulating hot or cold fluid in the jacket, and a pre-emulsified mixture of monomer, water, initiator, and surfactant is fed at a constant rate for a fixed period (usually 4.5 h). As mentioned earlier, the laboratory-scale reactor is a 3-L commercial reaction calorimeter (RC1; Mettler-Toledo). Calibrations with a resistance heater were used to determine that the initial value of the heat-transfer coefficient was U0 ) 117 W/K/ m2. The pilot-scale reactor is a 250-L metallic reactor (P250). The initial value of the heat-transfer coefficient is U0 ) 350 W/K/m2 and was determined using the method outlined in ref 22. All reactions consisted of three phases: (1) a batch seed creation or nucleation stage, (2) a semibatch particle growth stage, and (3) a finishing stage. The nucleation step is a batch reaction during which the
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Table 2. Recipes for the Terpolymerization component
3-L reactor mass (kg)
250-L reactor mass (kg)
water initiator emulsifier pre-emulsion
Initial Charge 0.0895 0.0005 0.0063 0.0995
53.730 0.030 0.420 6.630
water initiator pre-emulsion
Feed 0.0729 0.0039 1.4925
4.860 0.260 99.500
Pre-emulsion Composition water 0.3552 emulsifier 0.0700 styrene 0.4229 butyl acrylate 0.5073 methyl methacrylate 0.2114
27.860 0.0055 33.170 39.800 16.580
Figure 3. Estimated heat-transfer coefficient and instantaneous heat of polymerization for the laboratory-scale reactor.
Figure 4. Estimated overall conversion compared to the gravimetric one for the laboratory-scale reactor.
Figure 2. Reactor and jacket temperatures for the laboratoryscale reactor.
solids content was invariably between 9 and 10 wt %. This stage lasted approximately 20 min, long enough to fix the number of particles nucleated and to increase conversion to between 90 and 95 wt %. The semibatch stage usually lasted between 3 and 4.5 h. During this stage a concentrated mixture of monomer, surfactant, and water (60-80% monomer by weight) was fed to the reactor. A separate stream of an aqueous solution of a persulfate initiator was fed to the reactor at a rate that allowed us to maintain a constant radical concentration. The finishing stage typically lasted between 20 and 30 min. This stage is also a batch stage and is used to consume most of the residual monomer remaining in the reactor at the end of the second step. The runs carried out in the RC1 were identical to the pilot-plant runs in terms of relative concentrations and lengths of reaction. The recipes for the laboratory- and the pilot-scale reactors are given in Table 2. 5. Results and Discussion The results of a typical laboratory-scale run are shown in Figure 2, where we can see the temperatures, and Figure 3 shows the estimated dynamics of the rate of heat generation due to polymerization (Qr) and the estimated heat-transfer coefficient (U) compared to the ones calculated by the RC1. To validate these estimations, the estimated overall conversion obtained from this estimated value Qr is compared to offline gravimetric measurements from the same experiment in Figure 4. This and similar laboratory runs allowed us to identify a difficulty with the procedure outlined in Table 1. In our case, both heating and cooling the jacket
temperature perform the regulation of the reactor temperature. Therefore, the temperature of the jacket can punctually be greater than or equal to that of the reactor during the nucleation stage and initial moments of the semibatch stages, which means that the system becomes unobservable. According to Figure 2, the reactor and jacket temperature profiles cross three times during the nucleation and the beginning of the batch steps. During the nucleation step, the high monomer concentration and rapid generation of particles means that a significant amount of heat is generated, so the reactor temperature increases rapidly, causing the controller to tell Tj to decrease. Because the jacket temperature is higher than the reactor temperature before the polymerization begins (we need to heat the reactor to the appropriate temperature), Tr rapidly exceeds Tj. This rapid heat generation causes the regulator to decrease the jacket temperature significantly. However, the residual monomer is rapidly consumed, which in turn leads to a decrease in the amount of heat generated by the reaction, and the reactor temperature Tr drops below the set point. The jacket temperature subsequently exceeds Tr once again. Then, when the feed begins, Tr increases as the monomer begins to react, and Tj drops one last time to counteract this and then remains inferior to the reactor temperature. If we attempt to apply the procedure defined in Table 1 throughout the entire reaction under the conditions described above, more often than not the observer will not converge toward physically realistic values of Qr and U, regardless of the value of the gains specified. However, looking closely at the progression of the reaction shows us that the zone where Tr crosses Tj at the beginning of the reaction corresponds to a period where the solids content is lower than 10-15%. It is, therefore, reasonable to suppose that U remains con-
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4237 Table 3. Modified Algorithm Used for Cascade Estimation of UA and Qr step 1 step 2
step 3 step 4 step 5 step 6 step 7 step 8 step 9 step 10 step 11
using Qr ) 0 at time t ) 0 and ν1 ) U0A0 for t ∈ [0, δt], estimate Qr by integrating observer (O1) from t ) 0 to δt estimate UA using U0A0 as the initial state variable and take ν2 ) last estimated value of Qr for t ∈ [0, δt], by integrating observer (O2) from t ) 0 to δt calculate conversion and polymer composition using the last estimated value of Qr estimate Qr with ν1 ) U0A with the value of A modified to account for increased emulsion volume from t ) (k - 1)δt to kδt estimate UA with ν2 ) last estimated value of Qr from t ) (k - 1)δt to kδt calculate conversion and polymer composition using the last estimated value of Qr repeat steps 4-6 for every time stage [(k - 1)δt, kδt], k g 2, until the amount of solids content reaches 25% estimate Qr with ν1 ) last estimated value of UA from t ) (k - 1)δt to kδt estimate UA with ν2 ) last estimated value of Qr from t ) (k - 1)δt to kδt calculate conversion and polymer composition using the last estimated value of Qr repeat steps 8-10, for every time stage until the final reaction time
stant to within a small margin. Therefore, to overcome problems of observability during the early stages of the reaction, the value of U used as a signal input in the observer (O1) is taken to be equal to U0 until the solids content reaches 20%, which is justified by the measurements presented in Figure 1. The heat-exchange area is calculated as soon as the semibatch feed begins. The results reported in this paper confirm that this approach is valid under typical industrial conditions. For this reason, the procedure in Table 1 was modified as shown in Table 3. This modified algorithm was applied to the laboratory- and pilot-scale reactors. For the pilot-scale reactor, the inlet and outlet jacket temperatures (Tj,in and Tj,out, respectively) were available. To have a more precise jacket temperature Tj, the log-mean value of the jacket temperature was taken: Tj ) (Tj,out - Tj,in)/ln(Tj,out: Tj,in). Also, second-order filters were used to smooth the information provided by the temperature sensors. The tuning parameters for the observers, θ1 and θ2, were taken to be exactly the same as those for the laboratoryscale experiments. The online run for the terpolymerization of STY-BuA-MMA was performed using the recipe given in Table 2. The estimation results are shown in Figures 5 and 6 where we can see the estimated values of UA and Qr for the pilot-scale reactor and a comparison of the gravimetric (offline) and online estimates of conversion using Qr, respectively. Once again the estimated and measured conversions show a satisfactory agreement, which demonstrates that the estimator proposed here provides reliable values of Qr. The assumption of a constant UA during the batch run does not introduce any noticeable error, and the method outlined in Table 3 provides reliable estimates of both UA and Qr throughout the entire batch period. The estimated heat of polymerization is used to determine the terpolymer composition using the method proposed by Urretabizkaia et al.24 An open-loop observer for the material balance was used to estimate online the polymer composition. From latex samples taken during the course of the reaction, GC measurements (offline) of the STY, BuA, and MMA mass fractions were
Figure 5. Estimated heat-transfer coefficient and estimated instantaneous heat of polymerization for the pilot-scale reactor.
Figure 6. Estimated overall conversion compared to the gravimetric one for the pilot-scale reactor.
Figure 7. Estimated and GC polymer compositions for pilot(dashed lines) and laboratory-scale (solid lines) reactors. Points are GC measurements for the pilot-scale reactor.
performed for the pilot-scale experiment. Figure 7 shows the comparison between the estimated and GC measurements. A satisfactory agreement can be observed for all of the points with the exception of the next to the last sample (at a reaction time of 266 min). However, it is likely that this discrepancy comes from an error in analyzing the sample rather than from the observer output. 6. Applications to Scale-up In practice, new formulations are first carried out in laboratory-scale reactors and are then generally scaled up first on a pilot-scale reactor (hundreds to thousands of liters) and then on an industrial-scale reactor. Problems such as safety (in particular, thermal runaway
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Figure 8. Estimated normalized pilot- and laboratory-scale heats of polymerization.
Figure 9. Estimated heat of reaction with different initial values of the heat-exchange coefficient (U0).
or pressure buildup) or sensitivity of the product formulation to changes in characteristic times for physical and chemical processes can arise when the recipe for the pilot- and industrial-scale reactors is not well defined (e.g., formation of agglomerations in the latex in large reactors due to poor dispersion of initiator feeds in large reactors when this is not a problem in a much smaller reactor). Because it is usually not practical to withdraw samples from pilot- and industrial-scale reactors at regular times during the course of the reaction, only the final latex characteristics can be verified. This can create difficulties in terms of interpreting what occurred during the experiment. On the other hand, online sensors such as calorimetry can be a useful diagnostic tool in scale-up because monitoring UA and Qr not only allows one to obtain the information needed to follow and control the polymer composition and to help prevent thermal runaway but also allows us to have information on the state of the reactor itself (e.g., formation of deposits on the reactor wall reflected by unexpected decreases in UA) and to clearly identify occasions where the rates of reaction per unit volume (Rp) are different in the large and small reactors. In principle, the same polymerization recipe should yield Rp and thus a value of Qr per unit volume (Qrv) regardless of the size of the reactor. If it does not, then changing the reactor has an impact on the polymerization. One of the advantages of online calorimetry is that knowledge of how changing scales impacts the value of Qrv tells us a great deal about what is occurring during the polymerization. Because the software sensor presented above is independent of the scale and geometry of the reactor, it can easily be applied to this end. Values of Qrv are compared for the laboratory- and pilot-scale runs in Figure 8. The estimated and gravimetric conversions for both pilot- and laboratory-scale reactors are presented in Figures 4 and 6, respectively, and the estimated polymer composition for both scales of reactors are compared to the offline measurements of composition found from GC measurements of the pilot-scale samples in Figure 7. These results demonstrate that the estimation strategy given in Table 3 is efficient and can be used for the monitoring of both small- and medium-scale experiments. In addition, the recipes seem to be easily scaled-up in terms of conversion and composition, yet some small differences appear. Some of these differences are due to the means of treating the input measurements. For instance, the amplitude of the peak values during the nucleation stage at approximately 10 min is slightly higher than that for the RC1. This is due to the fact that the
temperature measurements used for the pilot-scale reactor were filtered. In addition, the curves of Qrv are less noisy for the RC1 runs than for the pilot-scale runs because the instrumentation of the former is more precise. However, given the fact that this information is integrated to find conversion and composition data, the noise in the temperature measurements has little to no impact on the estimated values of conversion and composition. Also, the duration of the stabilization during the nucleation step was 26 min for the pilot-scale reactor and 19 min for the laboratory-scale run. This can also be observed at the end of the semibatch step because the feeding time was exactly the same (4.5 h) for both experiments. 7. Sensitivity and Robustness The high-gain observer is a global observer; i.e., the initial value of the estimated variables can (in theory) be chosen arbitrarily, and the observer will converge. However, when the initial values of the estimated variables are far from the true values, observer convergence can be quite slow. It is, therefore, preferable to choose initial values for the estimated variables as close as possible to those of the process. However, this can often be difficult for many reasons, so it is necessary that the observer be insensitive to “reasonable” errors in the initial conditions as well as to process noise. Two high-gain observers are used sequentially in the estimation strategy presented here. This idea allowed us to overcome the unobservability of the whole system, but this is only possible if the initial value of the heatexchange coefficient is known (the initial value of Qr is always zero). For this reason, we need to explore the implications of uncertainties in the value of U0. As shown in Figure 9, when U0 is greater than 350 W/K/m2 (what we assume to be very close to the true value), the amplitude of the exothermic point due to the nucleation is higher and vice versa when it is lower. Although the three initial values eventually converge to the same value of Qr, the time it takes for this to happen is several hours. Also, if we consider Figure 10, it can be seen that the value of the heat-transfer coefficient estimated with U0 ) 450 W/K/m2 does not converge to the same value as the base case during the entire run. Therefore, to have confidence in the estimates provided with this technique, it is important to have a reliable value of the initial heat-exchange coefficient. Fortunately, this can be calibrated during the reactor preparation phase, as outlined in ref 22. Even if reliable values of U0 are available, it is still important that the estimation scheme be robust to
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4239
Figure 10. Estimated heat-exchange coefficient with different initial values for U0. Straight lines at the beginning of each curve correspond to the period where a constant U0 is imposed.
Figure 11. Evolution of temperature profiles and estimated value of the heat generation rate due to polymerization for a typical laboratory-scale run. In the first phase of the reaction, the polymerization is run in batch and semibatch as described above. There is a subsequent change to isoperibolic operation, followed by a loss of control of the jacket temperature. The observer is robust to the mild changes in reactor conditions.
process noise or to extremely rapid changes of the input values. Several tests were run with different types of changes, and the results of two of these are shown in Figures 11 and 12. It was found that the observer output was occasionally sensitive to rapid changes in either Qr or UA (because this type of operation invalidates the hypothesis that the dynamics of these estimated quantities are slow). To overcome this problem, a “sensitivity” test was applied to the estimated value of UA. Because UA should be insensitive to rapid changes in Qr, it was decided that if the output of UA at time tk (where k indicates the current time step) evolved by more than 5% with respect to the value found at tk-1, then UAk was set equal to UAk-1. This should allow us to account for physically realistic movements in UA(t), while eliminating unreasonable estimates because there is no physical way in which U should evolve more rapidly that that. As shown in Figure 11, this test is not necessary when the evolution of conditions inside the reactor are relatively mild. In the laboratory-scale run shown here, the polymerization was run under normal conditions for the first portion of the run and then the reactor was shifted into isoperibolic mode. At the end of isoperibolic operation, the intention was to shift the reactor back to isothermal mode, but we suffered a loss of control on Tj for reasons which are not clear. Nevertheless, it can be seen from this figure that when the dynamics of Tr (and
Figure 12. Sensitivity of estimations to rapid changes in conditions inside the reactor. Without the sensitivity test, the estimator does not converge to a reasonable final state.
thus Qr) are not particularly rapid, the cascade observer presented here is robust to this type of change in process conditions. On the other hand, when the conditions in the reactor change in such a way that Qr changes very quickly, it is necessary to apply the test described above. This is illustrated in Figure 12, where we have injected a shot of initiator during the semibatch stage. The new nucleation caused the temperature of the reactor to cross that of the jacket, thus rendering the system (Σ2) unobservable. A perturbation on the estimated U is observed without the test. However, when the test is used, the output of the estimator for U remains constant (as we would expect from Figure 1) and the heat of polymerization presents a new high exothermic point. The results of these two runs show that a simple, practical solution can be applied in a number of relevant cases to prevent loss of information when the conditions of observability are momentarily violated. This type of test allows us to ensure that the observer output remains reflective of what is happening in the reactor at a given time. However, it should be pointed out that although this approach allows us to render the estimation procedure robust to incidents that could occur under industrial conditions, it is not intended to be a fault detection system (quite the contrary in fact). 8. Conclusions We have developed and validated a simple type of cascade observer that allows us to use online calorimetry for polymerization reactions. The advantage of this observer is that it can be used without online sampling
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and it does not require a model for the evolution of UA with solid content. For this reason, it can be used in realistic industrial conditions, even when fouling occurs and causes UA to evolve differently from run to run. The only parameter that needs to be known well is the initial value of UA, but it has been shown in the literature that we can estimate its value fairly easily during the process preparation stage.22 Finally, a simple test that prevents UA from evolving too quickly due to a momentary loss of observability can be implemented easily.
dT ˆr
(O1)
Appendix A1: Estimation of Qr In this appendix, we give the high-gain observer synthesis for the estimation of Qr. Consider system (Σ1) and denote it by
()( )
r x x ) x1 ) T Q 2 r
{
Straightforward calculations give the observer for system (Σ1):
dt
)
1 (mCp)r
1
Q ˆr +
(mCp)r
(-ν1(T ˆ r - Tj ) +
∑i FiCp (Tfeed - Tˆ r) - Qloss) - 2θ1(Tˆ r - y) i
dQ ˆr dt
ˆ r - y) ) -θ12(mCp)r(T
Appendix A2: Estimation of UA In this appendix, we give the proof of the observability of the system (Σ2) and the synthesis of the high-gain observer for the estimation of UA. Consider system (Σ2) and denote it by
() ( )
x T x ) x1 ) r UA 2
and u ) (u1, ..., with u1 ) v1/(mCp)r, u2 ) v1Tj/(mCp)r, u3 ) ∑iFiCpi/(mCp)r, u4 ) ∑iFiCpiTfeed/(mCp)r, and u5 ) Qloss/(mCp)r. Using these notations, the system (Σ1) is in the observable canonical form u5)T
{
and u ) (u1, ..., u5)T with u1 ) Tj, u2 ) v2/(mCp)r, u3 ) ∑iFiCpi/(mCp)r, u4 ) ∑iFiCpiTfeed/(mCp)r, and u5 ) Qloss/(mCp)r. Using the notation above, the system (Σ2) is state affine up to input/output injection:
x˘ ) a(t) Ax + φ(u,y) y ) Cx ) x1
{
x˘ ) A(u,y) x + φ′(u,y) y ) Cx ) x1
where
where
a(t) )
A)
1 (mCp)r
a(t) )
( ) 0 1 0 0
A(u,y) )
C ) (1 0 )
(
)
Denote by θ1, θ1 > 0 a fixed parameter and
S)
( ) 1 θ1
-
1 θ12
1 2 - 2 θ1 θ13
the solution of the Lyapunov equation θ1S + ATS + SA ) CTC. The correction term of the observer is given by Γ(t) S-1CT(y(t) - Cxˆ (t)) with
( )
1 0 1 Γ(t) ) 0 a(t)
(
0 a(t) (y - u1) 0 0
)
and
and
1 (-u1y + u2 - u3y + u4 - u5) φ(u,y) ) (mCp)r 0
1 (mCp)r
φ′(u,y) )
(
u2 - u3y + u4 - u5 0
)
Consider two outputs y and yj respective to xu(t,x0) and xj u(t,xj0), the trajectories of system (Σ2) initialized at x0 and xj0, respectively, and let us show that if y ) yj, then x ) xj.
y ) yj T x1 ) xj1 T x˘ 1 ) xj1 T a(t) (y - u1)x2 + u2 - u3y + u4 - u5 ) a(t) (yj - u1)xj2 + u2 - u3yj + u4 - u5 T (y - u1)x2 ) (y - u1)xj2 T x2 ) xj2
for a(t) * 0
for (y - u1) * 0
Since a(t) ) 1/(mCp)r > 0, the system (Σ2) is observable if and only if y * u1, i.e., Tr * Tj. Let us denote by U the set of inputs such that Tr * Tj. System (Σ2) is U uniformly observable.
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4241
Consider the coordinate change
()
(x1 - Tj)x2 z x Φ(x) ) z ) z1 ) ( 1 )(mCp)r 2 and denote by θ2, θ2 > 0 a fixed parameter and S the solution of the Lyapunov equation θ2S + ATS + SA ) CTC. The gain of the observer is given by -1
(xˆ ) (∂Φ ∂x )
{
S-1CT
Hence, the observer for system (Σ2) is the following system:
dT ˆr
)-
dt
T ˆ r - Tj 1 U ˆA+ [v2 + (mCp)r (mCp)r
∑i FiCp (Tfeed - Tˆ r) - Qloss] - 2θ2(Tˆ r - y)
(O2)
[
i
]
(mCp)r U ˆA + θ22 (T ˆ r - y) ) 2θ2 dt T ˆ r - Tj T ˆ r - Tj
dU ˆA
Notation A ) heat-exchange area (m2) A0 ) initial value of the heat-exchange area (m2) Cpi ) specific heats of monomers (J/g/K) dmi/dt ) Fi feed flow rates of monomers (g/s) (mCp)r ) total specific heat of reaction mass (J/K) Qr ) rate of heat production due to chemical reaction (W) Q ˆ r ) estimated rate of heat production due to chemical reaction (W) Qrv ) volumetric rate of heat production due to chemical reaction (W/L) Tfeed ) temperature of the added medium (K) Tj ) jacket temperature (K) Tr ) reactor temperature (K) T ˆ r ) estimated reactor temperature (K) U ) heat-transfer coefficient (W/K/m2) U0 ) initial value of the heat-transfer coefficient (W/K/m2) UA ) global heat-transfer coefficient (W/K) U ˆ A ) estimated global heat-transfer coefficient (W/K) ν1, ν2 ) signal inputs θ1, θ2 ) tuning parameters for observers O1 and O2, respectively
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Received for review November 26, 2001 Revised manuscript received April 26, 2002 Accepted May 9, 2002 IE010948H
