Simultaneous Estimation of the Heat of Reaction and the Heat Transfer

Chemical Engineering Science 60 (2005) 4233–4248 www.elsevier.com/locate/ces

Simultaneous estimation ofthe heat ofreaction and the heat transfer coefficient by calorimetry: estimation problems due to model simplification and high jacket flow rates—theoretical development

Stefan Krämer∗, RalfGesthuisen

Innovene (BP Köln GmbH), Chemicals Production Köln, Postfach 750212, 50754 Köln, Germany

Received 11 December 2003; received in revised form 14 September 2004; accepted 4 February 2005 Available online 26 April 2005

Abstract

In batch and semi-batch reactors, the heat ofreaction Q˙ R is normally estimated using calorimetry. Ifall temperatures and volumes are measured correctly and the measurements are filtered sensibly, the results are usually very good. For many applications, the overall heat transfer coefficient k also needs to be known. In heat balance calorimetry k and Q˙ R can be calculated simultaneously, ifthe correct model is used. It is shown that the commonly used model simplifications pose serious problems for large reactors. Subsequently a sensible model extension is discussed. For this extension we propose the application ofan Extended Kalman Filter (EKF) to estimate the heat of reaction (Q˙ R) and the heat transfer coefficient (k) simultaneously, as the EKF can handle the model extension well. Our emphasis lies on three important factors. Firstly and mainly, the jacket ofa jacketed reactor is generally modelled as a stirred tank. When looking at real jacketed reactors, the jacket behaves more like a plug-flow reactor. We propose a model extension to overcome this problem. Secondly, the heat transfer coefficient (k) is for many reactions strongly dependent on the batch time and should therefore also be estimated. With the usual models, errors may result which can be corrected by the model extension. Thirdly, the flow rate through the jacket and the hold-up in the reactor strongly influence the estimation quality. With a lower jacket flow rate estimation quality increases but cooling decreases, a trade-off has to be made. Using an EKF, good estimation quality can still be achieved for high flow rates. However, the trade-off is considered and the tuning is adjusted to the flow rate. An optimal flow rate calculation is suggested. Finally, it will be shown that adding measurements in the jacket rather than in the reactor will improve calorimetric estimation for the proposed model extension. ᭧ 2005 Elsevier Ltd. All rights reserved.

Keywords: Chemical reactors; Control; Extended Kalman ﬁlter; State estimation; Heat transfer; Calorimetry

1. Introduction calorimetry is used frequently for exothermic reactors and their control (Gugliotta et al., 1995; deBuruaga et al., 1997; In batch and semi-batch stirred tank reactors, the heat of Bartanek et al., 1999; Tauer et al., 1999; McKenna et al., reaction is normally estimated using calorimetry. Calorimet- 2000; Vicente et al., 2001). ric measurements offer a cheap and reliable way of identi- A classical and widely applied method is reaction fyingthe heat ofreaction, because the temperatures ofthe calorimetry, which employs the energy balances around a reaction medium and the cooling fluid are usually measured process to identify the heat of reaction. For most processes frequently. An extensive overview ofthe subject ofcalorime- the determination ofthe heat ofreaction is sufficient,but try can be found in Weber (1974). Furthermore, heat balance for the safe operation of exothermic reactions, knowledge ofthe heat transfercoefficient,which governs the rate ∗ Corresponding author. Tel.: +49 2133 556578; fax: +49 2133 557 133. ofenergy transferredbetween reactor and jacket, is also E-mail addresses: [email protected], required. Often this value is needed for optimisation or [email protected] (S. Krämer). control.

When semi-batch polymerisation processes are consid- simultaneous strategy to estimate the unknown parameters. ered as an example, strong variation in the overall heat trans- This way, the global observability can be utilised and the lo- fer coefficient between reactor and jacket as well as the in- cal stability ofthe EKF is ensured. The EKF is known to give crease in heat transfer area have to be considered. good and robust estimates and can be easily extended by the There are a few approaches to overcome this problem. proposed model extension. However, it has to be noted that Schmidt and Reichert (1988) simply estimate the derivatives the proposed approach can be realised also with any other ofthe temperatures and use a model to identify kA. Oscil- nonlinear state observers (e.g., Nonlinear Luenberger Ob- lation calorimetry can be used on laboratory scale reactors server, High Gain Observer, Nonlinear reduced observer). (Tietze et al., 1995, 1996a,b; Carloff et al., 1994; Luca and As the EKF and calorimetry themselves are well known, Scali, 2002). It separates the process dynamics influenced the aims ofthis paper do not lie in explaining these concepts. by the change of kA (fast dynamics) from the process dy- We concentrate on the following points instead: namics influenced by the reaction rate (slow dynamics) and thus allows for their separate estimation. A more advanced 1. The jacket ofa jacketed reactor is generally modelled as heat balance calorimetric method was used by many authors a stirred tank. When looking at pilot or industrial scale (Guo et al., 2001; McKenna et al., 1996; Fevotte et al., 1998; jacketed reactors, the jacket behaves more like a plug- McKenna et al., 2000; Othman et al., 2002; Krämer et al., flow reactor. Does this pose a problem and how is it ˙ 2003a,b), who use nonlinear observers to estimate QR and overcome? kA successfully. Santos et al. (2001), BenAmor et al. (2002) As we focus on the problem of different jacked geome- as well as Freire et al. (2004) use a so called “cascaded” tries and dynamics, other effects such as nonideal mixing observer to estimate both parameters. In this approach only in the reactor and additional equipment (e.g. condenser) an energy balance for the reactor and dummy derivatives are not considered. for the heat of reaction and the heat transfer coefficient are 2. The heat transfer coefficient (k) depends for many re- used. Santos et al. (2001) state that this system ofdifferen- actions strongly on the batch time. Can it be estimated tial equations is unobservable from the reactor temperature well for different jacket geometries? measurement and therefore the authors have to assume that 3. The flow rate through the jacket and the hold-up in the re- neither the heat ofreaction nor the heat transfercoefficient actor strongly influence the eigenvalues ofthe linearised vary quickly in the process to allow the cascaded observer system matrices. Does this pose a problem for the EKF to function at all. On the one hand, this assumption does and how can we deal with it? not change the observability condition. On the other hand, it has to be noted that the stated system ofdifferentialequa- The main focusofthis paper is point 1, the extension ofthe tions used in Santos et al. (2001), BenAmor et al. (2002) and typically employed heat balance model to a model, in which Freire et al. (2004) is globally observable as can be shown non-ideal coolant flow regimes in the jacket are considered. by a nonlinear analysis ofthe system. Ifthe second deriva- Point 3 is also very important. Maximum cooling depends tive ofthe reactor temperature was used by the observer, on the jacket flow rate, as long as the heat transfer coefficient both parameters could be observed in theory. Santos et al. is high. Ifisothermal conditions are required in the reactor, (2001), BenAmor et al. (2002) and Freire et al. (2004) pro- a controller has to be used for the reactor temperature, the pose a different strategy to overcome this problem: They use performance of which requires a large flow rate, as it uses a “cascaded” observer, which assumes one parameter (e.g., the jacket inlet temperature as the manipulated variable. the heat ofreaction) to be constant at one sampling point to As we will show, the estimation, but also the plant model estimate the other parameter (e.g., the heat transfer coeffi- mismatch depends on the flow rate, which has to be low for cient) and vice versa for the following sampling point. From good estimation quality, and trade-offs have to be made. a theoretic point ofview in none ofthe mentioned papers The aims of this work are threefold: the observability or the stability ofthe cascaded approach Firstly, we will describe the typical heat balance model is investigated. Nonetheless, for the stated examples, they used around a stirred tank reactor, which considers the jacket ˙ estimate both k and QR well. behaving like a continuously stirred tank reactor (from now The advantage ofthis approach is, ofcourse, similar to on: CSTR). This implies that there is no spatial distribution of oscillation calorimetry, i.e., no jacket balance is required. If temperature within the jacket. We will then extend this model a jacket balance is used, the jacket is assumed to behave like by modelling the jacket as a plug flow reactor (from now a CSTR. on: PFR), which implies that there is a spatial temperature In this paper, we use the proposed approach by Krämer variation within the jacket from the inlet to the outlet. We et al. (2003b) to employ an Extended Kalman Filter (EKF) consider this the realistic case for industrial scale reactors, as (Jazwinski, 1970; Gelb, 1974) to estimate the heat ofre- jackets oflarge reactors oftenhave flow channels (as shown action and the heat transfer coefficient. The basis of the in Fig. 1 on p. 35 in the magnified section) or are realised analysis is formed by the classic equations of heat balance as welded on halfpipes. calorimetry, where a dynamic jacket heat balance is a re- Secondly, a state estimator is set up to estimate the heat quirement. In contrast to the cascaded observer we propose a ofreaction and the heat transferparameters. The state S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248 4235

TV 2.1. The jacket behaviour is modelled as a CSTR R, R,in M The mass and energy balances for a CSTR are set up as follows: T . Jout, mJ b dhR ˙ ˙ d Vessel: AB = VR, − VR, , (1) R dt in out

TJ d(CRTR) TV h R, R R dt h ˙ ˙ ˙ ˙ ˙ = QR,in − QR,out + QR − QR,loss − QJ , = c V˙ T − c V˙ T in p,in R,in R,in R p,R R,out R ˙ ˙ + QR − QR,loss − kA(TR − TJ ). (2) . . TmJ TVR, R,out T Jin, d J 1 ˙ ˙ ˙ ˙ Jacket = (Qin − Qout + QJ − QJ,loss) dt mJ cp Fig. 1. Typical stirred tank diagram with labels. The magniﬁed section 1 = (m˙ J cp,J (TJ,in − TJ ) explains the ﬂow channels inside the jacket, from which PFR behaviour mJ cp,J results in large tanks. ˙ + kA(TR − TJ ) − QJ,loss). (3)

with : A = AB + dRhR estimator will use a model assuming the jacket is a CSTR, 2 = dR + dRhR. (4) whereas simulation will assume a PFR jacket. By looking at 4 the linearisation and the estimation quality, important obser- The term CR describes the absolute heat capacity ofthe vations concerning the tuning will be reported. Additionally, reactor, its contents and inserts and has to be identified for the authors are unaware ofany formalcheck ofthe observ- each system separately. Ifthe densities and heat capacities ability conditions for a calorimetric estimator in the open can be considered the same, the balances can be significantly literature. Therefore, global observability conditions will be simplified. checked. Thirdly, the EKF model will be extended by the model for plug flow conditions in the jacket in order to improve the 2.2. The jacket behaviour is modelled as a PFR estimation quality resulting from plant model mismatch. The improved estimation will be shown in simulation studies. The problem is modelled by assuming that the jacket be- Finally, considerations and calculations will be presented haves like a PFR. Ifthe reactor is not completely filled, no considering how the estimation quality can be further im- heat is exchanged between reactor and jacket in the upper proved, both by adding additional temperature measure- part of the vessel, as the heat transfer from the gaseous phase ments, by sensor placement and by calculating an optimal to the jacket is negligible compared to the heat transfer from flow rate. the liquid phase to the jacket. Therefore, this upper part is modelled separately just considering the convective term, which means it is simply a time delay (modelling this time delay is important for good estimation). Balancing a finite 2. Models volume ofthe PFR yields the following equations (development details are given in the Appendix): Consider a typical stirred tank reactor as shown in Fig. 1 on p. 35, symbols are explained in the list ofsymbols. d(CRTR) Vessel : In all cases the reactor is assumed to be perfectly mixed dt and is therefore modelled as a CSTR. Differences in mod- ˙ ˙ ˙ ˙ ˙ = QR,in − QR,out + QR − QR,loss − QJ elling occur for the jacket part. Firstly, the jacket is assumed = c V˙ T − c V˙ T in p,in R,in R,in R p,R R,out R to be perfectly mixed and is also balanced as a CSTR. Sec- + Q˙ − Q˙ − Q˙ ondly, the jacket is not assumed to be perfectly mixed. Some R R,loss J , (5) general assumptions are made: jT : J,L Lower jacket jt • It is assumed that the jacket is completely filled, jTJ,L k • the fluid is incompressible, =−v + (T − T ) jz b c R J,L • the reactor is a cylinder, and J p,J • q˙J,loss(z) to model a semi-batch reactor, the outflows ofthe reactor − , (6) are simply set to zero. bJ cp,J 4236 S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248

jTJ,U jTJ,U q˙J, (z) to yield Upper jacket : =−v − loss , (7) jt jz bJ cp,J ˙ dTR VR,in L = max t m c + A h c : Q˙ = q˙ (z) z d S p,S B R R p,R Jacket loss J,loss J,loss d . × ( c T − c T ) 0 in p,in R,in R p,R R 1 The length ofthe upper part ofthe jacket is given by L − + (Q˙ − Q˙ ) max m c + A h c R R,loss L, whereas L corresponds to the length ofthe lower part and S p,S B R R p,R kA Lmax is the total length ofthe jacket. − (T − T ) m c + A h c R J . (14) A S p,S B R R p,R L = , (8) h For the sake ofclarity, we have assumed that mScp,S (heat A capacity ofthe reactor wall and possible inserts) is negligible L = max c = c max h . (9) and R p,R in p,in, an assumption which generally should not be made in practice. Ifthe heat capacities ofthe different Ifthis system is to be simulated, the manipulated variables components in the reactor and the feed are very different, for the jacket (TJ,in) as well as the heat flow are needed. For this simplification cannot be made. the lower part of the jacket, the manipulated variable is the The above stated simplifications yield the model ˙ boundary condition for Eq. (6) and QJ is found by integrat- ˙ ˙ ˙ ing the heat transfer of every balanced volume element dz dTR VR,in QR − QR,loss = (TR,in − TR) + up to length L. dt VR VRRcp,R kA Boundary : TJ,L(0,t)= TJ,in(t), (10) − (TR − TJ ) (15) VRRcp,R L ˙ Heat flow : QJ = kh(TR − TJ,L(z)) dz. (11) and the unaltered Eqs. (3) and (4) for the jacket and heat 0 transfer area. ˙ The upper part ofthe jacket is also modelled as a PFR. The The aim ofcalorimetry is to find QR. However, k is gener- boundary condition for Eq. (7) is given by: ally also unknown. Therefore, k is also observed. The model used in the observer is therefore extended by the following Boundary : TJ,L(L, t) = TJ,U(0,t). (12) ˙ two equations, which allow for the estimation of QR and k, As back mixing in the jacket can be neglected in most cases, which are considered parameters in the estimation problem. Eqs. (6) and (7) describe the ‘true’ behavior ofthe jacket ˙ dQR especially ifthe cooling device is realised as a halfpipe = t 0, (16) welded to the wall ofthe reactor. d dk = 0. (17) 2.3. Estimation model dt All models have been extended by a geometry factor to allow Generally, models where the jacket is assumed to be per- for scale-up, fG =1 means a 10 l-reactor is used. The model fectly mixed, i.e., behave like a CSTR, are used for calorime- including fG can be found in Appendix B. try. Ifcalorimetry is performedby a state estimator, the es- ˙ timated heat ofreaction QR and heat transfer coefficient k have to be treated as parameters. The model has to be 2.4. Model analysis separated into an observed and unobserved submodel. The unobserved submodel consists ofthe differentialequation 2.4.1. Observability for the liquid level in the tank, which is only simulated (or The first step in the observer design is to check the ob- measured). Ifthe liquid level is simulated or measured, it servability ofthe considered dynamic system. A system is (t = ) = ∈ can be considered an input to the state estimator model. For said to be observable ifan initial state x 0 x0 X0 (t) an EKF, sampled measurement systems are considered. The can be determined by the knowledge ofthe inputs u and the measurements y(t) in the interval 0

For the above model (15)–(17) as well as (3) and (4), This means that only the sum ofall the differentheat gen- the global observability can be checked (A is replaced by eration terms can be estimated. This estimation is indepen- r2 + 2VR R rR ). Using the method presented by e.g., Birk (1992), dent ofthe estimation of k. The loss terms can be identified a nonlinear observability map can be identified: during a period where no reaction takes place (for example while heating the tank) and then used as a bias term. For ˙ y1 = TR, (18) the estimation of QR, the distinction between the different origins ofthe heat loss is irrelevant. For a more advanced y2 = TJ , (19) technique to determine the heat losses, the interested reader is referred to Othman et al. (2002). ˙ ˙ ˙ VR,in QR − QR,loss The situation is different for the estimation of k. Here, a y˙1 = (TR,in − TR) + VR VRRcp,R heat loss from the tank has no influence, whereas a heat loss 3 from the jacket influences its estimation. In order to receive k(rR + 2VR) − (TR − TJ ), (20) good results, the considered tank has to be known well, a rRVR cp,R R situation which might pose problems in practice. Ifone of 3 the loss terms is negligible, the other can be estimated during m˙ k(rR + 2VR) y˙2 = (TJ,in − TJ ) + (TR − TJ ) a period with no heat ofreaction and used as a bias or input m rRmcp term. Ifthe terms both have a similar value which is not ˙ QJ, negligible, good estimation quality cannot be guaranteed. − loss . (21) mcp However, when the method is applied to a real reactor, these flaws can be identified before the method is applied. For an This leads to an equation system with six unknowns (TR, T, industrial jacketed tank, heat loss from the tank can probably ˙ ˙ ˙ QR, QR, , QJ, , k) and four equations. Assuming y and be neglected or modelled ifheat is lost through the lid or loss loss ˙ all its derivatives w.r.t. time are available from measurements by evaporation. Then only QJ,loss is estimated before the an equation system results which is solvable for the desired reaction starts. A well insulated jacketed tank probably has ˙ values QR and k to yield: generally negligible heat loss. We therefore develop our method with this problem ˙ ˙ mc˙ py˙ −˙mcp(TJ, − y ) + QJ, in mind but using the simplification QJ,loss = 0 and k = 2 in 2 loss , (22) ˙ 3 QR, = 0 without loss ofgenerality forthe proposed rR + 2VR loss (y1 − y2) estimator and jacket model extension, as these terms are rR linear bias terms, which can be added after the estimation if required. k(r3 +2VR) Q˙ −Q˙ =V c y˙ + R A further matter, which is of practical importance but does R R,loss R R p,R 1 r V c R R R p,R not affect the theoretical results, is the situation in small lab ˙ VR,in scale reactors. For very high flow rates in the jacket, the × (y1−y2)− (TR,in−y1) . (23) VR measurements ofthe jacket inlet and the jacket outlet stream appear identical, as their difference is beyond the measure- This system is globally observable for every TR and TJ ex- ment resolution. This does not change the global observ- cept for TR = TJ .Ifk is known, Eq. (22) becomes unneces- ability map and the observability ofthe system theoretically. sary and the system is globally observable. The result ofthe From a practical view point, however, observation will not inverse observability map can be used to compare its results work at high flow rates due to measurement limitations, i.e., and the results ofa well-tuned EKF ( Franke, 2000; Krämer too large a system noise. Then, k cannot be estimated in et al., 2001). practice. Additionally in order to receive sensible information, the The inverse observability map describes the solution an ˙ ˙ bias terms QR,loss and QJ,loss need to be known. observer is converging to. Incidentally, the results ofthe It is important to distinguish between the reactor and the global observability map are also the equations used in stan- ˙ Q˙ −Q˙ −Q˙ jacket heat loss. Let us first consider the estimation of QR. dard heat balance calorimetry. R R,loss J,loss is ob- This heat ofreaction cannot be estimated by itselfin the servable by itself, even for TR = TJ ; (TR − TJ ) should be above estimation scheme. However, ifEq. (22) is inserted eliminated by inserting k into Eq. (23). Therefore, for perfect in Eq. (23), the following results: signals, an observer would not be necessary ifthe inverse observability map can be identified, as the correct solution Q˙ − Q˙ − Q˙ can always be inferred. However, even if this is the case, the R R,loss J,loss mc˙ y˙ −˙mc (T − y ) definition space here has one exception (TR = TJ ). These = V c y˙ + p 2 p J,in 2 R R p,R 1 V c equations are therefore not useful to estimate k ifsituations R R p,R T ≈ T ˙ arise where R J . An observer cannot estimate k in that VR,in − (TR,in − y1) . (23a) region, either, but its smoothing and simulation capabilities VR avoid a division by zero. 4238 S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248

Eigenvalue 1 Eigenvalue 2

0 −0.05

−0.005 −0.1

−0.15 −0.01 10 10 0.2 0.2 5 0.1 5 0.1 f 0 m [kg/s] f 0 m [kg/s] G J G J

Fig. 2. Eigenvalues plotted vs. fG and m˙ J for a small range of these values.

Eigenvalue 1 Eigenvalue 2

0 0

−0.01 −0.5

−0.02 0.5 −1 0.5 1000 1000 500 500 f 0 m [kg/s] f 0 m [kg/s] G J G J

Fig. 3. Eigenvalues plotted vs. fG and m˙ for a large range of these values.

2.4.2. Eigenvalues of the linearised system eigenvalues, it means that the eigenvalue in question moves An EKF uses a linearised system for the estimation. It is closer towards zero, i.e., its absolute value decreases, the therefore interesting for the tuning of the filter to consider time constants ofthe system increase and it becomes slower! the change in the eigenvalues ofthe linearised system with As can be seen in Fig. 2, eigenvalue 1 decreases consid- design parameters. The design parameters considered here erably with an increase of m˙ J . This effect is not changed by are the jacket flow rate m˙ J and the geometry factor fG. a large geometry as can be seen in Fig. 3. This behaviour is Ofthese two factors, fG is a design factor decided when expected as a small m˙ J increases the residence time ofthe building the plant. m˙ J is normally constant for jacketed re- jacket fluid. The reactor dynamics are only influenced by the actors and TJ,in is regulated. However, m˙ J is easily changed jacket flow rate for small geometries due to the volume to in an existing plant. Its possible change needs to be consi- area ratio. It can be deduced that the observer gain needs to dered. be adjusted to the jacket flow rate. With this eigenvalue so The eigenvalues ofthe linearised system matrix pro- close to zero it governs the system dynamics, i.e., the flow vide information about the system’s dynamic behaviour in rate defines the slow and therefore governing dynamics of close proximity ofa considered state and dependent on the the system. system’s parameters, design parameters and states. Ifthey A similar effect can be seen when looking at the geometry change strongly, it may be necessary to adjust the weighting factor. Its increase decreases eigenvalue 2. As the absolute matrixes (covariance matrices) ofthe EKF. For simplicity, value is significantly larger than eigenvalue 1, it does not the following analysis is performed for model (15) and the govern the dynamics. The mass flow rate in the jacket has jacket balance (3). virtually no influence on this eigenvalue whereas the geom- Ifthe eigenvalues ofthe linearisation matrix are checked, etry increases the second time constant. it becomes obvious that the design variable m˙ J has a large influence on the dynamics ofthe system. As it can be manipulated over a large range, this influence has to be consid- 3. Simulation study I m˙ ered. It should be noted that changes in J do not influence 3.1. Simulation the global observability ofthe system but may influence estimation quality. An EKF is used here to estimate the process parameters ˙ ˙ In the system described, 2 eigenvalues are below zero and QR and k. QR and k are set as would be expected in a two eigenvalues are zero. When talking about an increase in typical semi-batch polymerisation process. In such a process S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248 4239

900 250 800 simulation m =0.4 200 J m increasing m =0.2 J 700 J m =0.1 J 150 m =0.05 600 J K] 2

[W] 100 simulation R m =0.4 J 500 Q m =0.2 J k [W/(m m =0.1 50 J m increasing m =0.05 400 J J 0 300

− 50 200

0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 time [min] time [min]

Fig. 4. Estimation for a 10 l reactor with constant covariance matrix Q. Please note that mJ in the figure is indeed m˙ j in kg/s. the heat transfer coefficient depends on the viscosity and The discrete linear Kalman Filter is based on the solution therefore on the conversion of the monomers. To show a ofthe minimization ofthe expected values ofthe estimation more realistic estimation problem, the temperature in the error and as such is optimal for the provided covariance reactor is controlled by a PI-controller, which can manipulate matrices. As the Kalman Filter can also be formulated as the jacket feed temperature. This manipulated variable itself an optimal state regulator problem, the covariance matrices is subject to a cascade controller within a heating and cooling Q and R can be generally considered as weighting matrices apparatus. To model this behaviour, the real Tin follows its ofthe estimation errors. The solution to this optimisation set point like a first order transfer function (PT1-behaviour) problem can be found in the Matrix–Riccati-equation. with a time constant of690 s. While the differentmass flow The EKF is similarly found. The linear model equations in rates were scaled for larger reactors, these values were kept the prediction are replaced by the nonlinear process model, the same. The model with the PFR jacket has been used for the prediction ofthe weighting matrix Pk+1|k is performed simulation. by Taylor-linearizing the process model. Its equations are written as: 3.2. Extended Kalman Filter equations Correction terms: T T −1 Kk = Pk|k− Hk|k− (Hk|k− Pk|k− Hk|k− + R) , (28) Mathematical modelling ofdynamic systems oftenyields 1 1 1 1 1 nonlinear differential equations off the form: xˆk|k = xˆk|k−1 + Kk(yk − h(xˆk|k−1)), (29) x˙ = f(x, u) + (t), Pk|k = (I − KkHk|k−1)Pk|k−1. (30) Prediction terms: x(0) = x0 + 0, (24) ˆ = (ˆ , ) y(t) = h(x) + (t), (25) xk+1|k F xk|k uk , (31) = T + where x, u, y, and are the n-dimensional state vector, the Pk+1|k Ak|kPk|kAk|k Q, (32) r-dimensional vector ofcontrol variables, the m-dimensional with: measurement vector and n-dimensional vectors ofthe model j and measurement errors, respectively. The EKF is one of = F Ak|k j , (33) the most frequently applied state estimation techniques in x xˆk|k chemical engineering. In this approach and are assumed jh to be zero mean random processes. = Hk|k−1 j . (34) In our work we consider sampled systems. Hence the or- x xˆk|k dinary differential equations have to be transformed to dif- Here, the model equations are (15)–(17) as well as (3) and ference equations by integration over one sampling interval: ˙ ˙ (4) with QJ, = 0 and QR, = 0. loss loss tk+1 = + ( ( , ) + ) t xk+1 xk f x u d , 3.3. Tuning of the EKF tk = F(xk, uk) + k, (26) Figs. 4 and 5 present results ofthe estimation. Only the es- ˙ yk = h(xk) + k. (27) timated ofthe unmeasured parameters QR and k are shown, 4240 S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248

900 300

800 250 simulation m =0.4 J 700 m =0.2 200 J m =0.1 J 600 m =0.05 150 J K)] 2

[W] 500 100 simulation R

Q m =0.4 J m =0.2 k [W/(m 50 J 400 m =0.1 J m =0.05 0 J 300

−50 200

−100 100 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 time [min] time [min]

Fig. 5. Estimation for a 10 l reactor with adjusted covariance matrix Q. Please note that mJ in the ﬁgure is indeed m˙ j in kg/s.

as the temperatures are measured. Measured data are TR and this adaptation improves the estimation result. The EKF is TJ which are disturbed by random measurement noise of now capable to produce good estimation results for different ±0.1 K. Initial conditions were offset by 100 W. The needed flow rates. quantities VR and hR are considered measured. The analysis ofthe adjustment of Q have all been per- The filter was tuned using the covariance ofthe measure- formed for a geometry factor of fG =1. As m˙ J is adjusted to ment error for matrix R and adjusting matrix Q by hand, un- the size scaling and only the equivalents to m˙ J = 0.05 kg/s til good agreement between estimation and simulation was and m˙ J = 0.2kg/s are used, m˙ J is from now on labelled reached. high and low. It can be clearly seen in Fig. 4 that the changing jacket flow rate results in a change in estimation quality ifthe 3.4. Estimation quality for plant model mismatch elements of Q are fixed. The observer dynamics should always be faster than the The simulations shown in this section are based on a system dynamics as an asymptotic behaviour towards the jacket, which behaves like a PFR, the EKF, however, uses real values is not possible otherwise. As the jacket flow rate the much simpler model, in which the jacket behaves like decreases the governing time constant ofthe system, the a CSTR. The covariance matrix Q has been adjusted as de- weighting matrix Q for the EKF has to be adjusted accord- scribed above. ingly. For further analysis, simulation studies with different ge- q33 and q44 in this example show a direct influence on the ˙ ometries as well as different jacket flow rates where con- two estimated parameters. As the estimation of QR is the ducted. Temperatures were simulated and random noise was more important parameter, emphasis was placed on q . The 33 added (±0.1K). Fig. 6 shows that a correct estimation of k lines in Fig. 4 identify the problem of the estimator when ˙ depends strongly on the jacket flow rate. The estimation im- flow rates are high. The estimation results for QR deviate proves with higher flow rate as the plant model mismatch is too far from the real data and would pose problems if these ˙ decreased while the adjusted EKF copes with the problem QR data were used for estimating concentrations with an ofa higher flow rate. additional open loop model. ˙ It is now important to consider how the reactor size in- The estimation of QR is strongly influenced by the jacket fluences the estimation quality. The considered 10 l reactor flow rate. We have therefore adapted the element q33 ofthe was therefore scaled by the geometry factor fG to volumes tuning matrix Q ofthe EKF by a factor m/˙ m˙ which com- min of2 and 20 m 3. Fig. 7 shows the results for these cases and pensates the eigenvalue change by increasing the amplifica- ˙ clearly demonstrates that a change in reactor size leads to an tion. This way, QR is corrected more strongly. This adjust- increasingly inaccurate estimation of the heat transfer coef- ment also improved the estimation quality of k. This also ˙ ficient k and the heat ofreaction QR. leads to a larger or smaller amplification ofmeasurement noise and may pose a problem with very large flow rates and a small TJ = TJ,in − TJ,out. 3.5. Intermediate conclusions The physical interpretation ofthis change can be given as follows: An increase in jacket mass flow rate reduces the Heat balance calorimetry is a means to estimate the heat resulting T s which is directly used in the state estimation ofreaction and the heat transfercoefficient.Alternatively, ˙ scheme and directly reflects QR. It can be clearly seen that the model can be used in a state estimation scheme to S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248 4241

f =1 m =low g J 400 900 k Simulation 800 k Estimation 300 700

200 K)] 2 600 [W]

R Q Simulation R

Q 500 100 Q Estimation

R k [W/(m 400 0 300

−100 200 0 50 100 150 200 250 0 50 100 150 200 250 Time [min] Time [min] f =1 m =high g J 400 800 k Simulation k Estimation 300 700 600

200 K)] Q Simulation 2 [W] R 500 R Q Estimation Q 100 R

k [W/(m 400

0 300

−100 200 0 50 100 150 200 250 0 50 100 150 200 250 Time [min] Time [min]

Fig. 6. Estimation using a 10 l reactor with low (top line) and high (bottom line) mass ﬂow rates. Please note that mJ in the ﬁgure is indeed m˙ j in kg/s.

f =200 m =high 4 g n x 10 8 1000 k Simulation k Estimation 6 800

4 K)] 600 2 [W] R

Q Q Simulation 2 R 400

Q Estimation k [W/(m R 0 200

−2 0 0 50 100 150 200 250 0 50 100 150 200 250 Time [min] Time [min] f =2000 m =high 5 g n x 10 8 1500 k Simulation k Estimation 6 1000 K)] 4 2

[W] 500 R Q Simulation Q R

Q Estimation k [W/(m 2 R 0

0 −500 0 50 100 150 200 250 0 50 100 150 200 250 Time [min] Time [min]

Fig. 7. Simulation and estimation ofa 2 m 3 (top line) 20 m3 (bottom line) reactor with high jacket mass ﬂow rates. 4242 S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248

˙ estimate k and QR as parameters. The model used is glob- expression: ally observable if TR = TJ . However, for large reactors the T m˙ jacket ofa jacketed reactors does not behave like a CSTR. d J =− (TJ − TJ (z = zp+1)). (39) Ifthis model mismatch is not considered in the estimation dt bh(Lmax − L) and the reactor size is increased, the estimation quality is This model is subsequently used in the EKF. dramatically decreased. It is therefore necessary to develop a model which can be used in the EKF and takes the PFR behaviour ofthe jacket into account. 5. Simulation study II

4. Model and estimation extension Using model (5)–(12) for simulation with the jacket equation discretisation shown in Eqs. (35)–(37) in the EKF, the The PFR behaviour in the jacket can be approximated well estimation for the 2 m3 reactor is repeated for two different by discretisation using orthogonal collocation (Villadsen and jacket flow rates. Note the improvement over the results in Michelsen, 1978; Krämer et al., 2003a). In order to yield Fig. 7. ˙ a simple and not too large model it is sufficient to use As Fig. 8 clearly shows, the estimation of k and QR is four internal collocation points resulting in five differential dramatically improved. It should be noted that the simula- equations forthe lower part ofthe jacket instead ofone. tion was also performed using orthogonal collocation, how- Such a discretised model shows good accuracy (Georgakis ever, the EKF model uses fewer collocation points and these et al., 1977). It has also been shown that orthogonal colloca- points do not coincide, which corresponds to a certain plant tion preserves observability (Gesthuisen and Engell, 2000). model mismatch. Therefore, this method is a suitable approach for the problem at hand. The application ofthe orthogonal collocation yields: 6. Additional considerations d m˙ m˙ X(t) =− BX(t) − Bz= x(t,z = 0) It has been shown above that the simultaneous estima- dt bhL bhL 0 k tion ofthe heat transfercoefficientand the heat ofreaction + F(X(t)) depends on the reactor geometry and the jacket flow rate. c b . (35) p These considerations lead to the question ifan optimal with flow rate calculation might be necessary. Furthermore, addi-   tional sensor placement might be considered to improve the TJ (z = z1)  T (z = z )  estimation quality. Ifadditional sensors are used, this fur-  J 2  X(t) =  .  (36) thermore leads to the question where they should be placed. . And finally, the considerations so far have been concerned TJ (z = zp+1) with near ideal jacket configurations, either a CSTR be- and haviour (bath) or a PFR behaviour (welded halfpipes). How   can other geometries be accounted for? (TR − TJ (z = z1))  (T − T (z = z ))   R J 2  F(X(t))= . , (37) 6.1. Optimal flow rate calculation  .  (T − T (z = z )) R J p+1 We address the practical problem ofthe decreasing esti- p is the number ofinternal collocation points and z describes mation quality with high jacket flow rates by calculating an the space domain. B are the collocation matrices. Therefore, “optimal” m˙ J , which is highly relevant, as it poses a limit zp+1 corresponds to the effective length of the jacket pipe, beyond which observation becomes practically impossible. up to where exchanging heat is possible. The expression for Thus, if m˙ J is regulated, the optimal value provides an up- ˙ QJ , with per limit, beyond which observation is not possible in prac- L tice and beyond which increasing the flow rate has virtually ˙ QJ = kh(TR − TJ (z)) dz, (38) no influence. It remains to be noted that regulating m˙ J is 0 not the preferred choice for exothermic reactions in jacketed is left to be determined. There are two ways of calculating tanks, as heating and cooling has to be provided. The ma- the integral: Integrating the resulting polynomial or approx- nipulated variable ofchoice is thus the inlet temperature at imating by trapezoid or Simpson’s rule. Here the choice was an optimal flow rate ofcoolant. trapezoid rule. It would appear to be the case that an optimal flow rate The model describing the upper part ofthe jacket is in the jacket exists above which its increase only results in solved for one collocation point. Discretising the equation an infinitesimally small increase in heat removed. Assuming for the upper part of the jacket results in the subsequent a tank is cooled and TR is constant (i.e., in infinitely large S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248 4243

f =200 m =low 4 x 10 g J 8 800 k Simulation 700 k Estimation 6 600

4 K)] Q Simulation 2 500

[W] R

R Q Estimation Q 2 R 400 k [W/(m 300 0 200

−2 100 0 50 100 150 200 250 0 50 100 150 200 250 Time [min] Time [min] f =200 m =high 4 g J x 10 7 800 k Simulation 6 700 k Estimation 5 600 4 K)] Q Simulation 2 500

[W] 3 R R Q Estimation

Q R 400 2 k [W/(m 300 1 0 200

−1 100 0 50 100 150 200 250 0 50 100 150 200 250 Time [min] Time [min]

Fig. 8. Simulation ofand estimation a 2 m 3 reactor with low (top line) and high (bottom line) jacket mass ﬂow rates and a discretised jacket model. Note the improvement over Fig. 7.

tank), an analytical solution can be found for both jacket to achieve maximum heat removal through the jacket. The descriptions. equation for the heat removal can be written as For the CSTR-jacket the solution becomes: ˙ Qout =˙mJ cp(TJ,out − TJ,in) (43) m˙ c T + kAT J p J,in R m˙ c T + kAT T = T = J p J,in R m˙ J cp + kA with J,out . (44) m˙ J cp + kA m˙ c (T − T ) + kA(T − T ) + J p 0 J,in J,0 R m˙ J cp + kA It can be shown easily that the function is monotonously m˙ > 1 increasing for all J 0. This means that maximum heat × exp − (m˙ J cp + kA)t (40) removal rate will be at m˙ J →∞.As mJ cp ˙ lim Qout = kA(TR − TJ,in) (45) for the initial condition m˙ J →∞ this maximum exists and fractions thereof exist as well. T (t = ) = T J 0 J,0. (41) It would now be possible to include the estimation quality and the heat removal into one objective function and solve The solution for the PFR-jacket can be written as an optimization problem. However, it is still up to the user to define the importance ofthe estimation over the heat T(x)  removal rate. It seems therefore more appropriate to define k x m˙ TR − (TR − TJ,x ) exp − t t< , some guidelines on how to set J . 0 bc v = p There will be a maximum m˙ J defined by the plant itself. k (42)  x This can be used to calculate the fraction of the maximum TR − (TR − TJ,in) exp − x t v . bcpv that can actually be achieved. Furthermore, it should be considered that from For simplicity and for the sake of the argument, the steady state case for the CSTR-jacket is considered. The aim is m˙ J →∞ ⇒ TJ,in = TJ,out, (46) 4244 S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248

which means any heat balance calorimetry is impossible as 2. When the model is successfully fitted, it will improve the TJ,out only depends on the changes of TJ,in and thus the estimation quality. jacket balance equation ceases to exist. Then, k needs to be ˙ known to estimate QR correctly but cannot be estimated. The additional jacket sensors will improve estimation quality It thus seems sensible to consider the following: for both cases, for the case the jacket behave like a CSTR as well as for the case the jacket behaves like a PFR. In the • A certain fraction of the maximum heat removal has to be first case, it can be treated as a second measurement ofthe maintained, as it governs the batch time, we suggest 0.9. same value TJ,out, with which the standard deviation ofthe • This fraction should allow for a reasonable temperature random measurement error is reduced. For the second case, difference between TJ,in and TJ,out. the measurement provides additional information about the • In order to calculate this fraction, a good estimation of k jacket behaviour. is needed, which should be calculated at the beginning of When the placement ofonly one additional jacket sensor the reaction using a low mass flow rate. To avoid a sudden is considered, the ever changing volume in the tank for the change in k, m˙ J should guarantee a turbulent regime in case ofa semi-batch reactor has to be taken into account. The the jacket. most sensible would be to place the sensor in the first halfof the considered jacket channel. Ifthe semi-batch reactor is, Ifthe fractionconsidered is called the jacket flow rate can for example, half full to start with, a sensor placed on half be calculated in steady state as follows: the channel length ofthe considered channel at that point of kA operation will move to a quarter the length when the reactor m˙ J = is completely filled. − c . (47) 1 p,J When collocation is used, calculation is easier, ifthe sen- For the case ofthe PFR-jacket the behaviour is analysed sors are placed on the collocation points. As the liquid level differently. The steps are the same as above therefore the in the tank increases, the collocation points shift, whereas explanations are omitted. the sensor cannot. Maintaining the collocation point position decreases the discretisation accuracy, as their placement is ˙ Qout =˙mJ cp(TJ,out − TJ,in) (48) optimal for the given task. It is thus necessary to calculate k the polynomial describing the temperature along the length with TJ,out = TR − (TR − Tin) exp − L (49) coordinate and include it into the measurement function. bcp,J v m˙ v = J 6.3. Separation of the jacket into PFR and CSTR and bh (50) ˙ Not all jackets ofjacketed tank reactors behave like either ⇒ lim Q = khL(TR − TJ, ), (51) m˙ →∞ out in ideal CSTRs or PFRs. Consider a jacket which does follow the walls ofthe tank but not the Klöpperform 1 base ofthe −khL ⇒ =˙m c T − (T − T ) tank. For the Klöpperform base, the jacket will behave like 0 J p,J R R J,in exp c m˙ p,J J a CSTR, for the rest like a PFR. When the above models are considered, it is possible to combine them to suit this ×−T − khL(T − T ) J,in R J,in . (52) system or any other. For the above example, the jacket input temperature is For this case m˙ J has to be found numerically from Eq. (52). the input to a CSTR ofthe volume around the Klöpperform The two results yield different values for m˙ J whereby the base. The exit temperature is the inlet temperature to the PFR case requires a lower m˙ J for the same fraction of the PFR-model ofthe jacket. The outlet ofthat part is the inlet to maximum cooling capacity. the PFR-part for which there is no conductive heat transfer. The outlet ofthat part is the jacket outlet temperature. 6.2. Additional sensor placement Obviously, to fit the model to such a system, temperature sensors in the jacket would be very helpful. Ifadditional temperature sensors are placed on a process, they are normally placed inside the reactor to identify pos- 7. Simulation study III sible hot spots resulting from poor mixing. While this is a valid consideration, we think for calorimetric estimation, Using the above described extension to the measurement ˙ additional sensors could also be placed inside the jacket. function, an estimation of k and QR is performed using These sensors can help with two important points when us- an extra temperature sensor. Considered is a batch reac- ing calorimetry: tion where the extra temperature sensor is placed at halfthe

1. They can be used to determine the jacket behaviour, es- 1 Klöpperform is a generally agreed technical term for the speciﬁcally timate the jacket geometry and improve model ﬁtting. ellipsoid base oftank reactors. S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248 4245

900 300 Simulation 800 Est. T only J,out 250 Est. T and T 700 J,out J,internal 200 600 K)] 150 2 [W] 500 R Q 100 k [W/(m 400

50 300

0 200

−50 100 0 100 200 300 400 0 100 200 300 400 time [min] time [min]

Fig. 9. Comparison ofthe estimation using the jacket outlet temperature only with the estimation using an additional measurement at halfthe jacket channel length. length ofthe considered jacket channel. The point coincides differential equation describing the jacket temperature. Even with a collocation point. The simulation was performed us- for a small number of collocation points, the results are very ing five collocation points, the estimation uses three. good. It can be seen that the estimation using the extra mea- Thirdly, a direct extension ofthe considered jacket be- surement compensates the initial condition error much faster haviour onto adding measuring sensors to the system was and remains closer to the real value than the estimation us- performed. If the jacket behaves like a PFR, an additional ing one measurement. The exact same weighting matrices measurement in the jacket improves the estimation quality were applied. It is thus possible to tune the filter with the and provides an inside into the real channel geometry. It can extra measurement less aggressively and achieve a smoother also be used well in calibration and model fitting. estimate with either equivalent or better accuracy (Fig. 9). Finally, as higher mass flow rates in the jacket result in a degradation ofestimation quality, a practical approach to calculate the lowest sensible jacket flow rate has been pre- 8. Discussion and conclusions sented, which does, however, require an initial estimate of k. These points show that good model fitting and good esti- We have demonstrated that the heat ofreaction and the mation will be an iterative process which can be improved heat transfer coefficient for a batch, semi-batch or continuous with every reaction. stirred tank reactor can be estimated using an EKF. We have In the practical case, the reactor geometry should be an- furthermore shown that the jacket geometry has a major alyzed carefully. The model should then be fitted off line impact on the estimation ofthe heat transfercoefficient. to those conditions. After that, a simple calibration experi- Firstly, it can be seen that the EKF performs well for small ment should be run and model fitting needs to be performed. reactor sizes, even though a plant model mismatch exists From typical reaction data, model fitting can be enhanced between the real plant, where the jacket behaves more like and a first guess of k can be obtained. The minimum sensible a plug-flow reactor, and the EKF model. mass flow rate needs to be calculated and with typical reac- Nonetheless, careful tuning of the filter is paramount. The tion data the EKF needs to be tuned. These steps should be jacket flow rate has a strong influence on the quality of performed for a few historical batch runs preferably out of the observation. An adaptation ofthe tuning matrix Q to a large range of different processes to allow for continuous compensate the jacket flow rate influence was shown. A model fitting and filter tuning improvement. trade-off between good observer performance and maximum heat removal has to be made, i.e., it might be sensible in the 9. Future work plant to reduce the jacket medium temperature, rather than increase the flow rate. The large amplification necessary also As indicated in the title, this work considers the theoreti- decreases the filtering effect of the EKF. ˙ cal side ofthe estimation of QR and k. Following in a future The model assumption ofan ideally stirred jacket is rea- paper, we will describe applications ofthe state estimation sonable for state estimation only if the reactor size is small. scheme to different reactors of different sizes and geome- However, it becomes obvious for larger reactors that estima- tries. tion quality decreases considerably with reactor size. The future work will therefore cover the following points: Secondly, we developed a model extension to include the case in which the reactor jacket behaves like a PFR. It is • Efficient calibration: An efficient calibration scheme is achieved by applying orthogonal collocation to the partial absolutely necessary to be able to apply the scheme to 4246 S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248

different reactors. This will have follow the steps outlined xˆk|j estimated state vector at time tk based on above. measurements up to time tj • Controlled application: The calibration and state estima- z distribution coordinate, m tion scheme will be applied to well known and studied zi ith collocation point, laboratory and pilot scaled reactors using a deﬁned heat source to test it in real processes. Greek Letter • Real life application: The presented methods will be applied to laboratory and pilot scale reactors running ﬂuid density, 1000 kg/m3 exothermic reactions and subsequently to industrial scale reactors to show the usefulness of the method. Acknowledgements

Both authors formerly worked at the Chair of Process Notation Control, University ofDortmund, Germany. The financial support ofStefanKrämer by the Deutsche Forschungs- A heat transfer area, 0.2m2 gemeinschaft (project EN 152/31-1) during his work in AB base area ofthe reactor, m 2 Dortmund and by the Marie Curie Fellowship (Contract 2 Amax maximum heat transfer area, 0.2m HPMT-CT-2001-00227) during a stay at the Institute for AX cross sectional flow area, hb m2 Polymer Materials (POLYMAT) at the University ofthe b channel width, 0.02 m Basque Country is gratefully acknowledged. cp fluid heat capacity, 4000 J/kgK The authors also thank Prof. Asua of the Institute for Poly- cp,S heat capacity ofthe reactor wall material, mer Materials (POLYMAT) at the University ofthe Basque J/kgK Country and Prof. Engell, Chair of Process Control, Univer- CR total absolute heat capacity ofthe reactor, J/K sity of Dortmund, Germany for helpful discussions. Further- dR reactor diameter, m more, the discussion and help ofthe student Gerit Nigge- h channel height, 0.05 m mann is very much appreciated. hR level ofthe reaction medium in the reactor, m k heat transfer coefficient, 10 W/Km2 L maximum length ofthe location coordinate, Appendix A. Detailed development of the PFR jacket A equations h m mJ mass in the jacket, Ab kg m˙ J mass flow through the jacket, 0.05 kg/s The jacket is unravelled and approximated by a rectangu- mS mass ofthe reactor wall, kg lar flow channel, one side ofwhich is used forconvective p number ofcollocation points, heat transfer. The problem is modelled by looking at one channel element x to x + dx as shown below: q˙J,loss heat loss from the jacket in one channel element, W dQ ˙ ˙ ˙ ˙ = Qconv,x − Qconv,x+dx + QJ QJ,in heat flow into the jacket, W dt Q˙ J, out heat flow out ofthe jacket, W jTJ (t, x) Q˙ Q˙ = mc J conductiveheat flow through reactor wall, W with p jt , Q˙ J,loss heat loss from the jacket, W ˙ ˙ Qconv,x =˙mcpTJ (t, x), QR,in heat flow ofthe feed,W ˙ jTJ (t, x) QR,loss heat loss ofthe reactor, W Q˙ =˙mc T (t, x) +˙mc x ˙ conv,x+dx p J p jx d , QR,out heat flow out ofthe reactor, W ˙ ˙ QR,source heat ofreaction, W QJ = kA(TR − TJ (t, x)). TJ, jacket inlet temperature, K jT (t, x) in ⇒ mc J =˙mc T (t, x) −˙mc T (t, x) TJ temperature in the jacket, K p jt p J p J TJ,U temperature in the upper jacket part, K jT (t, x) T −˙mc J x + kA(T − T (t, x)) J,L temperature in the lower jacket part, K p jx d R J , TJ,out jacket outlet temperature, K TJ,z initial temperature along the z coordinate, K jTJ (t, x) jTJ (t, x) 0 hbcpdx =−vhbcp dx TJ,0 initial temperature at t = 0, K jt jx TR temperature in the reactor, K + kh x(T − T (t, x)) m˙ d R J , v fluid velocity, hb m/s ˙ 3 jTJ (t, x) jTJ (t, x) k VR,in inlet flow to the reactor, m /s =−v + (TR − TJ (t, x)). (A.1) ˙ 3 jt jx bcp VR,out outlet flow ofthe reactor, m /s S. Krämer, R. Gesthuisen / Chemical Engineering Science 60 (2005) 4233–4248 4247 3 2 Appendix B. Geometry factor Divide Eqs. (B.10) and (B.11) by fG and it becomes obvious, what the geometry factor influences: The models are extended by a geometry factor (fG), √ 3 whereby the ratio between the vessel radius and height is dhR fG ˙ ˙ = (VR, ,S − VR, ,S) (B.12) t 2 in out kept. d rR,S T V˙ B.1. CSTR model d R R,in,S 3 = fG(TR,in − TR) dt r2 hR R,S For the CSTR model presented in Eqs. (1)–(4) the geom- 1 1 2 3 −1 etry factor should increase the volume by the desired factor, + q˙R,source− + fG q˙R,loss Rcp,R hR rR,S i.e., V = fGVS. This leads to −k 1 + 2 3 f −1 (T − T ) 3 G R J , (B.13) rR = fGrR,S, (B.2) hR rR,S 3 T b = fGbS, (B.3) d J 1 3 −1 3 −1 2 = fG m˙ Scp(TJ,in − TJ ) + k fG rR,S dt mScp Q˙ =˙q h A R,source R,source R B , (B.4) 3 −2 W + 2 fG rR,ShR (TR − TJ ) . (B.14) [ l ] √ Q˙ =˙q A h 3 f R,loss R,loss , (B.5) The initial condition for R is also multiplied by G. [ W ] l B.2. PFR model for the geometric sizes and to For the PFR model the influences are the same. h = 3 f h R,0 G R,0,S, (B.6) However, it is questionable ifthe channel height in a real reactor would remain the same or is adjusted accordingly. m˙ = 3 f 2 f m˙ G PFR S, (B.7) The factor to keep it at the same level here, would be the velocity v. Therefore, a second design factor or relating con- V˙ = f V˙ R,in G R,in,S, (B.8) stant is included to relate the mass flow and the channel ˙ ˙ height, fPFR. VR,out = fGVR,out,S, (B.9) √ Ifthe channel height is increased by 3 fG, it means the for initial conditions and manipulated variables, where S ratio ofvessel height to channel height remains the same, f denotes the standard value. For a CSTR, PFR will be 1, √i.e., the length (L), one plug has to travel is increased by 3 for an explanation of this value, see below. √fG. As such, the mass flow would have to be decreased by IfEqs. (B.2)–(B.9) are inserted into the CSTR equations, 3 fG to maintain velocity. Therefore the following relations the following results: is added: hR 3 2 3 2 d ˙ ˙ h = fGfPFRhS. (B.15) rR,S f = fG(VR, ,S − VR, ,S) G dt in out A sensible value is 1 as the velocity remains the same. The TR hJ 2 3 2 d largest possible value is where hJ denotes the height of rR,ShR fG hS dt the wall covered by the jacket. = 1 c f V˙ (T − T ) c R p,R G R,in,S R,in R R p,R References 2 3 +˙qR, hRr f 2 source R,S G Bartanek, B., Gottfried, M., Korfhage, K., Pauer, W., Schulz, K., Moritz, 2 3 2 3 H.-U., 1999. Closed loop control ofchemical composition in free −˙qR,loss rR,S fG + 2rR,ShR fG radical copolymerization by online reaction monitoring via calorimetry and IR-Spectroscopy. In: RC User Forum Europe. pp. 1–13. 2 3 2 3 − k rR,S fG + 2rR,ShR fG (TR − TJ ) , BenAmor, S., Colombie, D., McKenna, T., 2002. Online reaction calorimetry. 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