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The use of calorimetry for on-line optimisation of isothermal semi-batch reactors
Chemical Engineering Science 56 (2001) 5147–5156
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The use of calorimetry for on-line optimisation of isothermal semi-batch reactors O. Ubricha , B. Srinivasanb , P. Lerenac , D. Bonvinb , F. Stoessela; c ; ∗ a Institut
de G enie Chimique, Ecole Polytechnique F ed erale de Lausanne, CH-1015 Lausanne, Switzerland d’Automatique, Ecole Polytechnique F ed erale de Lausanne, CH-1015 Lausanne, Switzerland c Swiss Institute for the Promotion of Safety & Security, CH-4002 Basel, Switzerland
b Institut
Received 7 April 2000; received in revised form 6 December 2000; accepted 7 May 2001
Abstract This paper considers the maximisation of conversion by manipulating the input 4owrate for an exothermic second-order esteri5cation reaction (propionic anhydride + 2-butanol) performed in a semi-batch reactor. Safety considerations impose bounds on the heat release rate in normal operation and on the maximum attainable temperature in the case of cooling failure. The optimal time-varying input 4owrate is calculated on-line using a heat balance over the reactor. Signi5cant improvement over the best constant feed rate can be achieved. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Process control; Safety; On-line optimisation; Isothermal operation; Calorimetry; Semi-batch reactors
1. Introduction An important operational objective in the chemical industry is to increase production or selectivity while guaranteeing safe processes. Since reactions are often exothermic, they may present a risk in the case of loss of control of the reactor. In order to enhance the control of the reaction, batch operation is often turned into semi-batch operation, thereby allowing to control the reaction rate via the feed rate of one of the reactants (Abel & Marquardt, 1998). The control and optimisation of semi-batch reactors have been discussed extensively in the literature. On the one hand, open-loop optimisation based on mathematical models has been proposed in the last decades (Lim, Tayeb, Modak, & Bonte, 1986; Arzamendi & Asua, 1989; Marchal-Brassely, Villermaux, Houzelot, Georgakis, & Barnay, 1989). It turns out that significant improvement can be obtained with well-chosen time-varying input 4owrates. On the other hand, predictive runaway regions using selected kinetic models have ∗
Corresponding author. E-mail address: [email protected] (F. Stoessel).
been developed (Strozzi & ZaldDEvar, 1994b; Strozzi, AlDos, & ZaldDEvar, 1994a; Strozzi, ZaldDEvar, Kronberg, & Westerterp, 1999; Wu, Morbidelli, & Varma, 1998). It appears that accidents can be foreseen and avoided this way. Also, some research has tried to couple the safety and optimisation aspects (Regenass, 1997). Gygax (1988) 5rst addressed the problem of controlling the accumulation of non-converted reactants by modulating the feed rate. The optimisation of productivity under safety constraints in semi-batch reactors, taking this problem of accumulation into account, was extensively studied by diHerent authors (Toulouse, Cezerac, Cabassud, Le Lann, & Casamatta, 1996; Abel, Helbig, Marquardt, Zwick, & Daszkowski, 2000; Abel & Marquardt, 2000). These methods require the availability of kinetics. However, in the 5ne chemical industry, models are very rarely available (Heinzle & HungerbIuhler, 1997). Thus, on-line measurements are used increasingly. Gas chromatography has been successfully utilised in polymer control and optimisation (Urretabizkaia, Leiza, & Asua, 1994). But this method only provides measurements approximately every 10 min. Spectroscopic methods can also be used (Rein, Phillips, & Yu, 1995). However, it possesses three main drawbacks. Firstly, most investigations are based on the Beer–Lambert law, which is valid
0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 1 8 3 - X
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only at low concentrations. In contrast, the concentrations are often high in production reactors. Secondly, for nonisothermal processes, the in4uence of temperature variations has to be considered. The relationship between absorbance and temperature is complex (WIulfert, Kok, & Smilde, 1998). Finally, safety considerations are diOcult to take into account. Indeed, safety considerations are often related to heat accumulation. It is possible to evaluate the heat production using spectroscopic data only if a kinetic model and thermal data are available. Reaction calorimetry represents an alternative experimental technique (SDaenz de Buruaga et al., 1997). The use of calorimetry for on-line optimisation under safety constraints presents some advantages. This method is based on easily available signals such as temperatures, 4ows and pressures (Schuler & Schmidt, 1992). It allows a control of the reactor based on the heat release (Regenass, 1985; Schimetzek & Giesbrecht, 1998). Moreover, its results can be directly used for scale-up. In this article, it is shown that calorimetry can be used to optimise a process on-line under safety constraints without the use of a detailed model and complex mathematical tools. The approach is based on the fact that, for feedback-linearisable systems, the optimal input will necessarily be on the constraints included in the optimisation problem. Thus, the approach is aimed at tracking active constraints. The paper is organised as follows. Section 2 formulates the problem, while Section 3 deals with calorimetric measurements. In Section 4, experimental implementation is presented. Some experimental results are proposed in Section 5. Finally, Section 6 concludes the paper.
2. Problem formulation 2.1. Chemical reaction The esteri5cation of propionic anhydride with 2-butanol is considered. The reaction, which is symbolised by A+B→C +D
(1)
is known to be a second-order reaction, i.e., 5rst- order in each reactant. This reaction is taken as a test reaction because its kinetic parameters are known (GalvDan, ZaldDEvar, HernDandez, & Molga, 1996). However, these will only be used for verifying the experimental optimisation results, and not for the purpose of optimisation.
Fig. 1. The runaway scenario (Gygax, 1988), showing the adiabatic temperature rise of the synthesis reaction (PTad ) after a cooling failure from the process temperature (Tr ) to the maximum temperature of the synthesis reaction (MTSR) and the further temperature increase due to the decomposition reaction triggered at the level MTSR.
2.2. Safety considerations Before scaling-up an isothermal process, and according to safety considerations, two important questions have to be answered: • Is the system capable of remaining isothermal in nor-
mal operation? In other words, can the heat released by the reaction be removed under normal operation? • Can the system be prevented from running away in the case of a cooling failure? In order to remain isothermal under normal operating conditions, the system has to produce less heat than can be removed by the cooling system qrx (t) 6 qex; max (t);
(2)
where qrx is the rate of heat production from the chemical reaction and qex; max the available rate of heat exchange. In the case of a cooling failure, the maximum attainable temperature has to remain inferior to a critical value. Runaways for exothermic reactions are often described by the scenario represented in Fig. 1. Following a cooling failure, the system can be considered adiabatic. Therefore, the heat produced by the chemical reaction can no longer be removed by the cooling system. This is especially true for industrial reactors where heat losses can be neglected. If unreacted material is still present in the reactor, the chemical reaction continues even if the feed is stopped immediately. This results in temperature increase up to Tcf (temperature in the case of cooling failure). The Tcf -value depends upon the process temperature, the amount of non-converted reactants and the total adiabatic temperature rise (Hugo, Steinbach, & Stoessel, 1988) Tcf (t) = Tr (t) + min[NA (t); NB (t)]
(−PHr ) : cp V (t)
(3)
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The term min[NA ; NB ] serves to calculate the maximum extent of reaction that could occur following the failure, and (−PHr )=cp V represents the adiabatic temperature increase. For safety considerations, the worst case scenario is considered. Therefore, the concept of MTSR (maximum temperature synthesis reaction) developed by Gygax (1988), is introduced MTSR = max Tcf (t):
(4)
t
This concept is used nowadays by several companies (Nomen, Sempere, & Serra, 1995; Elvers, Hawkins, & Russey, 1995). The MTSR-value is used to predict whether the boiling point of the system can be reached or if the reaction medium can trigger some exothermic decomposition reactions (Lerena, Wehner, Weber, & Stoessel, 1996). In this case, a cooling failure can lead to an accident with severe consequences. 2.3. Dynamic model
NA (0) = NA; 0 ;
·
N B = cB; in u − rV; ·
V = u;
NB (0) = NB; 0
V (0) = V0 ;
constraint simply states Tcf (t) 6 Tmax ;
(9)
where Tmax is the upper bound on the reactor temperature. 2.4. Optimisation problem The optimal control problem consists, for a given 5nal time, of adjusting the feed rate u(t) to maximise the production of the ester D whilst meeting speci5c operational and safety constraints. The constraints of this optimisation problem are: • The physical limitation of the volume:
V (t) 6 Vmax :
(10)
• The physical bounds on the feed rate:
umin 6 u(t) 6 umax :
(11)
• The safety constraints:
The reaction system can be modeled as follows: · N A = − rV;
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(5)
where r is the reaction rate, cB; in represents the concentration of B fed to the reactor, u is the volumetric feed rate, NA; 0 and NB; 0 are the initial mole numbers of A and B, respectively, and V0 the initial volume in the reactor. From System (5), the following expressions can be obtained for NA and NB : NA = NA; 0 (1 − xA );
(6)
NB = NB; 0 + cB; in (V − V0 ) − NA; 0 xA ;
(7)
where xA is the molar conversion of A. Since a single reaction is considered, molar and thermal conversions are equivalent. The thermal conversion can be evaluated on-line from the heat power generated by the chemical reaction t q (t ) d t 0 rx ; (8) xth (t) = min(NA; 0 ; NB; 0 )(−PHr ) where −PHr is the heat of reaction and qrx the heat rate (power) produced by the chemical reaction. In this work, qrx will be estimated via calorimetric measurements without knowledge of kinetic parameters. Thus, Eqs. (3) – (8) enable calculation of Tcf . The safety
qrx (t) 6 qex; max (t);
(12)
Tcf (t) 6 Tmax :
(13)
Visser, Srinivasan, Palanki, and Bonvin (2000) have shown that, in the terminal-cost optimisation of control-aOne systems, the solution either lies on path constraints (input bounds or state constraints) or is singular. Ubrich, Srinivasan, Stoessel, and Bonvin (1999a) showed that for an isothermal reaction system involving safety constraints, the solution has at most 5ve distinct arcs characterised by path constraints (see their Fig. 9). With the reaction system at hand, it is not possible to reach the operational safety constraint (12). Furthermore, the initial conditions NA; 0 and NB; 0 were so chosen that the system be initially on the safety constraint (13). Thus, the 5rst two arcs in Ubrich et al. (1999a) do not exist here, and conversion will be maximised for Tcf = Tmax . Furthermore, as soon as NA (t)¡NB (t); Tcf will no longer depend on NB , i.e., on the input u. Thus, if there is still material B to be fed, u(t) can be switched to umax until V (t) reaches Vmax , at which point u = umin . If the total number of moles of B that can be added is less than the number of moles of A; u(t) will switch to umin = 0 as soon as all the B has been added. This approach to dynamic optimisation is always applicable to control-aOne terminal-cost problems. Of course, only one constraint per available input can be active at any point in time, namely the most restrictive one. However, the number of constraints that can be active during the duration of the batch is not limited (see, e.g. the 5ve constraints that are followed in the example by Ubrich et al., 1999a).
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temperature Tr is given by qin = uin cpin (Tin − Tr );
(16)
where in and cpin are the density and the speci5c heat capacity of the feed stream. 3.3. Enthalpy of mixing
Fig. 2. Heat balance over the reactor. The following temperatures are used for the heat balance: reaction mixture (Tr ), average jacket temperature (Tj ), jacket inlet and outlet temperature (Tjin , Tjout ) and the feed temperature (Tin ). The diHerent terms of the heat balance are also indicated: reaction (qrx ), exchange (qex ), stirrer (qst ), calibration heater (qc ), feed (qin ) and losses (qloss ).
qmix =
3. Calorimetric measurements On-line calorimetric measurements will be used to estimate qrx (t) and evaluate xth according to Eq. (8). Reaction calorimetry is based on an energy balance over the reactor (see Fig. 2) d Tr = qrx + qin + qmix + qc + qst m(i) cp(i) dt i in4ow
qex + qloss − ;
out4ow
(i)
When two reactants, initially set at the same temperature, are mixed, a signi5cant change in temperature may occur. This is due to the enthalpy of mixing. In the reaction considered here, the time constant of mixing is assumed to be negligibly small compared to the reaction time constant. Thus, the heat eHect due to mixing can be modelled as a static term
(14)
cp(i)
where m and are the mass and the speci5c heat capacity of the ith component in the reaction mixture, qin , qmix and qloss are the power terms corresponding to: (i) feeding at a temperature diHerent from Tr , (ii) mixing enthalpy, and (iii) the heat lost to the surroundings. qc and qst are the heating power resulting from the calibration and the stirrer, respectively. The terms qrx , qin , qmix , qc and qst represent in4ow terms, while qex and qloss are the out4ow terms. Thus, in theory, qrx can be calculated from Eq. (14) if the other terms can be evaluated or measured with suOcient accuracy. The various terms in Eq. (14) are described below. 3.1. Accumulation of heat The reaction is performed isothermally. Therefore, the heat accumulation is negligible d Tr = 0: (15) m(i) cp(i) dt i 3.2. Heat e9ect of the feed stream The heating eHect due to the diHerence in temperature between the feed temperature Tin and the reactor
(−PHmix )u ; Vmol
(17)
where PHmix is the mixing enthalpy, and Vmol the molar volume of the added 2-butanol. 3.4. Calibration power Calibrations are performed before and after the experiment. Therefore, during the reaction, qc is zero qc = 0:
(18)
3.5. Stirrer power The heat dissipated by the stirrer is neglected here qst = 0:
(19)
3.6. Heat exchange to the coolant Assuming steady-state conditions, qex can be calculated from a heat balance on the jacket qex = F cool cpcool (Tjout − Tjin );
(20)
where F cool and cpcool are the mass 4owrate and the speci5c heat capacity of the cooling medium, Tjout and Tjin the outlet and inlet temperatures of the cooling medium, respectively. If the 4ow rate of the cooling medium is high, the temperature variation in the jacket is negligible and Tjout ≈ Tjin . This is especially the case in reaction calorimeters working following the heat 4ow principle. In this case, the heat exchanged through the reactor wall can be expressed as qex = UA(Tr − Tj ); where Tj is the temperature of the jacket.
(21)
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In industrial production or in reaction calorimeters working following the heat balance principle, Eq. (20), which is independent of U , is used preferably to Eq. (21). This is because U may vary and is diOcult to evaluate on-line. Recently, methods using “temperature oscillation mode” in the jacket have been proposed (CarloH, Pross, & Reichert, 1994; Bou Diab, Schenker, Marison, & von Stockar, 2000) but are not often used yet. However, the calorimeter used in this study is the RC1 J developed by Mettler Toledo. This instrument, working following the heat 4ow principle uses Eq. (21) to calculate the heat 4owrate. 3.7. Heat losses The heat losses are due mainly to evaporation of the reaction medium and to heat exchange with the environment. The second contribution, which is a function of reactor size and geometry, is less important in industrial reactors than in laboratory setups. This supports the use of on-line calorimetry in industrial production. In order to perform reaction calorimetry on-line, the base line needs to be known, i.e., the measured (or estimated) reaction rate in the case of no chemical reaction. With the experimental setup used in this work, i.e., constant surrounding temperature and non-heated reactor cover, the heat losses are mainly due to evaporation of the reaction medium. Some of the vapour produced may condense at the colder cover and cold droplets 4ow back to the reactor. This can be considered to be the major contribution to heat losses (Nomen, Sempere, & Lerena, 1993). Therefore it can be assumed that the heat losses are proportional to the total vapour pressure of the system xj Pj ; (22) qloss = where is a constant, xj and Pj are the molar fraction and the vapour pressure of component j, respectively. The vapour pressure can be estimated using Clausius– Clapeyron equation PHv 1 1 − ; (23) P(T1 ) = P(T2 ) exp R T1 T2 where PHv is the heat of vaporisation, R the molar gas constant, P(T1 ) and P(T2 ), the vapour pressure at temperature T1 and T2 , respectively. The factor of proportionality in Eq. (22) is evaluated by trial and error adjustments of the baseline using on-line experiments. 3.8. Integrated heat balance Integration of Eq. (14) for isothermal conditions gives Qrx (t) = Qex (t) + Qloss (t) − Qin (t) − Qmix (t);
(24)
Fig. 3. Experimental set-up. The reaction calorimeter is linked to an external computer used for the on-line heat balance and for the control of the feed.
where Qi (t) is the energy term that corresponds to the integral of the power term qi (t): t Qi (t) = qi (t ) d t : (25) 0
4. Experimental implementation 4.1. Experimental setup With the commercially available software of the calorimeter, it is not possible to calculate the optimal feed rate on-line. Thus, an external computer is used, which performs the on-line heat balance calculations and controls the addition of reactant. It is connected to a balance, which can read the mass increment Pm added to the reactor, and to a ProMinent gamma G=4b pump J (Fig. 3). This is a membrane pump that gives a stroke each time a pulse is applied. The calorimeter is used to control the reactor temperature at a constant value. 4.2. Choice of sampling time The mixing of 2-butanol with the reaction medium is accompanied by a temperature decrease due to the temperature diHerence between the feed and the reactor medium but also to the endothermic enthalpy of mixing. The eHect of adding 2-butanol to the reactor is depicted in Fig. 4. The temperature diHerence (Tr − Tj ) is represented for a sampling time of 8 s, which is large enough to allow the balance to stabilise after a feeding pulse but small enough to allow adequate representation of the temperature variations. The large and the small integrated areas correspond to Qex and (Qin + Qmix ) for a sampling period, respectively.
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Fig. 4. Evolution of Tr − Tj vs. time for the semi-batch process. At each stroke of the pump the endothermic eHect of the feed (Qin +Qmix ) is superimposed on the main signal (Qex ).
4.3. Optimisation algorithm Since the optimisation objective is to be on the safety constraint Tcf (t) = Tmax whenever possible, the following control law is introduced:
0 for Tcf (t)¿Tmax − PTs ; (26) u(t) = umax for Tcf (t) 6 Tmax − PTs ; where PTs is an imposed safety factor or back oH that is introduced to guarantee feasibility despite disturbances (Visser, 1999). The data processing and optimisation algorithm is depicted in Fig. 5. It is programmed within LabView J. The integration of the signals is performed using the rectangular approximation, which is justi5ed by the small sampling time. For example, at the discrete time tk ; Qi (k) can be expressed as tk qi (t) d t = Qi (k − 1) + PQi (k): Qi (k) = Qi (k − 1) + tk−1
(27)
Thus, the discrete expression for Qrx (k) follows from Eq. (24) and reads Qrx (k) = Qrx (k − 1) + PQex (k) + PQloss (k) − PQin (k) − PQmix (k):
(28)
5. Experimental results 5.1. Calibration In order to be able to implement the optimisation approach on-line, it is necessary to know a priori (for example, through experimental evaluation) certain thermal quantities. Their experimental evaluation is discussed next.
Fig. 5. Labview programme for implementing the optimisation.
5.1.1. Evaluation of the heat of reaction The proposed optimisation under safety constraints requires accurate knowledge of the heat of reaction PH . ◦ For its evaluation, an isothermal (70 C) batch run is performed in the calorimeter in which 2:5 mol of A are added to 2:5 mol of B. Integration of the reaction power gives (Fig. 6) Q = − 52:5 kJ= mol:
(29)
However, this value does not represent the correct heat of reaction. Indeed, on the one hand, when 2-butanol previously heated at the reaction temperature is added to propionic anhydride, there is a signi5cant drop in temperature. This is due to the negative mixing enthalpy. On the other hand, the reaction is very slow and therefore not totally 5nished after 20 h. With the help of spectroscopic measurements, it can be determined that the 5nal conversion is 90%. For these two reasons, the value given by the calorimeter has to be corrected (Regenass, 1997).
O. Ubrich et al. / Chemical Engineering Science 56 (2001) 5147–5156
Fig. 6. Determination of the reaction enthalpy: evolution of the power ◦ released by the reaction for an isothermal batch run at 70 C with a mixture of 2:5 mol of A with 2:5 mol of B, both initially charged.
Fig. 7. Determination of the mixing enthalpy of propionic anhydride ◦ and 2-butanol at 25 C as given by the C80-calorimeter.
In order to evaluate the mixing enthalpy, mixing of the ◦ two products is performed at 25 C in a sensitive calorimeter (Calvet calorimeter Setaram C80 J, 1996). At this temperature, the chemical reaction can be considered negligible (Ubrich, Srinivasan, Bonvin, & Stoessel, 1999b). The result of this experiment is depicted in Fig. 7 and gives PHmix = + 4:2 kJ= mol:
(30)
Therefore, the ‘corrected’ value of the heat of reaction is PHr = − 52:5=0:9 − 4:2 = − 62:5 ± 1 kJ= mol:
(31)
GalvDan et al. (1996) proposed −62:99 kJ= mol for this reaction. 5.1.2. Determination of heat transfer properties In order to implement the optimisation strategy, it is necessary to know the speci5c heat capacity (cp ), the overall heat transfer coeOcient (U ) and the density () of the reaction mass during the experiment.
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Fig. 8. Evolution of Tcf evaluated on-line (solid line) in relation to the constraint Tmax (dashed line).
The speci5c heat capacity is measured in the calorimeter for the initial and 5nal reaction masses. The values are approximately the same. Therefore, it is postulated that the speci5c heat capacity remains constant during the experiment. Similarly, the overall heat transfer coeOcient is evaluated from calibrations performed with the initial and 5nal reaction masses. Since the values are approximately the same, U is considered constant during the reaction. Finally, the densities of the four diHerent products are close to each other. The average value is taken as the constant density of the reaction medium. These properties are summarised in Table 1. 5.2. On-line results The reactor is initially 5lled with NA; 0 moles of propionic anhydride and heated to the desired temperature of ◦ 70 C for isothermal operation. In order to already be on the safety constraint at time t = 0, i.e., Tcf (0) = Tmax − ◦ PTs ; NB; 0 moles of 2-butanol preheated at 70 C are also added to the reactor, NB; 0 : cp V (0) : (32) NB; 0 = (Tmax − PTs − Tr (0)) (−PHr ) Then, the remaining B is added through the inlet feed according to the optimisation procedure. Fig. 8 shows the evolution of Tcf calculated on-line in relation to Tmax . As predicted, the system rides on the constraint until the latter cannot be reached anymore. Fig. 9 represents a comparison between: • (i) the on-line estimated chemical conversion; • (ii) the conversion estimated upon completion of the
reaction from the calorimeter data;
• (iii) the chemical conversion obtained by simulation
using the kinetic parameters of the reaction.
There is a maximum deviation of 5% between the three curves. The on-line energy balance is in good agreement with the other methods.
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Table 1 Numerical values of parameters and experimental conditions Symbol
Description
Value
Unit
cp PHr PHmix NA (0) cBin NB (0) P1 PHv1 P2 PHv2 P3 PHv3 P4 PHv4 in PTs Tin Tmax Tr (0)
factor of proportionality speci5c heat capacity reaction enthalpy mixing enthalpy initial number of moles of A concentration of B fed initial number of moles of B ◦ vapour pressure of the 2-butanol at 54:1 C vaporisation enthalpy of the 2-butanol ◦ vapour pressure of the propionic anhydride at 57:7 C vaporisation enthalpy of the propionic anhydride ◦ vapour pressure of the propionic acid at 65:8 C vaporisation enthalpy of the propionic acid ◦ vapour pressure of the propionic acid butyl ester at 79:5 C vaporisation enthalpy of the propionic acid butyl ester density of B safety factor feed temperature highest safe temperature initial reaction temperature sampling time overall heat transfer coeOcient maximum 4ow rate minimum 4ow rate initial volume of A molar volume of 2-butanol maximum volume
4 2000 −62500 +4200 4 10.7 0.94 100 −45:1 10 −49:0 40 −46:3 100 −42:4 808 1 298 405 343 8 170 51.2 0 0.52 0.091 0.88
(J= kg K ) (J=mol) (J=mol) (mol) (mol=l) (mol) (mm Hg) (kJ=mol) (mm Hg) (kJ=mol) (mm Hg) (kJ=mol) (mm Hg) (kJ=mol) (kg= m3 ) (K) (K) (K) (K) (s) (W = m 2 K ) (ml=h) (ml=h) (l) (l=mol) (l)
U umax umin V0 Vmol Vmax
Fig. 9. Comparison between the chemical conversion estimated on-line (solid line), that estimated upon completion of the reaction using calorimeter data (), and calculated from the kinetics (4).
Another comparison is possible. In industrial production, the feeding pro5les are often kept constant. In order to take into account the safety constraints, the input rate typically implemented is that for which Tcf reaches Tmax at just one point. For the experiment considered, the evolution of Tcf for the best constant input (u = 0:043 l= h) is depicted in Fig. 10. Fig. 11 shows that an increase of 20% in conversion can be obtained with the proposed method compared to the method generally used in production.
Fig. 10. Evolution of Tcf (solid line) compared to the constraint Tmax (dashed line) for the fastest constant input (u = 0:043 l= h) respecting the constraint.
5.3. Limitations of the method With calorimetric measurements, an error of about 10% on industrial processes can be expected. Indeed, there are many error sources. Some errors aHect the estimates of the speci5c heat capacity, the overall heat transfer coeOcient or the density. However, the most diOcult problem is the on-line evaluation of the base line, especially for slow reactions.
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Acknowledgements This work is part of the Ph.D. thesis being prepared at the EPFL by the 5rst author with the 5nancial support of the Corporate Heath, Safety and Environment Department of Novartis A.G. (Basel). The support of Dr. Eigenmann, Head of this Department, is particularly acknowledged. References
Fig. 11. Comparison of the conversion obtained with the on-line calculated time-varying input (solid line) and the constant input (dash line).
Nevertheless, calorimetry is capable of performing the optimisation on-line if, within the safety constraints, one takes into account the sources of error (in U; cp and ). For example, instead of Tmax , it is recommended to use Tmax − PTs (Eq. (26)). Additionally, as the model does not use the kinetic parameters for evaluating Tcf , the optimisation can be performed without the knowledge of the kinetic parameters. Hence, only the initial conditions have to be known with some accuracy. In case of doubt regarding the initial conditions, one has to overestimate the quantities involved in order to overestimate the risk. This way, one takes a safety margin in case of cooling failure. 6. Conclusion On-line optimisation under safety constraints has been presented for a second-order reaction taking place in a semi-batch reactor. Calorimetry was used to estimate the extent of reaction and the temperature that would result from a cooling failure. The main experimental diOculties were the identi5cation of an accurate base line and the correction for heat eHects associated with the feeding. It was shown via comparison of on-line, oH-line and simulated results that it is possible to follow the reaction and to determine the optimal feed rate on-line. The main advantage of this method is that, even if the kinetic parameters of the reaction system are unknown, it is possible to increase productivity without the risk of violating safety constraints. Also, conversion can be increased compared to the constant input pro5le policy normally used in production reactors. Moreover, the method could even be more useful in production than in the laboratory since heat balances can be easier to perform there. Indeed, in production, the heat 4ow can be directly evaluated using the temperature diHerence between cooling inlet and outlet.
Abel, O., Helbig, A., Marquardt, W., Zwick, H., & Daszkowski, T. (2000). Productivity optimization of an industrial semi-batch polymerization reactor under safety constraints. Journal of Process Control, 10(4), 351–362. Abel, O., & Marquardt, W. (1998). A model predictive control scheme for safe and optimal operation of exothermic semi-batch reactors. Fifth IFAC symposium on dynamics and control of process systems, Corfu, Greece. Abel, O., & Marquardt, W. (2000). Scenario-integrated modelling and optimization of dynamic systems. American Institution of Chemical Engineers Journal, 46(4), 803–823. Arzamendi, G., & Asua, J. M. (1989). Monomer addition policies for copolymer composition control in semicontinuous emulsion copolymerization. Journal of Applied Polymer Science, 38, 2019– 2036. Bou Diab, L., Schenker, B., Marison, I., & von Stockar, U. (2000). Improvement of continuous calibration based on temperature oscillation and application to biochemical reaction calorimetry. Chemical Engineering Journal, 3674, 1–15. CarloH, R., Pross, A., & Reichert, K. H. (1994). Temperature oscillation calorimetry in stirred tank reactors with variable heat transfer. Chemical Engineering Technology, 17, 406–413. Elvers, B., Hawkins, S., & Russey, W. (Eds.) (1995). Ullmann’s encyclopaedia of industrial chemistry. Vol. B8 (5th edn). Weinheim: VCH. GalvDan, I. M., ZaldDEvar, J. M., HernDandez, H., & Molga, E. (1996). The use of neural networks for 5tting complex kinetic data. Computers & Chemical Engineering, 20, 1451–1465. Gygax, R. (1988). Chemical reaction engineering for safety. Chemical Engineering Science, 43(8), 1759–1771. Heinzle, E., & HungerbIuhler, K. (1997). Integrated process development: The key to future production of chemicals. Chimia, 51, 176–183. Hugo, P., Steinbach, J., & Stoessel, F. (1988). Calculation of the maximum temperature in stirred tank reactors in case of a breakdown of cooling. Chemical Engineering Science, 43(8), 2147–2152. Lerena, P., Wehner, W., Weber, H., & Stoessel, F. (1996). Assessment of hazards linked to accumulation in semi-batch reactors. Thermochimica Acta, 289, 127–142. Lim, H. C., Tayeb, Y. J., Modak, J. M., & Bonte, P. (1986). Computational algorithms for optimal feed rates for a class of fed-batch fermentation: Numerical results for penicillin and cell mass production. Biotechnology and Bioengineering, 28, 1408– 1420. Marchal-Brassely, S., Villermaux, J., Houzelot, J. L., Georgakis, C., & Barnay, J. L. (1989). Une mDethode itDerative eOcace d’optimisation des pro5ls de tempDerature et de dDebit d’alimentation pour la conduite optimale des rDeacteurs discontinus. R ecents progrAes en G enie des proc ed es, Toulouse. Vol. 3(9) (pp. 441– 446). Nomen, R., Sempere, J., & Lerena, P. (1993). Heat 4ow calorimetry under re4ux conditions. Thermochimica Acta, 225(2), 263–276.
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Nomen, R., Sempere, J., & Serra, E. (1995). Temas de Segurudad. 3. PrevenciDon de la accumulaciDon de reactivos. ABnidad, 547, 155–161. Rein, J. A., Phillips, J. A., & Yu, K. (1995). Application of mid IR spectroscopy to bioreactions. Proceedings of First ReactIR User Forum Europe, October 5 – 6, Interlaken, Switzerland. Regenass, W. (1985). Calorimetric monitoring of industrial chemical processes. Thermochimica Acta, 95, 351–368. Regenass, W. (1997). The development of stirred-tank heat 4ow calorimetry as a tool for process optimisation and process safety. Chimia, 51, 189–200. SDaenz de Buruaga, I., EchevarrDEa, A., Armitage, P. D., de la Cal, J. C., Leiza, J. R., & Asua, J. M. (1997). On-line control of a semi-batch emulsion polymerization reactor based on calorimetry. American Institution of Chemical Engineers Journal, 43(4), 1069–1081. Schimetzek, R., & Giesbrecht, H. (1998). Safe operation of semi-batch reactors by supervising the accumulation of reactants through an on-line energy balance. Proceedings of the Nineth international symposium on loss prevention and safety promotion in the process industries, May 4 –7, Barcelona, Spain (pp. 523–545). Schuler, H., & Schmidt, C. U. (1992). Calorimetric-state estimators for chemical reactor diagnosis and control: Review of methods and applications. Chemical Engineering Science, 47(4), 899–915. Strozzi, F., AlDos, M. A., & ZaldDEvar, J. M. (1994a). A method for assessing thermal stability of batch reactors by sensitivity calculation based on Lyapunov exponents: Experimental veri5cation. Chemical Engineering Science, 49, 5549–5561. Strozzi, F., & ZaldDEvar, J. M. (1994b). A general method for assessing the thermal stability of chemical batch reactors by sensitivity calculation based on Lyapunov exponents. Chemical Engineering Science, 49, 2681–2688. Strozzi, F., ZaldDEvar, J. M., Kronberg, A., & Westerterp, K. R. (1999). Runaway prevention in chemical reactors using chaos
theory techniques. American Institution of Chemical Engineers Journal, 45, 2394–2408. Toulouse, C., Cezerac, J., Cabassud, M., Le Lann, M. V., & Casamatta, G. (1996). Optimization and scale-up of batch chemical reactors: Impact of safety constraints. Chemical Engineering Science, 51(10), 2243–2252. Ubrich, O., Srinivasan, B., Stoessel, F., & Bonvin, D. (1999a). Optimization of a semi-batch reaction system under safety constraints. European control conference 1999, Karlsruhe, Germany. Ubrich, O., Srinivasan, B., Bonvin, D., & Stoessel, F. (1999b). Optimal feed pro5le for a second order reaction in a semi-batch reactor under safety constraints. Experimental study. Journal of Loss Prevention in the Process Industries, 12, 485–493. Urretabizkaia, A., Leiza, J. R., & Asua, J. M. (1994). On-line terpolymer composition control in semi-continuous emulsion polymerization. American Institution of Chemical Engineers Journal, 40(11), 1850–1864. Visser, E. (1999). A feedback-based implementation scheme for batch process optimization. Thesis No 2097, Ecole Polytechnique FDedDerale de Lausanne, Lausanne, 1999. Visser, E., Srinivasan, B., Palanki, S., & Bonvin, D. (2000). A feedback-based implementation scheme for batch process optimization. Journal of Process Control, 10, 399–410. Wu, H., Morbidelli, M., & Varma, A. (1998). An approximate criterion for reactor thermal runaway. Chemical Engineering Science, 53(18), 3341–3344. WIulfert, F., Kok, W. Th., & Smilde, A. K. (1998). In4uence of temperature on vibrational spectra and consequences for the predictive ability of multivariate models. Analytical Chemistry, 70, 1761–1767.
www.elsevier.com/locate/ces
The use of calorimetry for on-line optimisation of isothermal semi-batch reactors O. Ubricha , B. Srinivasanb , P. Lerenac , D. Bonvinb , F. Stoessela; c ; ∗ a Institut
de G enie Chimique, Ecole Polytechnique F ed erale de Lausanne, CH-1015 Lausanne, Switzerland d’Automatique, Ecole Polytechnique F ed erale de Lausanne, CH-1015 Lausanne, Switzerland c Swiss Institute for the Promotion of Safety & Security, CH-4002 Basel, Switzerland
b Institut
Received 7 April 2000; received in revised form 6 December 2000; accepted 7 May 2001
Abstract This paper considers the maximisation of conversion by manipulating the input 4owrate for an exothermic second-order esteri5cation reaction (propionic anhydride + 2-butanol) performed in a semi-batch reactor. Safety considerations impose bounds on the heat release rate in normal operation and on the maximum attainable temperature in the case of cooling failure. The optimal time-varying input 4owrate is calculated on-line using a heat balance over the reactor. Signi5cant improvement over the best constant feed rate can be achieved. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Process control; Safety; On-line optimisation; Isothermal operation; Calorimetry; Semi-batch reactors
1. Introduction An important operational objective in the chemical industry is to increase production or selectivity while guaranteeing safe processes. Since reactions are often exothermic, they may present a risk in the case of loss of control of the reactor. In order to enhance the control of the reaction, batch operation is often turned into semi-batch operation, thereby allowing to control the reaction rate via the feed rate of one of the reactants (Abel & Marquardt, 1998). The control and optimisation of semi-batch reactors have been discussed extensively in the literature. On the one hand, open-loop optimisation based on mathematical models has been proposed in the last decades (Lim, Tayeb, Modak, & Bonte, 1986; Arzamendi & Asua, 1989; Marchal-Brassely, Villermaux, Houzelot, Georgakis, & Barnay, 1989). It turns out that significant improvement can be obtained with well-chosen time-varying input 4owrates. On the other hand, predictive runaway regions using selected kinetic models have ∗
Corresponding author. E-mail address: [email protected] (F. Stoessel).
been developed (Strozzi & ZaldDEvar, 1994b; Strozzi, AlDos, & ZaldDEvar, 1994a; Strozzi, ZaldDEvar, Kronberg, & Westerterp, 1999; Wu, Morbidelli, & Varma, 1998). It appears that accidents can be foreseen and avoided this way. Also, some research has tried to couple the safety and optimisation aspects (Regenass, 1997). Gygax (1988) 5rst addressed the problem of controlling the accumulation of non-converted reactants by modulating the feed rate. The optimisation of productivity under safety constraints in semi-batch reactors, taking this problem of accumulation into account, was extensively studied by diHerent authors (Toulouse, Cezerac, Cabassud, Le Lann, & Casamatta, 1996; Abel, Helbig, Marquardt, Zwick, & Daszkowski, 2000; Abel & Marquardt, 2000). These methods require the availability of kinetics. However, in the 5ne chemical industry, models are very rarely available (Heinzle & HungerbIuhler, 1997). Thus, on-line measurements are used increasingly. Gas chromatography has been successfully utilised in polymer control and optimisation (Urretabizkaia, Leiza, & Asua, 1994). But this method only provides measurements approximately every 10 min. Spectroscopic methods can also be used (Rein, Phillips, & Yu, 1995). However, it possesses three main drawbacks. Firstly, most investigations are based on the Beer–Lambert law, which is valid
0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 1 8 3 - X
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only at low concentrations. In contrast, the concentrations are often high in production reactors. Secondly, for nonisothermal processes, the in4uence of temperature variations has to be considered. The relationship between absorbance and temperature is complex (WIulfert, Kok, & Smilde, 1998). Finally, safety considerations are diOcult to take into account. Indeed, safety considerations are often related to heat accumulation. It is possible to evaluate the heat production using spectroscopic data only if a kinetic model and thermal data are available. Reaction calorimetry represents an alternative experimental technique (SDaenz de Buruaga et al., 1997). The use of calorimetry for on-line optimisation under safety constraints presents some advantages. This method is based on easily available signals such as temperatures, 4ows and pressures (Schuler & Schmidt, 1992). It allows a control of the reactor based on the heat release (Regenass, 1985; Schimetzek & Giesbrecht, 1998). Moreover, its results can be directly used for scale-up. In this article, it is shown that calorimetry can be used to optimise a process on-line under safety constraints without the use of a detailed model and complex mathematical tools. The approach is based on the fact that, for feedback-linearisable systems, the optimal input will necessarily be on the constraints included in the optimisation problem. Thus, the approach is aimed at tracking active constraints. The paper is organised as follows. Section 2 formulates the problem, while Section 3 deals with calorimetric measurements. In Section 4, experimental implementation is presented. Some experimental results are proposed in Section 5. Finally, Section 6 concludes the paper.
2. Problem formulation 2.1. Chemical reaction The esteri5cation of propionic anhydride with 2-butanol is considered. The reaction, which is symbolised by A+B→C +D
(1)
is known to be a second-order reaction, i.e., 5rst- order in each reactant. This reaction is taken as a test reaction because its kinetic parameters are known (GalvDan, ZaldDEvar, HernDandez, & Molga, 1996). However, these will only be used for verifying the experimental optimisation results, and not for the purpose of optimisation.
Fig. 1. The runaway scenario (Gygax, 1988), showing the adiabatic temperature rise of the synthesis reaction (PTad ) after a cooling failure from the process temperature (Tr ) to the maximum temperature of the synthesis reaction (MTSR) and the further temperature increase due to the decomposition reaction triggered at the level MTSR.
2.2. Safety considerations Before scaling-up an isothermal process, and according to safety considerations, two important questions have to be answered: • Is the system capable of remaining isothermal in nor-
mal operation? In other words, can the heat released by the reaction be removed under normal operation? • Can the system be prevented from running away in the case of a cooling failure? In order to remain isothermal under normal operating conditions, the system has to produce less heat than can be removed by the cooling system qrx (t) 6 qex; max (t);
(2)
where qrx is the rate of heat production from the chemical reaction and qex; max the available rate of heat exchange. In the case of a cooling failure, the maximum attainable temperature has to remain inferior to a critical value. Runaways for exothermic reactions are often described by the scenario represented in Fig. 1. Following a cooling failure, the system can be considered adiabatic. Therefore, the heat produced by the chemical reaction can no longer be removed by the cooling system. This is especially true for industrial reactors where heat losses can be neglected. If unreacted material is still present in the reactor, the chemical reaction continues even if the feed is stopped immediately. This results in temperature increase up to Tcf (temperature in the case of cooling failure). The Tcf -value depends upon the process temperature, the amount of non-converted reactants and the total adiabatic temperature rise (Hugo, Steinbach, & Stoessel, 1988) Tcf (t) = Tr (t) + min[NA (t); NB (t)]
(−PHr ) : cp V (t)
(3)
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The term min[NA ; NB ] serves to calculate the maximum extent of reaction that could occur following the failure, and (−PHr )=cp V represents the adiabatic temperature increase. For safety considerations, the worst case scenario is considered. Therefore, the concept of MTSR (maximum temperature synthesis reaction) developed by Gygax (1988), is introduced MTSR = max Tcf (t):
(4)
t
This concept is used nowadays by several companies (Nomen, Sempere, & Serra, 1995; Elvers, Hawkins, & Russey, 1995). The MTSR-value is used to predict whether the boiling point of the system can be reached or if the reaction medium can trigger some exothermic decomposition reactions (Lerena, Wehner, Weber, & Stoessel, 1996). In this case, a cooling failure can lead to an accident with severe consequences. 2.3. Dynamic model
NA (0) = NA; 0 ;
·
N B = cB; in u − rV; ·
V = u;
NB (0) = NB; 0
V (0) = V0 ;
constraint simply states Tcf (t) 6 Tmax ;
(9)
where Tmax is the upper bound on the reactor temperature. 2.4. Optimisation problem The optimal control problem consists, for a given 5nal time, of adjusting the feed rate u(t) to maximise the production of the ester D whilst meeting speci5c operational and safety constraints. The constraints of this optimisation problem are: • The physical limitation of the volume:
V (t) 6 Vmax :
(10)
• The physical bounds on the feed rate:
umin 6 u(t) 6 umax :
(11)
• The safety constraints:
The reaction system can be modeled as follows: · N A = − rV;
5149
(5)
where r is the reaction rate, cB; in represents the concentration of B fed to the reactor, u is the volumetric feed rate, NA; 0 and NB; 0 are the initial mole numbers of A and B, respectively, and V0 the initial volume in the reactor. From System (5), the following expressions can be obtained for NA and NB : NA = NA; 0 (1 − xA );
(6)
NB = NB; 0 + cB; in (V − V0 ) − NA; 0 xA ;
(7)
where xA is the molar conversion of A. Since a single reaction is considered, molar and thermal conversions are equivalent. The thermal conversion can be evaluated on-line from the heat power generated by the chemical reaction t q (t ) d t 0 rx ; (8) xth (t) = min(NA; 0 ; NB; 0 )(−PHr ) where −PHr is the heat of reaction and qrx the heat rate (power) produced by the chemical reaction. In this work, qrx will be estimated via calorimetric measurements without knowledge of kinetic parameters. Thus, Eqs. (3) – (8) enable calculation of Tcf . The safety
qrx (t) 6 qex; max (t);
(12)
Tcf (t) 6 Tmax :
(13)
Visser, Srinivasan, Palanki, and Bonvin (2000) have shown that, in the terminal-cost optimisation of control-aOne systems, the solution either lies on path constraints (input bounds or state constraints) or is singular. Ubrich, Srinivasan, Stoessel, and Bonvin (1999a) showed that for an isothermal reaction system involving safety constraints, the solution has at most 5ve distinct arcs characterised by path constraints (see their Fig. 9). With the reaction system at hand, it is not possible to reach the operational safety constraint (12). Furthermore, the initial conditions NA; 0 and NB; 0 were so chosen that the system be initially on the safety constraint (13). Thus, the 5rst two arcs in Ubrich et al. (1999a) do not exist here, and conversion will be maximised for Tcf = Tmax . Furthermore, as soon as NA (t)¡NB (t); Tcf will no longer depend on NB , i.e., on the input u. Thus, if there is still material B to be fed, u(t) can be switched to umax until V (t) reaches Vmax , at which point u = umin . If the total number of moles of B that can be added is less than the number of moles of A; u(t) will switch to umin = 0 as soon as all the B has been added. This approach to dynamic optimisation is always applicable to control-aOne terminal-cost problems. Of course, only one constraint per available input can be active at any point in time, namely the most restrictive one. However, the number of constraints that can be active during the duration of the batch is not limited (see, e.g. the 5ve constraints that are followed in the example by Ubrich et al., 1999a).
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temperature Tr is given by qin = uin cpin (Tin − Tr );
(16)
where in and cpin are the density and the speci5c heat capacity of the feed stream. 3.3. Enthalpy of mixing
Fig. 2. Heat balance over the reactor. The following temperatures are used for the heat balance: reaction mixture (Tr ), average jacket temperature (Tj ), jacket inlet and outlet temperature (Tjin , Tjout ) and the feed temperature (Tin ). The diHerent terms of the heat balance are also indicated: reaction (qrx ), exchange (qex ), stirrer (qst ), calibration heater (qc ), feed (qin ) and losses (qloss ).
qmix =
3. Calorimetric measurements On-line calorimetric measurements will be used to estimate qrx (t) and evaluate xth according to Eq. (8). Reaction calorimetry is based on an energy balance over the reactor (see Fig. 2) d Tr = qrx + qin + qmix + qc + qst m(i) cp(i) dt i in4ow
qex + qloss − ;
out4ow
(i)
When two reactants, initially set at the same temperature, are mixed, a signi5cant change in temperature may occur. This is due to the enthalpy of mixing. In the reaction considered here, the time constant of mixing is assumed to be negligibly small compared to the reaction time constant. Thus, the heat eHect due to mixing can be modelled as a static term
(14)
cp(i)
where m and are the mass and the speci5c heat capacity of the ith component in the reaction mixture, qin , qmix and qloss are the power terms corresponding to: (i) feeding at a temperature diHerent from Tr , (ii) mixing enthalpy, and (iii) the heat lost to the surroundings. qc and qst are the heating power resulting from the calibration and the stirrer, respectively. The terms qrx , qin , qmix , qc and qst represent in4ow terms, while qex and qloss are the out4ow terms. Thus, in theory, qrx can be calculated from Eq. (14) if the other terms can be evaluated or measured with suOcient accuracy. The various terms in Eq. (14) are described below. 3.1. Accumulation of heat The reaction is performed isothermally. Therefore, the heat accumulation is negligible d Tr = 0: (15) m(i) cp(i) dt i 3.2. Heat e9ect of the feed stream The heating eHect due to the diHerence in temperature between the feed temperature Tin and the reactor
(−PHmix )u ; Vmol
(17)
where PHmix is the mixing enthalpy, and Vmol the molar volume of the added 2-butanol. 3.4. Calibration power Calibrations are performed before and after the experiment. Therefore, during the reaction, qc is zero qc = 0:
(18)
3.5. Stirrer power The heat dissipated by the stirrer is neglected here qst = 0:
(19)
3.6. Heat exchange to the coolant Assuming steady-state conditions, qex can be calculated from a heat balance on the jacket qex = F cool cpcool (Tjout − Tjin );
(20)
where F cool and cpcool are the mass 4owrate and the speci5c heat capacity of the cooling medium, Tjout and Tjin the outlet and inlet temperatures of the cooling medium, respectively. If the 4ow rate of the cooling medium is high, the temperature variation in the jacket is negligible and Tjout ≈ Tjin . This is especially the case in reaction calorimeters working following the heat 4ow principle. In this case, the heat exchanged through the reactor wall can be expressed as qex = UA(Tr − Tj ); where Tj is the temperature of the jacket.
(21)
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In industrial production or in reaction calorimeters working following the heat balance principle, Eq. (20), which is independent of U , is used preferably to Eq. (21). This is because U may vary and is diOcult to evaluate on-line. Recently, methods using “temperature oscillation mode” in the jacket have been proposed (CarloH, Pross, & Reichert, 1994; Bou Diab, Schenker, Marison, & von Stockar, 2000) but are not often used yet. However, the calorimeter used in this study is the RC1 J developed by Mettler Toledo. This instrument, working following the heat 4ow principle uses Eq. (21) to calculate the heat 4owrate. 3.7. Heat losses The heat losses are due mainly to evaporation of the reaction medium and to heat exchange with the environment. The second contribution, which is a function of reactor size and geometry, is less important in industrial reactors than in laboratory setups. This supports the use of on-line calorimetry in industrial production. In order to perform reaction calorimetry on-line, the base line needs to be known, i.e., the measured (or estimated) reaction rate in the case of no chemical reaction. With the experimental setup used in this work, i.e., constant surrounding temperature and non-heated reactor cover, the heat losses are mainly due to evaporation of the reaction medium. Some of the vapour produced may condense at the colder cover and cold droplets 4ow back to the reactor. This can be considered to be the major contribution to heat losses (Nomen, Sempere, & Lerena, 1993). Therefore it can be assumed that the heat losses are proportional to the total vapour pressure of the system xj Pj ; (22) qloss = where is a constant, xj and Pj are the molar fraction and the vapour pressure of component j, respectively. The vapour pressure can be estimated using Clausius– Clapeyron equation PHv 1 1 − ; (23) P(T1 ) = P(T2 ) exp R T1 T2 where PHv is the heat of vaporisation, R the molar gas constant, P(T1 ) and P(T2 ), the vapour pressure at temperature T1 and T2 , respectively. The factor of proportionality in Eq. (22) is evaluated by trial and error adjustments of the baseline using on-line experiments. 3.8. Integrated heat balance Integration of Eq. (14) for isothermal conditions gives Qrx (t) = Qex (t) + Qloss (t) − Qin (t) − Qmix (t);
(24)
Fig. 3. Experimental set-up. The reaction calorimeter is linked to an external computer used for the on-line heat balance and for the control of the feed.
where Qi (t) is the energy term that corresponds to the integral of the power term qi (t): t Qi (t) = qi (t ) d t : (25) 0
4. Experimental implementation 4.1. Experimental setup With the commercially available software of the calorimeter, it is not possible to calculate the optimal feed rate on-line. Thus, an external computer is used, which performs the on-line heat balance calculations and controls the addition of reactant. It is connected to a balance, which can read the mass increment Pm added to the reactor, and to a ProMinent gamma G=4b pump J (Fig. 3). This is a membrane pump that gives a stroke each time a pulse is applied. The calorimeter is used to control the reactor temperature at a constant value. 4.2. Choice of sampling time The mixing of 2-butanol with the reaction medium is accompanied by a temperature decrease due to the temperature diHerence between the feed and the reactor medium but also to the endothermic enthalpy of mixing. The eHect of adding 2-butanol to the reactor is depicted in Fig. 4. The temperature diHerence (Tr − Tj ) is represented for a sampling time of 8 s, which is large enough to allow the balance to stabilise after a feeding pulse but small enough to allow adequate representation of the temperature variations. The large and the small integrated areas correspond to Qex and (Qin + Qmix ) for a sampling period, respectively.
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Fig. 4. Evolution of Tr − Tj vs. time for the semi-batch process. At each stroke of the pump the endothermic eHect of the feed (Qin +Qmix ) is superimposed on the main signal (Qex ).
4.3. Optimisation algorithm Since the optimisation objective is to be on the safety constraint Tcf (t) = Tmax whenever possible, the following control law is introduced:
0 for Tcf (t)¿Tmax − PTs ; (26) u(t) = umax for Tcf (t) 6 Tmax − PTs ; where PTs is an imposed safety factor or back oH that is introduced to guarantee feasibility despite disturbances (Visser, 1999). The data processing and optimisation algorithm is depicted in Fig. 5. It is programmed within LabView J. The integration of the signals is performed using the rectangular approximation, which is justi5ed by the small sampling time. For example, at the discrete time tk ; Qi (k) can be expressed as tk qi (t) d t = Qi (k − 1) + PQi (k): Qi (k) = Qi (k − 1) + tk−1
(27)
Thus, the discrete expression for Qrx (k) follows from Eq. (24) and reads Qrx (k) = Qrx (k − 1) + PQex (k) + PQloss (k) − PQin (k) − PQmix (k):
(28)
5. Experimental results 5.1. Calibration In order to be able to implement the optimisation approach on-line, it is necessary to know a priori (for example, through experimental evaluation) certain thermal quantities. Their experimental evaluation is discussed next.
Fig. 5. Labview programme for implementing the optimisation.
5.1.1. Evaluation of the heat of reaction The proposed optimisation under safety constraints requires accurate knowledge of the heat of reaction PH . ◦ For its evaluation, an isothermal (70 C) batch run is performed in the calorimeter in which 2:5 mol of A are added to 2:5 mol of B. Integration of the reaction power gives (Fig. 6) Q = − 52:5 kJ= mol:
(29)
However, this value does not represent the correct heat of reaction. Indeed, on the one hand, when 2-butanol previously heated at the reaction temperature is added to propionic anhydride, there is a signi5cant drop in temperature. This is due to the negative mixing enthalpy. On the other hand, the reaction is very slow and therefore not totally 5nished after 20 h. With the help of spectroscopic measurements, it can be determined that the 5nal conversion is 90%. For these two reasons, the value given by the calorimeter has to be corrected (Regenass, 1997).
O. Ubrich et al. / Chemical Engineering Science 56 (2001) 5147–5156
Fig. 6. Determination of the reaction enthalpy: evolution of the power ◦ released by the reaction for an isothermal batch run at 70 C with a mixture of 2:5 mol of A with 2:5 mol of B, both initially charged.
Fig. 7. Determination of the mixing enthalpy of propionic anhydride ◦ and 2-butanol at 25 C as given by the C80-calorimeter.
In order to evaluate the mixing enthalpy, mixing of the ◦ two products is performed at 25 C in a sensitive calorimeter (Calvet calorimeter Setaram C80 J, 1996). At this temperature, the chemical reaction can be considered negligible (Ubrich, Srinivasan, Bonvin, & Stoessel, 1999b). The result of this experiment is depicted in Fig. 7 and gives PHmix = + 4:2 kJ= mol:
(30)
Therefore, the ‘corrected’ value of the heat of reaction is PHr = − 52:5=0:9 − 4:2 = − 62:5 ± 1 kJ= mol:
(31)
GalvDan et al. (1996) proposed −62:99 kJ= mol for this reaction. 5.1.2. Determination of heat transfer properties In order to implement the optimisation strategy, it is necessary to know the speci5c heat capacity (cp ), the overall heat transfer coeOcient (U ) and the density () of the reaction mass during the experiment.
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Fig. 8. Evolution of Tcf evaluated on-line (solid line) in relation to the constraint Tmax (dashed line).
The speci5c heat capacity is measured in the calorimeter for the initial and 5nal reaction masses. The values are approximately the same. Therefore, it is postulated that the speci5c heat capacity remains constant during the experiment. Similarly, the overall heat transfer coeOcient is evaluated from calibrations performed with the initial and 5nal reaction masses. Since the values are approximately the same, U is considered constant during the reaction. Finally, the densities of the four diHerent products are close to each other. The average value is taken as the constant density of the reaction medium. These properties are summarised in Table 1. 5.2. On-line results The reactor is initially 5lled with NA; 0 moles of propionic anhydride and heated to the desired temperature of ◦ 70 C for isothermal operation. In order to already be on the safety constraint at time t = 0, i.e., Tcf (0) = Tmax − ◦ PTs ; NB; 0 moles of 2-butanol preheated at 70 C are also added to the reactor, NB; 0 : cp V (0) : (32) NB; 0 = (Tmax − PTs − Tr (0)) (−PHr ) Then, the remaining B is added through the inlet feed according to the optimisation procedure. Fig. 8 shows the evolution of Tcf calculated on-line in relation to Tmax . As predicted, the system rides on the constraint until the latter cannot be reached anymore. Fig. 9 represents a comparison between: • (i) the on-line estimated chemical conversion; • (ii) the conversion estimated upon completion of the
reaction from the calorimeter data;
• (iii) the chemical conversion obtained by simulation
using the kinetic parameters of the reaction.
There is a maximum deviation of 5% between the three curves. The on-line energy balance is in good agreement with the other methods.
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Table 1 Numerical values of parameters and experimental conditions Symbol
Description
Value
Unit
cp PHr PHmix NA (0) cBin NB (0) P1 PHv1 P2 PHv2 P3 PHv3 P4 PHv4 in PTs Tin Tmax Tr (0)
factor of proportionality speci5c heat capacity reaction enthalpy mixing enthalpy initial number of moles of A concentration of B fed initial number of moles of B ◦ vapour pressure of the 2-butanol at 54:1 C vaporisation enthalpy of the 2-butanol ◦ vapour pressure of the propionic anhydride at 57:7 C vaporisation enthalpy of the propionic anhydride ◦ vapour pressure of the propionic acid at 65:8 C vaporisation enthalpy of the propionic acid ◦ vapour pressure of the propionic acid butyl ester at 79:5 C vaporisation enthalpy of the propionic acid butyl ester density of B safety factor feed temperature highest safe temperature initial reaction temperature sampling time overall heat transfer coeOcient maximum 4ow rate minimum 4ow rate initial volume of A molar volume of 2-butanol maximum volume
4 2000 −62500 +4200 4 10.7 0.94 100 −45:1 10 −49:0 40 −46:3 100 −42:4 808 1 298 405 343 8 170 51.2 0 0.52 0.091 0.88
(J= kg K ) (J=mol) (J=mol) (mol) (mol=l) (mol) (mm Hg) (kJ=mol) (mm Hg) (kJ=mol) (mm Hg) (kJ=mol) (mm Hg) (kJ=mol) (kg= m3 ) (K) (K) (K) (K) (s) (W = m 2 K ) (ml=h) (ml=h) (l) (l=mol) (l)
U umax umin V0 Vmol Vmax
Fig. 9. Comparison between the chemical conversion estimated on-line (solid line), that estimated upon completion of the reaction using calorimeter data (), and calculated from the kinetics (4).
Another comparison is possible. In industrial production, the feeding pro5les are often kept constant. In order to take into account the safety constraints, the input rate typically implemented is that for which Tcf reaches Tmax at just one point. For the experiment considered, the evolution of Tcf for the best constant input (u = 0:043 l= h) is depicted in Fig. 10. Fig. 11 shows that an increase of 20% in conversion can be obtained with the proposed method compared to the method generally used in production.
Fig. 10. Evolution of Tcf (solid line) compared to the constraint Tmax (dashed line) for the fastest constant input (u = 0:043 l= h) respecting the constraint.
5.3. Limitations of the method With calorimetric measurements, an error of about 10% on industrial processes can be expected. Indeed, there are many error sources. Some errors aHect the estimates of the speci5c heat capacity, the overall heat transfer coeOcient or the density. However, the most diOcult problem is the on-line evaluation of the base line, especially for slow reactions.
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Acknowledgements This work is part of the Ph.D. thesis being prepared at the EPFL by the 5rst author with the 5nancial support of the Corporate Heath, Safety and Environment Department of Novartis A.G. (Basel). The support of Dr. Eigenmann, Head of this Department, is particularly acknowledged. References
Fig. 11. Comparison of the conversion obtained with the on-line calculated time-varying input (solid line) and the constant input (dash line).
Nevertheless, calorimetry is capable of performing the optimisation on-line if, within the safety constraints, one takes into account the sources of error (in U; cp and ). For example, instead of Tmax , it is recommended to use Tmax − PTs (Eq. (26)). Additionally, as the model does not use the kinetic parameters for evaluating Tcf , the optimisation can be performed without the knowledge of the kinetic parameters. Hence, only the initial conditions have to be known with some accuracy. In case of doubt regarding the initial conditions, one has to overestimate the quantities involved in order to overestimate the risk. This way, one takes a safety margin in case of cooling failure. 6. Conclusion On-line optimisation under safety constraints has been presented for a second-order reaction taking place in a semi-batch reactor. Calorimetry was used to estimate the extent of reaction and the temperature that would result from a cooling failure. The main experimental diOculties were the identi5cation of an accurate base line and the correction for heat eHects associated with the feeding. It was shown via comparison of on-line, oH-line and simulated results that it is possible to follow the reaction and to determine the optimal feed rate on-line. The main advantage of this method is that, even if the kinetic parameters of the reaction system are unknown, it is possible to increase productivity without the risk of violating safety constraints. Also, conversion can be increased compared to the constant input pro5le policy normally used in production reactors. Moreover, the method could even be more useful in production than in the laboratory since heat balances can be easier to perform there. Indeed, in production, the heat 4ow can be directly evaluated using the temperature diHerence between cooling inlet and outlet.
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