A general criterion to define runaway limits in chemical reactors

Journal of Loss Prevention in the Process Industries 16 (2003) 187–200 www.elsevier.com/locate/jlp

A general criterion to define runaway limits in chemical reactors J.M. Zaldı´var a,∗, J. Cano b, M.A. Alo´s b 1 1, J. Sempere b, R. Nomen b, D. Lister c, G. Maschio c, T. Obertopp d 2 2, E.D. Gilles d, J. Bosch e, F. Strozzi e a

European Commission, Joint Research Centre, Institute for Environment and Sustainability, Ispra, Italy b Chemical Engineering Department, Institut Quı´mic de Sarria´—URL, Barcelona, Spain c Dipartimento di Chimica Industriale ed Ingegneria dei Materiali, Universita` di Messina, Messina, Italy d Institut fu¨r Systemdynamik und Regelungstechnik, Universita¨t Stuttgart, Stuttgart, Germany e Engineering Department, Carlo Cattaneo University, Quantitative Methods Group, Castellanza, Italy

Abstract A general runaway criterion valid for single as well as for multiple reaction types, i.e. consecutive, parallel, equilibrium, and mixed kinetics reactions, and for several types of reactors, i.e. batch reactor (BR), semibatch reactor (SBR) and continuous stirred tank reactor (CSTR) has been developed. Furthermore, different types of operating conditions, i.e. isoperibolic and isothermal (control system), have been analysed. The criterion says that we are in a runaway situation when the divergence of the system becomes positive (div ⬎ 0) on a segment of the reaction path. The results show that this is a general runaway criterion than can be used to calculate the runaway limits for chemical reactors. The runaway limits have been compared with previous criteria. A considerable advantage, over existing criteria, is that it can be calculated on-line using only temperature measurements and, hence, it constitutes the core of an early warning runaway detection system we are developing.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Runaway; Nonlinear dynamics; Safety; Batch; Reaction engineering

1. Introduction It is well-known that for certain values of the parameters in the mass and energy balance equations that represent the dynamic behaviour of chemical reactors, the system becomes very sensitive to the values of the initial conditions and the parameters of the system (Varma, Morbidelli, & Wu, 1999). Parametric sensitivity in this context describes the situation in which a small change in the inlet conditions, as well as to any of the other physicochemical parameters of the system, induces a large change in the temperature profile of the reactor. There have been many runaway/parametric sensitivity studies in chemical reactors (Adler & Enig, 1964; Alo´s, Nomen, Sempere, Strozzi, & Zaldı´var, 1998; Alo´s, Strozzi, & Zaldı´var, 1996; Balakotaiah, 1989; BalakoCorresponding author. Fax: +39-0332-789328. E-mail address: [email protected] (J.M. Zaldı´var). 1 Present address: Hyprotech Europe, Pg Gracia 56, Barcelona, Spain. 2 Present address: BASF Aktiengesellschaft, Engineering Services Process Control, Ludwigshafen, Germany. ∗

taiah, Kodra, & Nguyen, 1995; Barkelew, 1959; Hagan, Herskowitz, & Pirkle, 1987; Hagan, Herskowitz, & Pirkle, 1988; Morbidelli & Varma, 1988, 1989; Rajadhyaksha, Vasudeva, & Doraiswamy, 1975; Steensma & Westerterp, 1990, 1991; Strozzi & Zaldı´var, 1994; Thomas, 1961; Vajda & Rabitz, 1992; Van Welsenaere & Froment, 1970; Westerterp & Ptasinski, 1984; Westerterp & Overtoom, 1985; Westerterp & Westerink, 1990; Wilson, 1946; Wu, Morbidelli, & Varma, 1998) aiming to derive runaway boundaries for different types of reactions and different types of reactors. For a general overview see the work by Varma et al. (1999). At the beginning, since numerical calculations were difficult, these criteria were based on some geometrical property in the temperature profile versus time or conversion (Adler & Enig, 1964). Although some of them give fundamentally correct descriptions of thermal runaway, they do not give any measure of its extent or intensity (Varma et al., 1999). Moreover, some of them were too conservative or were derived separately for each type of reactor and kinetic scheme. In order to overcome these limitations, a new series of criteria were developed based on the concept of parametric sensitivity (Morbidelli & Varma,

0950-4230/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0950-4230(03)00003-2

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Nomenclature B Cp C Da E Hj k n r q R Rj Rij St S t tref T Tw uA uB U V z

(⫺⌬H)Cig r·Cp·Ti mean specific heat of reaction mixture kJ/(K kg) Concentration of reactant, kmol/m3 Damko¨ hler number, Da = k(Ti)(Ci)(n⫺1)tref activation energy, kJ/mol heat of reaction ratio, ⌬Hj/⌬H1 reaction rate constant, (kmol/m3)(1⫺n)/s reaction order reaction rate, kmol/(m3 s) volumetric flow, m3/s universal gas constant dimensionless reaction rate j⫺n1)k (Ti) C(n Ai j reaction rate constant ratio, i k1(T ) U·S·tref Stanton number, St = r·V·Cp heat exchange surface area, m⫺2 time, s V reference time = residence time, tref = qd temperature, K jacket temperature, K dimensionless concentration of A, CA / CiA dimensionless concentration of B, CB / CiA heat transfer coefficient, kJ/(m2 K s) volume of reaction conversion, z = 1-u dimensionless heat of reaction parameter, B =

Greek Symbols δ ⌬Hi ⌬Hr ⑀ ρ γj θ τ τd

cooling dimensionless time. d = (V·r·Cp)w / (V·r·Cp) heat of the ith reaction, kJ/kmol heat of the propagation reaction fraction of dosed volume, e = Vd / V0 density of the mixture Ej dimensionless activation energy, RTi T dimensionless temperature in the reactor, q = i⫺1 g1 T dimensionless time, t/tref dosing time, td = min(1,t)

冉 冊

Superscripts i sp

initial condition set-point

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Subscripts c d w

critical condition dosing jacket

1986, 1988, 1989; Vajda & Rabitz, 1992, etc.). However, these new criteria require a significant amount of computation and they are implicit and, hence, it is not possible to apply them on-line. Finally, in despite of all these studies, there are still few criteria derived for predicting runaway boundaries that can be used for several types of reactors or for systems with multiple reactions. For example, general criteria that have been applied to different processes and to multiple reactions are the MV criterion (Morbidelli & Varma, 1988) based on the concept of parametric sensitivity and Balakotaiah et al. (1995) criterion based on the analysis of reaction paths in the temperature-conversion plane. Sensitivity to initial conditions is a well-known characteristic of chaotic phenomena. To study such systems, researchers have developed powerful methods for extracting physical quantities from theoretically or experimentally obtained signals (Abarbanel, 1996; Kantz & Schreiber, 1997). In a series of studies carried out previously (Alo´ s et al., 1996; Strozzi & Zaldı´var, 1994), it was shown, theoretically and experimentally, that some of the chaos theory techniques, developed to measure the parametric sensitivity in strange attractors could be used to identify critical regions in which thermal runaway can occur. The advantage of this approach is that, in principle, a runaway criterion based on one invariant of the system may be calculated on-line using only temperature measurements by applying state space embedding reconstruction techniques (Abarbanel, 1996). Hence, we first defined an appropriate early warning detection criterion, based on an invariant property under phase space reconstruction. The criterion says that we are in a runaway situation when the divergence of the system becomes positive (div ⬎ 0) on a segment of the reaction path (Strozzi, Zaldı´var, Kronberg, & Westerterp, 1999). We recall that the divergence is a scalar quantity defined at each point as the sum of the partial derivatives of the mass and energy balances with respect to the corresponding state variables—temperature and conversions—i.e. (dT / dt) / T + Σi∂(dzi / dt) / zi. The criterion was compared with several existing criteria for the case of a first order and an autocatalytic reaction carried out in an isoperibolic (constant jacket temperature) batch reactor (BR). It was shown that the runaway boundaries obtained were between the ones produced by conservative criteria (Barkelew, 1959) and the maximum para-

metric sensitivity criteria which normally is less conservative. Furthermore, the use of the divergence with an EKF was applied by Obertopp, Alo´ s, Zaldı´var, Strozzi, and Gilles (1998a) as an alternative approach. However, in order to develop a robust early warning detection system, which is the main goal of this investigation, it is necessary first to assess the validity of the divergence criterion for more kinetic schemes and operating conditions; second to assess that it is possible to reconstruct theoretically the divergence form only temperature measurements for all those kinetic schemes and operating conditions; and finally to show their feasibility in ‘realistic’ situations, i.e. in industrial plants. In this work we have assessed the validity of the divergence criterion to define runaway boundaries. For this purpose we have first applied the divergence criterion for the case of multiple reactions for which few studies exist (Balakotaiah et al., 1995). Several kinetic schemes, i.e. consecutive, parallel, equilibrium, and mixed cases have been considered. Secondly, we have analysed several type of reactors, i.e. batch, semibatch reactor (SBR) and continuous tank stirred reactor (CSTR) and finally we have compared different operating conditions, i.e. isoperibolic, and isothermal (controlled jacket temperature to maintain constant reactor temperature). It is shown that the criterion based on div ⬎ 0 holds true in all these cases and it can be used not only to identify critical parametric regions and boundary diagrams between runaway and non-runaway, but also others in which the performance of the reactor is optimal. The results are compared with several runaway criteria such as the maximum parametric sensitivity criterion (Morbidelli & Varma, 1988). It is shown that the same behaviour observed for the cases of a first order and autocatalytic reactions (Strozzi et al., 1999) holds true, i.e. the divergence criterion is more restrictive than the maximum sensitivity criterion. Finally, a considerable advantage of this criterion over existing criteria is that it can be calculated on-line based only on temperature measurements without the necessity of a model of the process and, hence, it constitutes the core of an early warning detection system we are developing. The work has been divided as follows: in Section 2 we present the divergence criterion, whereas in Section 3 the application to BRs with multiple reactions is

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developed; in Section 4 the intercomparison between BRs, SBRs and CSTRs are studied; in Section 5 we discuss the influence of the control system, and, finally, in Section 6 we present some conclusions and future work.

2. Definition of the runaway criterion According to the Liouville’s theorem (Arnold, 1973) there is a relation between the state space volume of a ddimensional dynamical system and its divergence. This relation can be expressed as V(t) ⫽ V(0) exp

冋冕

t

0

div{F[x(t)]} dt

册

(1)

where div{F[x(t)]} ⫽ ⫹

∂F1[x(t)] ∂F2[x(t)] ⫹ ⫹… ∂x1 ∂x2

(2)

∂Fd[x(t)] ∂xd

Hence, for a dynamical system, see Fig. 1, the rate of change of an infinitesimal volume V(t) following an orbit x(t) is given by the divergence of the flow, which is locally equivalent to the trace of the Jacobian of F. Even though for dissipative systems, like the chemical reactors we are studying, the divergence will decrease as t→⬁, it has been observed (Strozzi et al., 1999) that in the case of a runaway there is, during a certain period of time, a state space volume expansion. This means that trajectories originating from nearby starting points would diverge and this, of course, is correlated with the

parametric sensitivity shown by these types of reactors. For example, Fig. 2 illustrates this behaviour for the case of a consecutive reaction carried out batchwise in an isoperibolic reactor. In the case of non-runaway the divergence is always negative, however, at a certain point in the parameter space—in this case by increasing the dimensionless heat of reaction parameter, B—it becomes positive. Therefore, taking into account the correlation between parametric sensitivity and the divergence of nearby trajectories, Strozzi et al. (1999) defined a runaway criterion as when the divergence of the reactor becomes positive on a segment of the reaction path, i.e. div[F(q,uA,uB,…)] ⬎ 0

(3)

For the case of a single reaction there are two components to the divergence, the exponential temperature dependence that will tend to increase the divergence between nearby trajectories and the effect of reactant conversion plus the heat transfer to the jacket that will tend to decrease the divergence between nearby trajectories. For the case of multiple reactions, the situation is similar. However, as we are studying thermal runaway behaviour, we should include in the divergence calculation only the reactions that contribute to the thermal behaviour of the system. Otherwise, if we have a really fast reaction that does not produce/consume heat, we could have a volume contraction in the d-dimensional state space considering that reaction, but an expansion in the d⫺1 dimensional space without considering that reaction. Furthermore, from the point of view of state space reconstruction using only temperature measurements, which is the final objective of our work, reactions that do have zero heat of reaction are not normally observed, i.e. not linked directly to the dynamic behaviour of the temperature.

3. Kinetic schemes In this section, we present the application of the runaway criterion, div ⬎ 0, to several kinetic schemes. This allows the extension and validation of the criterion for the case of multiple reactions where, at present, there is very little guidance (Balakotaiah et al., 1995). The application of the runaway criterion is illustrated in detail for consecutive reactions and the results for several other kinetic schemes are briefly summarised. 3.1. Consecutive reactions The model for the case of two consecutive reactions r1

Fig. 1. State space evolution of an infinitesimal volume V as a function of time.

r2

(A→B→C) carried out batchwise in an isoperibolic reactor can be obtained by solving the mass and heat balances differential equations

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Fig. 2. Simulated temperatures, conversions and divergence for one (a) non-runaway and (b) a runaway scenarios, for a consecutive reaction carried out batchwise in an isoperibolic reactor. Parameters: n1 = 1; n2 = 1; g1 = 20; g2 = 40; H2 = 1; Ri2 = 0.6; (st / Da) = 30. For (a) B = 10, for (b) B = 12.

duA ⫽ ⫺DaR1 dt

(4)

duB ⫽ Da(R1⫺Ri2R2) dt

(5)

dq ⫽ BDa(R1 ⫹ H2Ri2R2)⫺St(q⫺qw) dt

(6)

with initial conditions uA = 1, uB = 0, and q = 0 at t = 0. In this case we have a three-dimensional state space given by the temperature and the concentrations of species A and B, respectively, i.e. (q, uA, uB). Where uA ⫽ CA / CiA; uB ⫽ CB / CiA and q ⫽

冉 冊

T ⫺1 g1 Ti

(7)

The reactions are assumed to be nith order with respect to reactant and to follow Arrhenius temperature dependence, i.e. rj = kjCnj i. The dimensionless rate expressions can be written as R1 ⫽ exp

冉 冊

冉 冊

g1q g 2q unA1; R2 ⫽ exp un2 g1 ⫹ q g1 ⫹ q B

Fig. 3. State space (uA, uB, q) for the case of two consecutive reactions (A→B→C) in a BR when changing the dimensionless heat of reaction parameter (B) from 10 to 12 with a constant step of 0.2. The critical value of B, i.e. when the sensitivity is maximum, is Bc = 11.4 (Morbidelli & Varma, 1989) and the critical value obtained using the criterion div ⬎ 0 is Bc = 11.0. Other parameter values are: n1 = 1; n2 = 1; g1 = 20; g2 = 40; H2 = 1; R2i = 0.75; (st / Da) = 30, and qi = qw = 0.

(8)

where Ri2 is the reaction rate constant ratio (see Nomenclature for a complete definition of all variables and parameters), H2 the heat of reaction ratio (⌬H2 / ⌬H1), and Da and St are the Damko¨ hler and Stanton numbers, respectively. Fig. 3 shows the state space representation while changing the dimensionless heat of reaction parameter (B), i.e. when increasing the exothermicity of the system. As can be seen the dynamic

behaviour of the system changes as the system moves from a non-runaway to a runaway situation. In order to calculate the divergence criterion, it is necessary to calculate the sum of the diagonal elements of the Jacobian, J, of the system, i.e. div = j11 + j22 + j33, where j11 ⫽ ⫺Da exp

冉 冊

g1q n u(n1⫺1) g1 ⫹ q 1 A

(9)

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冉 冊

(10)

BDa (R g2 ⫹ H2Ri2g1g2)⫺St (g1 ⫹ q)2 1 1

(11)

j22 ⫽ ⫺DaRi2 exp j33 ⫽

g 2q n u(n2⫺1) g1 ⫹ q 2 B

The parametric sensitivity of this system was studied by Morbidelli and Varma (1989) as a function of its main parameters. In this work we have calculated the boundary diagrams in the B–Ri2 parameter plane as a function of several parameters of the system: the activation energy of the second reaction, g2; the heat of reaction ratio parameter, H2; and the order of the second reaction, n2 (Fig. 4). As can be seen the boundary diagrams are similar, but always more conservative, than those described by Morbidelli and Varma (1989). These results are in-line with the results obtained for the nthorder and autocatalytic reactions (Strozzi et al., 1999). Apart from that, practically all the same general trends and conclusions as in the work by Morbidelli and Varma (1989) are observed: 앫 The second exothermic reaction always enlarges the region of reactor runaway. 앫 For increasing values of the activation energy g2 and the heat of reaction ratio H2 the runaway region enlarges. 앫 For increasing values of the reaction order n2 the runaway region shrinks. 앫 In the case where the heat generated by the second reaction is zero, H2 = 0, the critical value of B becomes independent of the second reaction rate. 앫 The dependence of the reaction order n2 for the case of n2 = 0.5 is not constant for Ri2 ⬎ 0.75 as in the work by Morbidelli and Varma (1989). 3.2. Multiple reactions A similar approach has been carried out to assess the validity of the criterion for the following cases: r1

r2

1. parallel(A→B, A→C),

冉

r2

冊

→ 2. equilibrium A→B r3 C , ← r1

Fig. 4. Regions of thermal parametric sensitivity for the case of two consecutive reactions using the divergence criterion: (a) effect of the activation energy of the second reaction, g2, discontinuous line: values from Table 2 in the study by Morbidelli and Varma (1989); (b) effect of heat of reaction ratio parameter, H2; (c) effect of order of the second reaction, n2. In each case, the region above the curve is sensitive, div ⬎ 0, while the region below is non-sensitive, divⱕ0. For a more detailed comparison see Fig. 2 in the study by Morbidelli and Varma (1989).

J.M. Zaldı´var et al. / Journal of Loss Prevention in the Process Industries 16 (2003) 187–200 r1

r2

r3

3. consecutive–parallel(A→B→C, A→D),

冉

r1

冊

→ r2 4. equilibrium–consecutive A r3 B, A→C ←

冉

r1

冊

→ r2 5. parallel–equilibrium A r3 B, A→C reactions. ← Table 1 summarises the pertinent equations representing the mass and energy balances whereas the divergence has been calculated in the same way as previously described. For the case of parallel reactions, the parametric sensitivity was studied by Morbidelli and Varma (1989) as a function of its main parameters. In this article we also have calculated the boundary diagrams in the B–Ri2 parameter plane as a function of several parameters of the system. For example Fig. 5 shows the influence of the activation energy of the second reaction, g2. As can be seen the boundary diagram is similar to those described by Morbidelli and Varma (1989). As in the case of consecutive reactions, the values calculated using the divergence criterion are more conservative than the values obtained with the parametric sensitivity criterion. Furthermore, as found by Morbidelli and Varma, in this case the presence of a second reaction intrinsically less sensitive than the first (i.e. H2 ⬍ 1, g2 ⬍ g1 or n2 ⬎ n1) can lead to a shrinking of the runaway region. The equilibrium reaction, r3, will always reduce the runaway area since its contribution to the divergence is always negative. This can be seen by comparing Fig. 6, where boundaries for the dimensionless heat of reaction parameter as a function of the reaction rate constant ratio of the second reaction have been calculated, with Fig. 4(b), for different values of the heat of reaction ratio parameter. Furthermore, the generalised criterion based on the parametric sensitivity presented by Morbidelli and Varma (1988) has been also applied to the case of equilibrium reactions in order to compare both criteria. As in the previous cases studied, general agreement has been found between both criteria, being the divergence criterion always more conservative. It is clear that the cases of two consecutive and two parallel reactions studied previously would be limiting cases (i.e. Ri2 = 0 and Ri3 = 0) for the case of consecutive– parallel reactions. In order to study the effect of the combination of both types of reactions in one kinetic scheme, we have calculated the boundary diagrams as a function of several parameters of the system. Fig. 7 shows the evolution of the boundary diagram between runaway and non-runaway as a function of the reaction rate constant ratios. As can be seen, the addition of other exothermic reactions always enlarges the region of reactor runaway, i.e. further decreases the safe region.

193

For the cases of equilibrium–consecutive and parallel– equilibrium reactions, the existence of an equilibrium reaction, r3, with a negative contribution to the divergence will always enlarge the safe region when compared with the case of consecutive or parallel reactions, respectively (results not shown). 4. Batch, semibatch and CSTR reactors In this section, we present a unified analysis that allows the comparison between different types of reactors, i.e. BR, SBR and CSTR. Some results of this section have been presented in an article by Obertopp, Alo´ s, Zaldı´var, Strozzi, and Gilles (1998b), which is not easily accessible and where emphasis was placed on the application of model-based (Kalman filtering) techniques. We extend these results and present them in a more general form. The mass and energy dimensionless balances (z, q) that describe the dynamic behaviour of a well-stirred batch, semibatch and continuous stirred tank reactors in r1

whose the reaction: A→B,—for BR and CSTR—or the r2

reaction: A + B→C—for SBR with stoichiometric dosing at t = 1—occur are summarised in Table 2. Parametric sensitivity and divergence criteria have been applied to BR, SBR and CSTR. The variation of Tw, Da and B allows the determination of the runaway boundary regions for each criterion and for each reactor type. For the case of SBR, Steensma and Westerterp, (1990, 1991) defined and studied several possible temperature behaviours: non-ignition, marginal ignition, runaway and quick onset, fair conversion and smooth temperature profile (QFS) in SBRs, and how to distinguish between them. For example, in Fig. 8, the reactor temperature, q, simulated profiles as a function of dimensionless time have been plotted for several values of the jacket temperature, qw. When the reaction rate is much lower than the dosing rate, the accumulation is large and the reactor operates as a batch process under subcritical conditions (non-ignition). There is no interest in carrying out the process under these conditions because of the length of time and dosing serves no purpose. As the jacket temperature increases, the reaction rate becomes faster and a runaway occurs. In the case of lower exothermicity instead of runaway we will be in a situation of marginal ignition. As the jacket temperature continues to increase, the runaway occurs earlier. When qmax takes place during the dosing period, i.e. t(qmax) ⬍ td, the accumulation starts to decrease and, hence, the maximum temperature decreases. When the jump of q is due to a little accumulation, which occurs close to the start-up, the semibatch process operates at the optimal values of the design parameters, since the reaction will proceed quasi-instan-

⫺Da(R1 + Ri2R2) DaR1

duA/dt duB/dt duD/dt

R2 = exp(g2q / g1 + q)unA2

As (6)

dq/dt

Parallel

Consecutive–parallel

BDa(R1 + H2Ri2R2⫺H3R3i R3)⫺ BDa(R1 + H2Ri2R2 + H3Ri3R3)⫺St(q⫺qw) St(q⫺qw) ⫺DaR1 ⫺Da(R1 + Ri3R3) Da(R1⫺Ri2R2 + Ri3R3) Da(R1⫺Ri2R2) DaRi3R3 R3 = exp(g3q / g1 + q)unC3 R2 = exp(g2q / g1 + q)unB2 R3 = exp(g3q / g1 + q)unA3

Equilibrium

Table 1 Energy and mass balances for the kinetic schemes analysed Parallel–equilibrium

R3 = exp(g3q / g1 + q)unB3

⫺Da(R1⫺Ri3R3) Da(R1⫺Ri2R2⫺Ri3R3)

⫺Da(R1 + Ri2R2⫺Ri3R3) Da(R1⫺Ri3R3)

BDa(R1 + H2Ri2R2⫺H3R3i R3)⫺St(q⫺qw) BDa(R1 + H2Ri2R2⫺H3Ri3R3)⫺St(q⫺qw)

Equilibrium–consecutive

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Fig. 5. Regions of thermal parametric sensitivity for the case of two parallel reactions: effect of activation energy of the second reaction, g2. In each case, the region above the curve is sensitive, div ⬎ 0, while the region below is non-sensitive, divⱕ0 (for comparison see Fig. 11 in the study by Morbidelli & Varma, 1989).

Fig. 6. Boundary diagrams for an equilibrium reaction where g1 = 20, g2 = 40, g3 = 30, Ri3 = 0.5, (St/ Da) = 30 and H2 = H3 = 0.5, 0.8, 1, 1.5; using the divergence criterion (continuous line) and the maximum parametric sensitivity criterion (discontinuous line).

taneously. In these conditions the q–t profile is in accordance with the meaning of the acronym QFS: quick onset, fair conversion and smooth profile. For larger qw, qmax occurs later but the thermal profile remains that of the QFS (Fig. 8). Consequently, boundary lines for a SBR should discern between non-ignition and runaway regions and between runaway and QFS regions. The runaway region is always situated, for highly exothermic processes, between non-ignition and QFS and, therefore, two boundary lines are necessary into the diagram. At low

195

Fig. 7. Boundary diagram between runaway (top) and non-runaway (bottom) regions in the case of consecutive–parallel reactions using the divergence criterion. Parameters are: n1 = 1; n2 = 1; n3 = 1; g1 = 20; g2 = 40; g3 = 40; H2 = 1; H3 = 1; (st / Da) = 30; Ri2 = 0–2; Ri3 = 0–2.

exothermicities, runaway area becomes marginal ignition area and therefore this area separates QFS and non-ignition area. Alo´ s et al. (1998) have shown that the maximum parametric sensitivity criterion of Morbidelli and Varma (1988) was able to discern in SBRs only between non-ignition and runaway conditions, and that an additional criterion to distinguish between runaway and QFS processes was necessary. The combination of both criteria allowed the comparison of the boundaries between the different regions in the parameter space for homogeneous and heterogeneous semibatch processes with the boundaries developed by Steensma and Westerterp (1990) with a satisfactory agreement. When we apply the divergence criterion to SBRs (Fig. 9), we can see that it is able to distinguish between nonignition and runaway at low reactive conditions, between runaway and marginal ignition at medium reactive conditions and between runaway and QFS at high reactive conditions. The criterion is more conservative than the maximum parametric sensitivity for the case of runaway and than the earliest ignition time of the process (Alo´ s et al., 1998) for the case of QFS. The differences are more pronounced in the last case. However, it is the only criterion that distinguishes between non-ignition, marginal ignition, runaway and QFS. In other words, the divergence criterion confines the runaway zone. In the case of a CSTR, like the BR, a single boundary differentiates between a safe process and a runaway. In this case the parameters were selected in such a way as there is no possibility of periodic behaviour, and, hence, the possibility of a runaway is restricted to the start-up. As can be seen in Fig. 10, both criteria produce similar results. In order to compare the three reactors under similar

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Table 2 Energy and mass balances for the types of reactor analysed

dθ/dτ dz/dτ d ln V/dτ

BR

SBR

CSTR

BDa1R1⫺St(q⫺qw) Da1R1

BDa2R2⫺St(q⫺qw)⫺(q⫺qd)(d ln V / dt) Da2R2 + (1⫺z)(d ln V / dt) e /(1 + et) if tⱕ1 or 0 if t ⬎ 1 R2 = (1⫺z)(1⫺z⫺((1⫺td)/ (1 + etd))) exp(gq / g + q)

BDa1R1⫺St(q⫺qw)⫺(q⫺qd) Da1R1⫺z

R1 = exp(gq / g + q)(1⫺z)n

R1 = exp(gq / g + q)(1⫺z)n

Fig. 8. Profile temperature for different values of qw = qd. Parameters are: Tref = 300 K, Tw = 276⫺312 K, g = 42, Da = 1.8, St = 4, e = 0.4.

Fig. 10. Runaway boundaries for a CSTR using the divergence (continuous) and Morbidelli and Varma’s (1988) criteria (discontinuous). Parameters are: g = 20, St = 1.

Fig. 9. Runaway boundaries for a SBR using the divergence (continuous line), maximum parametric sensitivity (discontinuous line) and Alo´ s et al.’s (1998) criteria (-.). Simulation parameters are: g = 42, Da = 1.8, St = 4, e = 0.4.

process conditions we have calculated the boundary diagrams for a BR, a CSTR, Fig. 11. For the SBR, as one can expect, when we decrease e, i.e. we decrease the dosed volume, the runaway region enlarges; in the limit, i.e. e→0, the boundaries for BR and SBR will coincide. However, BRs do not possess the boundary between run-

Fig. 11. (a) Runaway boundaries for divergence criteria for a BR (continuous line); (b) runaway boundaries for divergence criteria for a CSTR reactor (discontinuous line, -.); Parameters are: g = 42, Da = 1.8, St = 4; (c) CSTR, same parameters but with Da = 0.9 (discontinuous line, --).

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away and QFS, which is characteristic of SBRs. As expected, for the CSTR, when the Damko¨ hler number is halted the runaway region shrinks. Comparing BRs and CSTRs for the same parameters, it is possible to see that the runaway region for a BR is always larger than for a CSTR. This is also clear by comparing their divergences since divCSTR = divBR⫺2. A similar comparison between SBR and CSTR is not possible since the conversion behaviours are different.

5. The influence of the control system An important aspect of definition of runaway boundaries in chemical reactors, that has received little attention, is the influence of the control system. Normally, all the studies have been carried out assuming isoperibolic behaviour, i.e. constant jacket temperature, and only recently some studies introducing the control system dynamics have been carried out. For example, Ratto and Paladino (2001) have studied using the divergence criterion, the start-up of a CSTR with the coolant temperature controlled by a PI device. They have shown that the divergence criterion can be applied to the study of the sensitivity of transient trajectories and it provides useful information about the controllability of many different start-up strategies. In this study, we have studied the applicability of the divergence criterion to a BR controlled with a master/slave proportional and PI controllers, which are typical for these types of installations (Zaldı´var, Herna´ ndez, Nieman, Molga, & Bassani, 1993). 5.1. Two proportional control loops In this case, the control configuration may be summarised as follows, when the process is carried out in isothermal conditions, the reactor temperature is maintained at its desired value, set-point, by adjusting the temperature set-point for the heat transfer fluid recirculating through the reactor jacket. This is accomplished by the master controller. The temperature of the heat transfer fluid, which circulates through the reactor jacket, is controlled using a slave controller. Hence, the controller of the outer loop corrects deviations of the reactor temperature, q, from the set value, qsp, following a proportional criterion sp ⫹ Kp1(qsp⫺q) qsp w ⫽ q

(12)

providing the set value for the heat transfer fluid temperature, qsp w , which is adjusted by the inner loop controller by means of another proportional controller. Expressing this in terms of heat balance, it is possible to write QR ⫽ Kp2(qsp w ⫺qw)

(13)

where q = 1. By substituting Eq. (12) into Eq. (13) we sp

197

obtain the heat flux necessary to keep the temperature of the reactor close to the pre-defined set-point. Then the differential equation, that describes the jacket dynamic behaviour of a controlled BR can be written, in dimensionless form, as dqw ⫽ δSt(q⫺qw) ⫹ QR dt

(14)

where d is the cooling dimensionless time of the heat transfer fluid system. In order to consider control saturation during transients, the manipulated variable, QR, is constrained as follows: if the required QR is higher than the maximum heating/cooling capacity of the system, Qmax R , this parameter takes this maximum value and if QR is lower than the minimum heating/cooling capacity then the parameter is equal to Qmin R . We have calculated the concentration, temperature and divergence profiles for isoperibolic and isothermal simulation runs using the conditions indicated in Fig. 12. As can be seen (Fig. 12(b) and (c)) when the control is applied the process changes from runaway (isoperibolic) to non-runaway, i.e. the process becomes safe. With the control system, the divergence never becomes positive in any interval of time and the maximum dimensionless temperature reached in the reactor falls from approximately 9–0.3, i.e. near to isothermal conditions. In general terms the introduction of a control system reduces the runaway region. Of course, an ‘ideal’ control system, i.e. instantaneous response and infinite power will always produce safe operating conditions. In reality, there is always a certain delay and a finite amount of power than can be used to compensate the temperature increase due to the exothermic reaction that occurs in the reactor. When the system is studied under different conditions it is possible to identify a critical set of parameters that distinguishes between safe and runaway regions. This boundary surface is defined as the first at which the divergence of the system becomes positive. Boundary values of the B parameter have been calculated for a PP-controlled BR with first order kinetics, varying the Damkho¨ ler number and for different values of the control parameters. As can be seen in Fig. 13, the divergence criterion discerns between runaway and non-runaway conditions. When the proportional control parameter increases the boundary surface increases that means the zone of non-runaway increases. This behaviour changes at high Damko¨ hler numbers, since in this case the reaction is faster and the controller system has no time to act. In these conditions all the proportional values in the control system produce similar safe regions. We have also calculated the boundary diagram for the case of a system whose maximum power capacity, Qmax R , has been doubled (Fig. 14). In general, there are two main parameters for controlled BRs that determine if we are in safe or runaway situations. These are the

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Fig. 12. Concentration, temperature and divergence profile for: (a) an isoperibolic BR; (b) an isothermal PP-controlled reactor (c) a PI-controlled BR. Conditions of simulation are: Tref = 298 K, Tw = 298 K, Da = 0.05, B = 1.8, g = 10, St = 1; for PP-control: Kp1 = 3, Kp2 = 5; for PI-control: Kp = 5, Ki = 1.

Fig. 13. Runaway boundaries for divergence criteria for a PP-controller BR for different values of Kp1. Conditions of simulation: Tref = 298 K, Tw = 298 K, Da = 0.02⫺0.3, g = 10, St = 1, Kp1 = 1–20, Kp2 = 5.

power capacity of the cooling system and the rate of reaction, given by the Damko¨ hler number. For slow reaction rates (low values of Damko¨ hler number) the determination of adequate control parameters, in this case the proportional gains in the inner and outer loops, are essential to keep the reaction temperature under control. In this case a good control design will attain safe operating conditions. For fast reaction rates the power capacity is the principal factor (Fig. 14), since in this case we are going to reach saturation in the control sys-

Fig. 14. Comparison between a control system with a maximum (continuous lines) and with 2Qmax power capacity of Qmax R R (discontinuous lines), (1) Kp1 = 20, (2)Kp1 = 10, (3) Kp1 = 5.

tem. In this situation, the appropriate solution would be to change the type of process from batch to semibatch since the temperature control system is not adequate to guarantee that we are going to work on safe conditions. 5.2. PI control In this part we present the analysis of a PI-controlled BR with a nth-order exothermic reaction, following the control strategy developed by Pellegrini and Biardi

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(1990) and analyzed using the divergence criterion by Ratto and Paladino (2001) for a CSTR. In this case the control action can be formulated as QR ⫽ Kp(q⫺qw)⫺Kij

(15)

where

冕

t

j⫽

(q⫺qw) dt

(16)

0

Therefore, the differential equations for the jacket in this system can be written, in dimensionless form, as dqw ⫽ δSt(q⫺qw) ⫹ QR dt

(17)

dj ⫽ q⫺qw dt

(18)

As before, control saturation has to be considered, mainly, during transients. In this case QR may change and Qmin between a certain range, i.e. Qmax R R . Applying PI control (Fig. 12(c)) to the isoperibolic system (Fig. 12(a)), the divergence never becomes positive in any interval of time and the maximum dimensionless temperature reached in the reactor falls from approximately 9–0.2, i.e. near to isothermal conditions. The boundary values of the dimensionless heat of reaction number, B, between runaway and safe conditions, have been calculated for the PI-controlled BR with first order kinetics. The boundary lines are represented in Fig. 15. As it can be seen, similar behaviour as in the PP-controlled BR can be observed, i.e. the same relationships between the control parameters, the Damko¨ hler number and the cooling power capacity of the system.

199

6. Conclusions In this study it has been shown that the divergence criterion, div ⬎ 0, can be applied for developing boundary diagrams able to distinguish between runaway and non-runaway situations for the cases of multiple reactions and for several types of reactors, i.e. BR, SBR and CSTR, with or without the presence of a control system. The criterion is more conservative than the maximum sensitivity criterion developed by Morbidelli and Varma (1988), but less conservative than other previously developed criteria as was shown by Strozzi et al. (1999) (e.g. Barkelew, 1959; Van Welsenaere & Froment, 1970; etc.), and it works for reaction schemes and operating procedures where several other criteria fail. Apart from that, practically all the same general trends and conclusions as in the study by Varma et al. (1999) are observed for multiple reactions and for BRs and CSTRs. Concerning SBRs, the criterion is able to distinguish between non-ignition and runaway and, to a lesser extent, between runaway and QFS regions. As far as we know, this is the only existing criterion able to delimit the runaway zone. Normally, one needs two separate criteria (Alo´ s et al., 1998; Steensma & Westerterp, 1990). Furthermore, the criterion can be extended, in a coherent mode, to assess the influences on the runaway boundaries when there is a control system operating in the plant aiming at working in isothermal conditions. An advantage of this criterion over existing criteria is that it is possible to reconstruct, using non-linear time series analysis techniques, the divergence of the system from temperature measurements without the necessity to have a model of the process, and hence, all the results and conclusions obtained from an off-line analysis may be extended and applied on-line to develop a general early warning detection device, based on a robust criterion. Our research is currently moving along these lines.

Acknowledgements This research has been supported by the EU funded project AWARD (Advanced Warning and Runaway Disposal, contract G1RD-CT-2001-00499) in the GROWTH programme of the European Commission.

References

Fig. 15. Runaway boundaries for divergence criteria for a PI-controller BR for different values of Kp. Conditions of simulation are: Tref = 298 K, Tw = 298 K, g = 10, St = 1, Ki = 5 and (1) Kp = 20, (2) Kp = 15, (3) Kp = 10, (4) Kp = 4, and isoperibolic (discontinuous line).

Abarbanel, H. D. I. (1996). Analysis of observed chaotic data. New York: Springer. Adler, J., & Enig, J. W. (1964). The critical conditions in thermal explosion theory with reactant consumption. Combustion and Flame, 8, 97–103. Alo´ s, M. A., Nomen, R., Sempere, J., Strozzi, F., & Zaldı´var, J. M. (1998). Generalised criteria for boundary safe conditions in semi-

200

J.M. Zaldı´var et al. / Journal of Loss Prevention in the Process Industries 16 (2003) 187–200

batch processes: simulated and experimental results. Chemical Engineering and Processing, 37(5), 405–421. Alo´ s, M. A., Strozzi, F., & Zaldı´var, J. M. (1996). A new method for assessing the thermal stability of semibatch processes based on Lyapunov exponents. Chemical Engineering Science, 51(11), 3089–3094. Arnold, V. I. (1973). Ordinary differential equations. Cambridge: Mitt Press. Balakotaiah, V. (1989). Simple runaway criteria for cooled reactors. AICHE Journal, 35(6), 1039–1043. Balakotaiah, V., Kodra, D., & Nguyen, D. (1995). Runaway limits for homogeneous and catalytic reactors. Chemical Engineering Science, 50(7), 1149–1171. Barkelew, C. H. (1959). Stability of chemical reactors. Chemical Engineering Progress Symposium Series, 25(55), 37–46. Hagan, P. S., Herskowitz, M., & Pirkle, C. (1987). Equilibrium temperature profiles in highly sensitive tubular reactors. SIAM Journal on Applied Mathematics, 47(6), 1287–1305. Hagan, P. S., Herskowitz, M., & Pirkle, C. (1988). Runaway in highly sensitive tubular reactors. SIAM Journal on Applied Mathematics, 48(6), 1437–1450. Kantz, H., & Schreiber, T. (1997). Nonlinear time series analysis. Cambridge: Cambridge University Press. Morbidelli, M., & Varma, A. (1988). A generalized criterion for parametric sensitivity. Chemical Engineering Science, 43(1), 91–102. Morbidelli, M., & Varma, A. (1986). Parametric sensitivity in fixedbed reactors: the role of interparticle transfer resistances. AICHE Journal, 32(2), 297–306. Morbidelli, M., & Varma, A. (1989). A generalized criterion for parametric sensitivity: application to a pseudohomogeneous tubular reactor with consecutive or parallel reactions. Chemical Engineering Science, 44(8), 1675–1696. Obertopp, T., Alo´ s, M. A., Zaldı´var, J. M., Strozzi, F., & Gilles, E. D. (1998a). Model-based on-line monitoring of chemical processes using a new intrinsic runaway criterion related to the divergence of the system. In Barcelona, 4–7 May 1998. Proceedings of the ninth international symposium on loss prevention and safety promotion in the process industries (pp. 641–651). Obertopp, T., Alo´ s, M. A., Zaldı´var, J. M., Strozzi, F., & Gilles, E. D. (1998b). New intrinsic runaway criteria and their application to the model-based on-line monitoring of continuous and discontinuous reactors. In P. S. Dhurjati, M. Kramer, & S. Cauvin (Eds.), IFP, Solaize (Lyon) France, 4–5 June 1998. Proceedings of the third IFAC workshop on-line fault detection and supervision in the chemical process industries. Oxford: Elsevier. Pellegrini, L., & Biardi, G. (1990). Chaotic behaviour of a controlled CSTR. Computers and Chemical Engineering, 14(11), 1237–1247. Rajadhyaksha, R. A., Vasudeva, K., & Doraiswamy, L. K. (1975).

Parametric sensitivity in fixed-bed reactors. Chemical Engineering Science, 30(11), 1399–1408. Ratto, M., & Paladino, O. (2001). Controllability of start-ups in CSTRs under PI control. Chemical Engineering Science, 56(4), 1477– 1484. Steensma, M., & Westerterp, K. R. (1990). Thermally safe operation of a semibatch reactor for liquid–liquid reactions. Slow reactions. Industrial and Engineering Chemistry Research, 29(7), 1259–1270. Steensma, M., & Westerterp, K. R. (1991). Thermally safe operation of a semibatch reactor for liquid–liquid reactions. Fast reactions. Chemical Engineering and Technology, 14(6), 367–375. Strozzi, F., & Zaldı´var, J. M. (1994). A method for assessing thermal stability of batch reactors by sensitivity calculation based on Lyapunov exponents. Chemical Engineering Science, 49(16), 2681– 2688. Strozzi, F., Zaldı´var, J. M., Kronberg, A., & Westerterp, K. R. (1999). On-line runaway prevention in chemical reactors using chaos theory techniques. AICHE Journal, 45(11), 2429–2443. Thomas, P. H. (1961). Effect of reactant consumption on the induction period and critical conditions for a thermal explosion. Proceedings of Royal Society of London Series A, 262(1309), 192–206. Vajda, S., & Rabitz, H. (1992). Parametric sensitivity and self-similarity in thermal explosion theory. Chemical Engineering Science, 47(5), 1063–1078. Van Welsenaere, R. J., & Froment, G. F. (1970). Parametric sensitivity and runaway in fixed bed catalytic reactors. Chemical Engineering Science, 25(10), 1503–1516. Varma, A., Morbidelli, M., & Wu, H. (1999). Parametric sensitivity in chemical systems. Cambridge: Cambridge University Press. Westerterp, K. R., & Overtoom, R. R. M. (1985). Safe design of cooled tubular reactors for exothermic, multiple reactions; consecutive reactions. Chemical Engineering Science, 40(1), 155–165. Westerterp, K. R., & Ptasinski, K. (1984). Safe design of cooled tubular reactors for exothermic, multiple reactions; parallel reactions— I. Development of criteria. Chemical Engineering Science, 39(2), 235–244. Westerterp, K. R., & Westerink, E. J. (1990). Safe design and operation of tanks reactors for multiple-reaction networks: uniqueness and multiplicity. Chemical Engineering Science, 45(1), 307–316. Wilson, K. B. (1946). Calculation and analysis of longitudinal temperature gradients in tubular reactors. Transactions of the Institution of Chemical Engineers (London), 24, 77–83. Wu, H., Morbidelli, M., & Varma, A. (1998). An approximate criterion for reactor thermal runaway. Chemical Engineering Science, 53(18), 3341–3344. Zaldı´var, J. M., Herna´ ndez, H., Nieman, H., Molga, E., & Bassani, C. (1993). The FIRES project: experimental study of thermal runaway due to agitation problems during toluene nitration. Journal of Loss Prevention in the Process Industries, 6(5), 319–326.