Cabeq 2011-01 verzija 3 - FKITsilverstripe.fkit.hr/cabeq/assets/Uploads/Cabeq-2011-01-02.pdf · The present study is focused on analysing com-bined methods for detecting the critical

Evaluation of Critical Operating Conditions for a Semi-batch Reactorby Complementary Use of Sensitivity and Divergence Criteria

G. Maria* and D.-N. Stefan

Department of Chemical Engineering,University Politehnica of Bucharest, Romania

This paper presents a comparison of several effective methods of deriving the criti-cal feeding conditions for the case of a semi-batch catalytic reactor used for theacetoacetylation of pyrrole with diketene in homogeneous liquid phase. The reaction isknown to be of high risk due to the very exothermic (polymerisation) side-reactions in-volving reactive diketene. In order to perform the sensitivity analysis, both theMorbidelli-Varma sensitivity criterion and div-methods were used, the latter of which arebased on the system’s Jacobian and Green’s function matrix analysis. The combined ap-plication of such criteria allows the optimal and safe setting of the reactor’s nominal op-erating conditions. Extended sensitivity analysis reveals possible QFS (quick onset, fairconversion and smooth temperature profile) regions, under severe operating conditionscharacterized by fast enough main reaction that prevents the co-reactant accumulation,and leads to a quasi-insensitive semi-batch reactor behaviour.

Key words:

Critical conditions, semi-batch reactor, sensitivity, divergence, pyrrole, diketene

Introduction

Setting the chemical reactor’s technologicalconstraints and runaway boundaries of the operat-ing conditions are important for both risk assess-ment and over-design avoidance. Whenever the re-actor’s economic performance is the most importantfactor, the frequent solution entails a risky opera-tion very close to the safety limits, where the run-away can occur due to either a malfunction in thecooling system or highly exothermic side reactions(even when implementing an advanced control sys-tem).

Knowledge of the safety limits and their confi-dence region is important not only for optimallysetting the nominal operating conditions of the re-actor, but also for implementing a real-time on-linealgorithmic sensor to detect system instability in theproximity of critical conditions (early warning de-tection), thus preventing the process runaway. Theprocess analysis must also consider the variabilityof the process conditions and sometimes the uncer-tainty in model parameters in order to increase theconfidence of the predicted critical conditions.

The present study is focused on analysing com-bined methods for detecting the critical conditionsin a semi-batch reactor (SBR). Such reactors aremore and more preferred due to the possibility ofcontrolling an exothermic chemical reaction and re-

actor temperature by the policy of adding one of thereactants to the other components, which have al-ready been fully loaded to the reactor. As a conse-quence, the probability of co-reactant’s accumula-tion is drastically reduced; moreover, in case of ab-normal behaviour of the reactor, simply reducingthe dosing rate can stop the process. On the otherhand, it is the current trend to move production of alot of chemicals from the stable continuous plantsto multi-product (semi-)batch reactors, as they aremore flexible and easily adaptable to market re-quirements. A certain optimal feeding policy canensure production maximization, even if frequentperturbations in the operating parameters, raw-ma-terials and recycling conditions, catalyst character-istics and presence of impurities from previousbatches, all require periodical model, feeding pol-icy, and safety limits updates.

One simple but approximate way to determinethe operation safety limits is to use explicit meth-ods, i.e. simple relationships derived from experi-mental observations on the reactor’s thermal sensi-tivity, or from simplifying more complicated mo-del-based criteria. Simple engineering numbers(such as Damköhler-Da, Stanton-St, or Lewis), orsafety indices may give an approximate idea on therunaway limits and many times replace the system-atic model-based safety analysis of the process.1–6

Even if quickly applicable, such calculations arenot sufficiently accurate for an advanced optimiza-tion of the process or for implementing an on-lineinstability detector.

G. MARIA and D.-N. STEFAN, Evaluation of Critical Operating Conditions for a …, Chem. Biochem. Eng. Q. 25 (1) 9–25 (2011) 9

*Corresponding author; Mail address: P.O. 35–107 Bucharest, Romania;

Email: [email protected]

Original scientific paperReceived: April 26, 2010

Accepted: September 26, 2010

Model based evaluations of critical operatingconditions, even if more laborious and requiring asteady effort to up-date the model parameters toprocess changes, offer a quite accurate prediction ofthe safety region and are generally applicable to ev-ery reactor type. According to Adrover et al.,7 suchcriteria used in characterization and diagnosis of re-actor runaway and explosion can be classified intofour categories: geometry-based criteria, parametricsensitivity-based criteria, divergence-based criteria,and stretching based criteria.

Geometry-based methods (GM) interpret theshape of the temperature or heat-release rate profileover the reaction (contact) time. Critical conditionscorrespond to an accelerated temperature increase,i.e. to an inflexion point before the curve maximumin a temperature – time plot T t( ).

Sensitivity-based methods (PSA) detect unsafeconditions as those characterized by high paramet-ric sensitivities of state variables xi with respect tooperating parameters �j, i.e. s x xi j i j( ; )� � ��(in absolute terms), that is, where “the reactor per-formance becomes unreliable and changes sharplywith small variations in parameters”.2 Local sensi-tivity analysis (developed for every state variableand parameter), or global sensitivity analysis (ex-tended over the whole reactor and operation time,by accounting for concomitant variations of severalinput/process parameters) eventually lead to globalrunaway conditions of the reactor.4–6

Divergence (div-)based criteria identify any in-stability along the system/process evolution and de-tect any incipient divergence from a reference(nominal condition) state-variable trajectory overthe reaction time x ti ( ). Any increased sensitivity ofthe system stability in the proximity of runawayboundaries in the parametric space is detected fromanalysing the eigenvalues of the process modelJacobian (J) and Green’s (G) function matrices,evaluated over the reaction time.8,9 More elaboratedversions use more sophisticated div-indices to char-acterize the expansion of volume elements inphase-space (having state variables as coordinates),e.g. Lyapunov exponents based on the analysis ofthe time-dependent J J

T matrix.10,11

Stretching based analysis (SBA), recently in-troduced by Adrover et al.,7 combines sensitivityand div-methods by investigating the dynamics ofthe tangent components to the state-variable trajec-tory. Critical conditions are associated with the sys-tem’s dynamics acceleration (‘stretching rates’ ofthe tangent vectors), corresponding to a sharp peakof the normalized stretching rate due to the acceler-ate divergence from the nominal trajectory.

Each critical condition method presents advan-tages and limitations related to precision, real-timeapplication, involved computational effort, and lo-cal applicability. Several classification criteria canbe defined to assess the generality of each method,7

such as: objectivity (validity independent on partic-ular choice of the system or operating conditions ofa checked process); generality (general validity, in-dependent on the process or reactor type); real-timeapplicability (as on-line algorithmic sensor to detectinstabilities and runaway conditions); locality (theuse of local quantities or model linearizations).According to such a classification, all mentionedmethod types are objective, all methods are gener-ally applicable excepting GM, all methods can beon-line used for real-time runaway detection ex-cepting PSA, and only PSA and SBA are globalmethods following the current x( )t vector evolu-tion.

The combined use of several methods is rec-ommended to increase the prediction precision butalso the applicability area. Because geometricalmethods tend to overestimate the stability region, orsometimes they are too conservative thus producingunnecessary warnings,2,8,11 PSA methods are pre-ferred, by offering precise predictions of runawayboundaries due to the possibility of choosing be-tween process parameters when calculating the sen-sitivity functions. However, PSA precision limita-tions in low sensitive operating regions,5,8,12 and im-possibility of an on-line use require supplementarychecks by means of div-methods. Even if precisionproblems are inherent due to model linearization inthe proximity of high sensitive operating regions,the possibility to on-line diagnose the process foran early stage runaway compensate the div-methoddrawbacks.10,31

However, SBR may pose serious operatingproblems when highly exothermic reactions areconducted, especially when the primary reaction isslow vs. addition time, while secondary reactionsquickly occur at any incidental increase of reactiontemperature. Runaway occurs when the rate of heatgeneration becomes faster than the rate of heat re-moval by the design cooling system. In the homo-geneous reaction case, the accumulation of theco-reactant at low temperatures, leads to an expo-nential rise in the reaction rates for any temperatureincrease, which in turn will generate large reactionheat fluxes overstepping the heat removal possibili-ties of the cooling system. The situation is wors-ened when secondary exothermic chain / polymeri-sation reactions are inducted by high temperatures,leading to a quick increase in temperature or pres-sure with eventually the same effect.1–3 Conditionsof runaway are also derived for heterogeneous liq-uid-liquid SBR.13 Such safety considerations when

10 G. MARIA and D.-N. STEFAN, Evaluation of Critical Operating Conditions for a …, Chem. Biochem. Eng. Q. 25 (1) 9–25 (2011)

conducting highly exothermic reactions encouragethe reactor overdesign (with additional costs), andoperation with a conservative, under-optimal oper-ating strategy to prevent accumulation of the co-re-actant in the reactor.14–16 Therefore, separate super-vision of each SBR, periodical / on-line updatesof the safety margins for the operating variablesbased on a process model, and a combined applica-tion of diagnosis / runaway criteria become neces-sary.

The sensitivity based analysis can be com-pleted with detection of the so-called “Quick onset,Fair conversion and Smooth temperature profile”(QFS) super-critical operating regions, character-ized by a high level but quite ‘flat’ temperatu-re-over-time-profile, and small sensitivities of statevariable vs. operating conditions. Such regions canbe of economic interest and should be accountedfor determining the optimal operating policy of theindustrial SBR. Classification of the operating re-gions in non-ignition, marginal ignition, runaway,and QFS is not easy, and depends on the accumula-tion of the co-reactant, temperature regime, and re-action characteristics.32,33 Steensma and Wester-terp32 indicated some criteria to characterise the op-erating region, and pointed-out that “the obtainingof a smooth, stable, and realistic target temperatureis more important than a sharp limitation of the ac-cumulated mass, or a limitation of the maximumconversion rate”.

The present paper aims at evaluating, com-paring, and investigating coupling possibilities ofsome sensitivity and div-criteria to offer precisepredictions of critical conditions and their confi-dence intervals for a SBR together with the possi-bility of an early detection of the runaway proxim-ity. Such a combination allows a more precise SBRoptimization and implementation of an on-linealgorithmic sensor to detect any system instability.The examined SBR is the bench-scale jacketedreactor for the acetoacetylation of pyrrole withdiketene to PAA (2-acetoacetyl pyrrole) in homo-geneous liquid phase, used by Ruppen et al.17

to identify the optimal isothermal feeding policiesthat maintain the reactor within technological limitsand lead to an acceptable PAA product yield.Recently, Maria et al.6 approached this high-riskSBR case study and have used a non-isothermalreactor model to derive the critical conditions bymeans of the generalized sensitivity criterion ofMorbidelli-Varma (MV).2 The present study ex-tends the analysis by accounting for several run-away criteria completed with identification ofpossible QFS operating regions of economic inter-est.

Sensitivity and divergence criteriaof critical operating conditions

The current study is focused on evaluating thecritical conditions for a semi-batch reactor by usingthe MV sensitivity criterion, and three div-criteria(denoted with div-J, div-SZ, and div-LY, see below).These runaway criteria are chosen to combine theevaluation precision/robustness with the sensitivityin detecting any system small instability of chemi-cal process referring to a nominal evolution.

The generalized sensitivity criterion MV. Thiscriterion associates the critical operating conditionswith the maximum of sensitivity of the hot spot( )maxT T� 0 in the reactor, evaluated over the reac-tion time, in respect to a certain operating parame-ter �j. In other words, critical value of a parameter�j,c corresponds to:

MV criterion:

� ��j c jjs T, maxarg max ( ; ) ,�

�

�

��

��

or � ��j c jjS T, maxarg max ( ; ) ,�

�

�

��

��

(1)

S T Tj j( ; ) ( )max*

max*� ��

� �s T T Tj j j( ; ) ( )( ),max*

max*

max� � � ��

[where: S T j t( )max � – relative sensitivity functionof Tmax vs. parameter �j; ‘*’ – nominal operatingconditions (set point) in the parameter space; t – re-action time].

According to the MV criterion, critical condi-tions induce a sharp peak of the normalized sensi-tivity S T j( ; )max � evaluated over the reaction timeand over a wide range of �j. The robustness andeffectiveness of the MV-criteria derives from thegeneral validity, irrespectively to the complex reac-tion pathway, reactor configuration, or the consid-ered operating parameter �j (e.g. T0, cj,0, Qinlet, cj,inlet,Ta, B, Da, St, …, or a combination of them, see thenotation list for symbol definitions). The sensitivityfunctions s(xi; �j) of the state variables xi (includingthe reactor temperature) can be evaluated by usingthe so-called ‘sensitivity equation’ solved simulta-neously with the reactor model:2

d

d

s

ts

j

jj

( ; )( ; ) ;

x g

xx

g� �

��

�

��

s x xjt

j( ; ) ( ),� � ��

00 (2)

d dx g x x xt t t� ��( , , ), ,f 0 0


(where the Kronecker delta function � �( )j x� 0

takes the value 0 for � j x� 0 , or the value 1 for� j x� 0). Evaluation of derivatives in (2) can beprecisely performed by using the analytical deriva-tion or, being less laborious, by means of numericalderivation. A worthy alternative, also used in thepresent study, is the application of the numerical fi-nite difference method, which implements a certaindifferentiation scheme (of various precision and com-plexity) to estimate the derivatives of s x i j t( ; )� atvarious reaction times.18 However, the method be-comes computationally costly when approachingthe runaway boundaries because precise evaluationof the state’s high sensitivities requires smalldiscretization steps in the parameter space.6 Also,the runaway boundary predictions become approxi-mate for very severe operating conditions wheresuper-criticality can induce a quasi-insensitivity ofthe temperature maximum to operating condi-tions.12

Div-J criterion. The divergence criteria are de-rived from the dynamical systems theory, fromcharacterization of chaotic attractors in dynamicalsystems.7 In the div-J variant of Hegdes andRabitz,19 one considers a reference solution x t* ( ),

usually known as set point or nominal conditions,and one investigates the effect of any perturbationin the initial conditions x i,0 or parameter � i byinspecting the eigenvalues of the system Jacobian� i ( ).J These eigenvalues prescribe how pertur-bations behave for small time intervals near everyconsidered moment. When the real part of only oneeigenvalue becomes positive at a certain time,this perturbation induces system instability anddivergence of the state-variable time-trajectoryfrom the reference solution. In the risk assessment,such instability is associated with the occurrenceof critical conditions determining process run-away. Zaldivar et al.9 introduced an early detectionof loss of stability as those corresponding to occur-rence of the Jacobian trace positiveness, i.e. for

Trace(J) � �� ii0. Starting from the Vajda and

Rabitz8 observation that critical conditions corre-spond to the positive extreme point of Re( ( ))� Jevaluated at the temperature peak, in the presentwork the critical value of a checked parameter � j c,

is estimated based on:

div-J criterion:

� �j c j, min ( ),�

for which max (max (Re( ( ( ))))) ,i t i j� �J � 0

where:

J g x� ( ) ;� � t d dx g xt t� ( , , ),f x xt� �0 0 , (3)

(when more severe conditions correspond tosmaller � j , then max ( )� j must be taken in the cri-terion, e.g. for � j � Da, or � j � Da/St). The eva-luation rule starts by computing Re( ( ))� i J valuesat various reaction times for a certain small� j value, and keeping the other parameters at no-minal values. If the condition (3) is not satisfied,the parameter � j is increased with a small incre-ment (to ensure a reasonable evaluation precision)and the procedure is repeated until the critical valueis identified. Evaluation of the J-matrix elements ismade analytically with high precision, for instanceby applying commercial software for symbolic cal-culation, such as Maple package in the presentwork.20

Div-SZ criterion. A convenient div-criterion al-ternative detects the possible loss of system stabil-ity by investigating the expansion of volume ele-ments V(t) in phase-space having state variables ascoordinates (the so-called Strozzi-Zaldivar SZ-cri-terion).10,11 The ellipsoidal volume having the pointx(t) in the centre is calculated at every moment t, byconsidering the volume expansion from an initialstate x0 due to variations �x of the state-vector com-ponents. Such volume elements’ expansion / con-traction corresponds to the solution x(t) divergence /convergence toward the reference trajectory. Check-ing evolution of ellipsoidal volume semi-axes,

through the so-called Lyapunov exponents~

,� i

better monitors the system evolution after a para-metric perturbation:

~( ) log

| ( )|

| ( )|log ( ) ,� �i

i

ii

Ttt

q t

q� �

1

02 2 J J

i� �1, , ( ).size x(4)

(where: q – orthogonalized vectors of x; size (x) –dimension of the vector x). If the volume semi-axisexpands, the corresponding Lyapunov exponentwill be positive. Consequently, for a higher dimen-sion of state vector, the critical conditions corre-spond to the extreme of the sensitivity function ofthe sum of Lyapunov exponents:

� ��j c t jjs V t, arg (max (max ( ( ( ); ))));�

V ti

it

( ) .

~( )

��

2�

(5)

An equivalent form of the Lyapunov exponentcriterion was proposed in this paper, by associatingcritical condition � j c, to a sharp change of slope inthe plot of positive maximum of Lyapunov expo-nents vs. � j , that is:


div-SZ criterion:

� �j c j, min ( ),�

for which max ( ; ) ,maxS L j j� ��

where:

L i t iT

max max max ( ) ,��

�

��

��

�

�

��

�� J J

S L Lj j( ; ) ln ( ) ln ( ) ,max max� � � �� (6)

J g x( ) ( ) ;� � �j t� d dx g xt t� ( , , ),f x xt� �0 0 .

The relative tolerance � j of sensitivity of theSZ-criterion with respect to the j-th parameter canbe chosen in relation with the known standard devi-ation (in relative terms) of the parameter value dueto (assumed) normal fluctuations,21 but other a-pri-ori values can also be adopted.

Div-LY criterion. When the number of reac-tions is much larger than that of the considered spe-cies in the process model, an alternative runawaydiagnostic method is the div-LY criteria based on theanalysis of the eigenvalues of the Green’s functionmatrix G.2 The Green’s functions G x t xij i i� � �( ) 0

are in fact the sensitivities of the state variables x tothe initial conditions, being evaluated from themodel Jacobian, by using the differential equationintegrated simultaneously with the reactor model:19

d dG J x G/ ( ( )) ,t t� G ( ) ,0 � 1 J g x� � �/ ,

d dx g x/ ( , , ),t t� f x x( ) .0 0�(7)

At critical conditions, at least one eigenvalueof G [denoted as � j ( )G ] diverges from zero or,equivalently, the asymptotic Lyapunov stabilitynumbers � i t( ) diverge. In the present study, anequivalent formulation is proposed, by associatingthe critical condition � j c, to a sharp change ofslope in the plot of positive maximum of Lyapunovnumbers vs. � j , which is equivalent to:

div-LY criterion:

� �j c j, min ( ),�

for which arg (max ( ( ; ))) ,max� � �j

S LY j j�

where:

LY ti t imax max (max ( ( ))),� �

S LY LYj j( ; ) ln ( )/ ln ( ),max max� � � �� (8)

� �j jt t t( ) ln | ( )|/( ).� �G 0

Finally, it is worth mentioning that regardlessof the method used, the safety analysis of the SBRcontinues to be of high interest due to the frequentproblems associated with the variability in operatingconditions inducing non-linear behaviours of com-plex processes and the requirement to up-date/opti-mise the reactor operation when the raw-mate-rial/catalyst properties change. On the other hand,the chosen nominal operating point tries to limit thehot spot during the batch, and thus avoid an exces-sive sensitivity to variations in the process parame-ters.

This is why other safety criteria continue to bedeveloped by combining the sensitivity and loss oflocal stability criteria, by analysing the characteris-tics of the system Jacobian, Green’s function ma-trix, Lyapunov numbers, tangent components to thestate-variable trajectory, or other combinations ofthem during the process evolution.7,22–24

QFS region detection. Another interesting as-pect to be investigated is the detection of theso-called “Quick onset, Fair conversion andSmooth temperature profile” (QFS) operating re-gion in the parametric space. In QFS regions, usu-ally located in the super-critical / high severity op-erating area, the SBR temperature displays a quite‘flat’ time-profile, even if at a higher level. TheQFS region actually belongs to the ignition behav-iour, and is explained by the fact that, under highseverity conditions the reactions occur quasi-instan-taneously, and accumulation of the co-reactant is nolonger possible. Thus, the temperature profile be-comes quasi-flat, SBR displays a low thermal sensi-tivity, and its operation in such a region makessense if other technological constraints are not ac-tive. Moreover, investigation of the QFS location atsevere conditions becomes important when estab-lishing the optimal operating “set-point” of the re-actor from both economic and safety perspectives.

For a single reaction case, Alos et al.12,30 pro-posed to determine the boundary of the QFS regionbased on evaluation of the time to reach the maxi-mum temperature in the reactor (denoted here witht max). Starting from the observation that t max � �curve passes through a minimum in a region whereit is quite insensitive to � (irrespectively to the �parameter choice), the condition for starting theQFS region becomes:12,24

� �QFS s t� �arg ( ( ; ) ),max 0

(i.e. t max ( )� presents a minimum).(9)

By repeating this rule, and every time deter-mining � QFS for different values of a certain operat-ing parameter (e.g. �), the QFS boundary curve


� �QFS ( ) results in the parametric plane � �� . Fora single reaction case, the limit of the QFS regionusually corresponds to SBR running conditions thatare even more severe than the runaway limit. Even-tually, the quasi-insensitive QFS region has to beconfirmed by the quasi-flatness of the temperatureevolution over the batch time, and its maximum oc-currence later than the minimal t max value. The va-lidity of the QFS-criterion was experimentally con-firmed for simple reaction cases.12 Since theMV-sensitivity criterion cannot distinguish betweenthe runaway and the QFS regions because in bothcases the reaction is ignited, the use of the QFS-cri-terion in locating alternative operating region ofeconomic interest is well supported. This rule issimilar12 to the so-called “target temperature crite-rion” of Steensma and Westerterp32,33 which usestwo dimensionless numbers (reactivity and exother-micity) together with a definition of the target tem-perature of the SBR to frame each running pointin the non-ignition, marginal ignition, runaway, orQFS region.

Acetoacetylation of pyrrole –process characteristics

The acetoacetylation of pyrrole (P) withdiketene (D) is conducted in homogeneous liquidphase (toluene), at around 50 °C and normal pres-sure, using pyridine as catalyst, for producingpyrrole derivates such as PAA used in the drug in-dustry. The process is of high thermal risk due tothe tendency of the very reactive diketene to poly-merise at temperatures higher than 60–70 °C, or inthe presence of impurities that can initiate highlyexothermic side reactions difficult to be con-trolled.25 The complex process kinetics have beeninvestigated by Ruppen et al.17 in a bench-scale iso-thermal SBR operated at 50 °C using a high excessof toluene as solvent.

The proposed kinetic model from Table 1 ac-counts for only four exothermic reactions: (a) thesynthesis of PAA (P + D � PAA) is accompaniedby several side-reactions of diketene, leading to itsdimmer (b)(2D � DHA; DHA = dehydroaceticacid), and oligomers (c)(nD � Dn), or to a by-prod-uct denoted by G (d)(PAA + D � G; the intermedi-ate reaction of diketene with DHA has been ne-glected from the model). Because the co-reactantdiketene presents extreme reactivity and hazardousproperties, the temperature regime must be strictlycontrolled and the diketene and DHA concentra-tions in the reactor kept lower than certain criticalthresholds (empirically determined, see Table1).17,26 The rate constants have been evaluated byRuppen et al.17 at 50 °C and [PAA] > 0.1 mol L–1.

Maria et al.6 completed the model by includ-ing the Arrhenius dependence of the main rateconstants, and by adopting an activation energy ofE R/ � 10242 K for all reactions of diketene, byanalogy with the diketene derivate polymerisation,and with the initiation energy of olefin polymeriza-tion. The resulting Arrhenius constants (Ai, Ei) aredisplayed in Table 1. All reactions are moderatelyexothermic, except for the diketene oligomerizationof standard heat around –1423 kJ mol–1.

Based on these thermodynamic-kinetic data,Maria et al.6 performed a quick assessment of thereaction hazard, which indicates reactions (a) and(c) as being dangerous even under the nominal op-erating conditions displayed in Table 1, presenting:�Tad � 62 K (reaction a) and �Tad � 83 K (reac-tion c), i.e. larger than the threshold 50 K; B = 6(reaction a) and B = 8.1 (reaction c), i.e. larger thanthe threshold 5; Da = 1.7 (reaction b) and Da = 4.2(reaction c), i.e. smaller than the threshold 50–100(slow reactions).1 Calculations have been madeusing the common relationships:1

� �T H c cad j p� �( ) /( ),,0 � B T E RTad� � /( ),02

Da v r cj j D j� �( ) ( )/ ,� 0

(10)

(where: �Tad – adiabatic temperature rise; B – reac-tion violence index; Da – Damköhler number forthe key reactant j; cj,0 – initial concentration of keyspecies; � – reacting mixture density; c p – averagespecific heat; T0 – initial/cooling agent temperatureof the reaction; R – universal gas constant; ( )�v j –stoichiometric coefficient of reactant j; �D – co-re-actant adding time).

For a quick process simulation, a simple SBRmodel was adopted, corresponding to a perfectlymixed vessel, with no mass and heat transferresistances in the liquid.27 The solution of diketenein toluene is continuously added with a variable fedflow-rate Q t( ) over the continuously stirred pyrrolesolution (including impurities) initially loaded tothe jacketed reactor, and the reaction heat is contin-uously removed through the reactor wall. The massand heat balance equations, presented in Table 1,explicitly account for the liquid volume and heattransfer area increase during the batch. The contin-uous catalyst dilution is accounted for when cor-recting the reaction rates, except for reaction (c)presumed to be promoted not by pyridine but bysome impurities (of quasi-constant concentration).To speed-up the computational steps, the physicalproperties of the reaction mixture have been ap-proximated to those of the toluene solvent, and aconstant overall heat transfer coefficient has beenevaluated (Table 1).



T a b l e 1– Process and semi-batch reactor model, nominal operating conditions, and technological constraints.

(Notations: D = diketene; P = pyrrole; PAA = 2-acetoacetyl pyrrole; DHA = dehydroacetic acid; Py = pyridine).6

Process main reactions and reduced kinetic model (Footnote a):

P D PAAPy

� ��ka ,

; r k c ca a P D� ; k Ta � � �3 1324 10 10242 412. exp ( . / ), L mol–1 min–1;

�H a ��132 69. kJ mol–1; �Tad � 61 8. K

D D DHAPy

� ��kb ,

; r k cb b D� 2 ; k Tb � � �7 5651 10 10242 412. exp ( . / ), L mol–1 min–1;

�H b ��91 92. kJ mol–1 ; �Tad � 5 3. K

n Dk

ncD � �� ( ) ; r k cc c D� ; k Tc � � �1 6549 10 10242 412. exp ( . / ), min–1;

�H c ��1426 12. kJ mol–1 ; �Tad � 83 0. K

PAA D GPy

� ��kd ; r k c cd d D� PAA ; k Td � � �1 7731 10 10242 412. exp ( . / ), L mol–1 min–1;

�H d ��132 69. kJ mol–1; �Tad � 7 7. K

Differential balance equations:

– species mass balance:d

d

c

tr r r r c c

Q t

V t

Da b c d D in D� � � � � �( ~ ~ ~ ~ ) ( )

( )

( );,2

d

d

c

tr c

Q t

V t

Pa P�� ~ ( )

( );

d

d

PAA

PAA

c

tr r c

Q t

V ta d� � �(~ ~ )

( )

( );

d

d

DHA

DHA

c

tr c

Q t

V tb� �~ ( )

( ); volume variation:

d

d

V

tQ t� ( );

– reaction rate correction with the catalyst dilution: ~

( );r r

V

V tj j� 0 j � a,b,d

– heat balance:d

d

T

t

Q t T T

V t

H r V t U A T T

c V

inj j r aj

p

��

� � ��( ) ( )

( )

( ) ~ ( ) ( )�

� ( ),

tj � a,b,c,d

Observations: i) at t c c T Tj j� � �0 0 0, , ;, ii) rC is not corrected with the catalyst dillution, the reaction displaying anothermechanism (the stoichiometric coefficient was included in the rate constant)

Model hypotheses:

– semi-batch reactor model with perfect mixing and uniform concentration and temperature field

– cylindrical reactor of variable liquid volume and heat transfer area: Ad

dV tr

r

r

� � 2

4

4( )

– overall heat transfer coefficient evaluated with criterial formula27 (approximate value for nominal conditions is U � 581 W m–2 K–1)

– heat of solvent vaporisation in the reactor is neglected

– specific heat capacity and density of fed solution are the same with those of reactor content, � �in p in pc c, �

Nominal operating conditions and range of variation

initial liquid volume (V0, L):

reactor inner diameter (dr, m):

stirrer speed (r min–1):

liquid physical properties:

inlet [D] (mol L–1):

initial [P] (mol L–1):

initial [D] (mol L–1):

1

0.1

640

toluene solvent

4 � 5.82 � 6

0.4 � 0.72 � 0.8

0.005 � 0.09 � 0.14

initial [PAA] (mol L–1):

initial [DHA] (mol L–1):

fed D solution flow rate (Q · 100, L min–1):

batch time (tf, min):

initial temperature (�0, °C):

cooling agent temperature (�a, °C):

feeding solution (�in, °C):

0.08 � 0.10 � 0.20

0.01 � 0.02 � 0.04

0.05 � 0.15 � 0.20

120 � 145 � 378

40 � 50 � 60

50

50

Process constraint expression: Significance

c fDHA , . ,� �0 15 0 (mol L–1)

c fD , . ,� �0 025 0 (mol L–1)

max ( ( )) ,� t � �70 0 (�, °C)

Prevent precipitation of DHA at room temperature (solubility at 50 °C is 0.20 mol L–1)17

Avoid high concentrations of toxic D in product;17 empirical critical runaway condition.26

Prevent toluene solvent excessive vaporization, pressure increase, and dangerous exothermicside-reactions (Footnote b)

Footnotes:

(a) � �T H c cad j p� �( ) / ( ).,0 �

(b) Empirically predicted by adding 2 20� ��Tc K to the nominal temperature, where �T RT Ec � 02 / correspond to the critical conditions of Semenov

for zero-order reactions.1

Simulations of the reactor dynamics reveal ahigh thermal sensitivity due to side-reaction ther-mal effect.6 Starting from feed levels approximatelyhigher than Q = 0.070–0.0080 L min–1, the tempera-ture T t( ) profile not only exhibits values higherthan the critical threshold of 70 °C (technologicalconstraint of Table 1), but tends to become oscilla-tory. Higher feed flow rates produce amplificationof oscillations, toluene vaporization, a dangerouspressure increase, and eventually the reactor run-away. Such an effect can be explained by the slowsecondary reactions (b-c) (diketene exothermicoligomerization), which become dangerous whenthe co-reactant D is accumulating at low tem-peratures. For small feeding Q-levels the main reac-tion consumes the co-reactant, and the side-reac-tions are negligible. Contrariwise, at high inputQ-levels the slow side-reactions lead to the co-reac-tant D accumulation, which will generate moreenergy increasing the reactor’s temperature, whichin turn will lead to the rapid consumption of D.Depletion of D will slow down the reaction ratesand diminish the generated heat leading to a tem-perature decrease. But at low temperatures, the ac-cumulation of D is again possible, and the tempera-ture will rise again. The result is a continuous oscil-lation of the reaction temperature and D-concen-tration in the reactor, with amplitudes larger asthe Q-level is higher. When exceeding a certaincritical Qc value, the temperature oscillationsare higher than a tolerable limit, leading to the pro-cess runaway due to the impossibility to quickly re-move the heat. The study of Maria et al.6 alsorevealed that the critical values of feeding ratesQc ( )f depend on the operating parameter vectorf � !Q T c c Ta D in P, , , ,, ,0 0 around which the evalu-ation is made.

Predictions of critical operatingconditions by various methods

Evaluation of critical feeding conditions Qc ( )f

(for a fixed D inlet concentration) starts with apply-ing the div-J, by computing Re ( ( ))� i J at variousreaction times for sub-critical (Fig. 1), critical (Fig.2), and super-critical (Fig. 3) conditions. Except for

the 6th eigenvalue of J, which is zero all the time,corresponding to the liquid volume increase modelequation (Table 1), all other eigenvalues presentlarge variations according to the operating condi-tions (especially the 2nd–4th eigenvalues related toindividual D reactions, and the 5th eigenvalue re-lated to the temperature dynamics). According tothe div-J criterion, there are no positive Re ( ( ))� i J

values for inlet Q < 0.0080 L min–1 (Fig. 1, and Fig.4-middle), there is an occurrence of positive valuesaround Q = 0.0080 L min–1 (for the 2nd and 3rd

J-eigenvalues in Fig. 2, and Fig. 4-middle), and ofvery sharp positive peaks at super-critical condi-tions for Q > 0.0080 L min–1 (for the 2nd to 5th

J-eigenvalues in Fig. 3, and Fig. 4-middle). Theplot of max (max (Re ( ( ))))i t i� J -vs.-Q in Fig. 4(middle) lead to the precise prediction of the criticalcondition Qc = 0.0078 L min–1 [div-J criterion (3)]for the nominal conditions of SBR, when usingcD in, = 5.82 mol L–1.

The same analysis has been repeated by usingthe SZ stability criterion (6), and computing

Re ( ( ))� iT

J J at various reaction times for sub-crit-

ical (Fig. 1), near critical (Fig. 2), and super-critical(Fig. 3) conditions. By plotting the Lmax index vs.inlet Q-levels (Fig. 4-right), and by taking a small�j tolerance (ca. 4–5), critical condition of Qc =0.0083 L min–1 is thus estimated.


F i g . 1 – Variation of real part of Jacobian J eigenvalues (left) and JTJ eigenvalues (right) with the reaction time at sub-criticalconditions (Q = 0.0055 L min–1)

A quite precise evaluation of the runawayboundaries in the parametric space in sensitive re-gions is offered by the MV-sensitivity criterion (1).By evaluating the absolute sensitivity of the tem-perature maximum vs. the feed flow rate Q (controlvariable), i.e. s T Q t( ; )max time-dependent functionat nominal conditions (Fig. 4), one can observe asharp increase of the curve for a certain Q exceed-ing a critical value Q Qc� . The obtained criticalcondition of Qc = 0.0084 L min–1 is practically thesame with those predicted by the div-SZ criterion,and higher than those predicted by the div-J crite-rion. Being more conservative, the div-J estimationmethod seems to be more suitable for early warningof any incipient instability of the process whenlarge perturbations in operating parameters occur.

Evaluations of the derivatives required bythe MV method have been performed by using

the finite difference numerical method, by re-placing the derivatives with finite differences ofs T T tj t j( ; ) ( )/� �� type, and using repeatedsimulations of the reactor model under various ope-rating conditions. To keep a satisfactory evaluationaccuracy of high sensitivities under severe operatingconditions, small discretization steps in the para-metric space have been used, i.e. ( )/max min� �� nwith n = 500, while a small time-discretization stephas been set (tf/5000) to detect all temperaturepeaks in the critical operating region.

The comparative evaluation of critical input Qc

has been repeated for less severe operating condi-tions, i.e. for a more diluted diketene feeding solu-tion of cD in, = 4 mol L–1. A comparative plot of re-sults obtained with MV, div-J, and div-SZ criteriaof Fig. 5 reveal quite similar predictions of criticalQc given by MV (Qc = 0.0128 L min–1) and div-SZ


F i g . 2 – Variation of real part of Jacobian J eigenvalues (left) and JTJ eigenvalues (right) with the reaction time at critical condi-tions (Q = 0.0080 L min–1)

F i g . 3 – Variation of real part of Jacobian J eigenvalues (left) and JTJ eigenvalues (right) with the reaction time at super-criticalconditions (Q = 0.0090 L min–1)

(Qc = 0.0129 L min–1), and a more conservativerunaway boundary predicted by the div-J criterion(Qc = 0.0113 L min–1). Such results confirm the rec-ommendation to use the div-J analysis method foron-line detection of any incipient process instabilityand divergence from the reference trajectory.

The runaway analysis has been repeated byusing the div-LY stability criterion (8), by com-puting LY max index based on Lyapunov numbersevaluated at various reaction times. By plotting theLY max-vs.-Q, a linear increase results for the presentcase study, for two checked operating conditions

(Fig. 6). Such a result indicates a continuous deteri-oration of the system stability, but the absence of aclear ‘break-point’, where the plot slope might dis-play a dramatic change, leads to the impossibility oflocalizing the critical conditions.

The comparative runaway analysis of theSBR continues with an evaluation of the safetyboundaries under broader operating conditions,represented in separate parametric coordinates.Systematic determination of the safety limitscan be made in this case for the main operatingparameters � � !Q T c ca D in P, , ,, ,0 of the process


F i g . 4 – Evaluation of the critical inlet conditions (Qc = critical feed flow rate of diketene solution) for the semi-batch reactor atnominal conditions ([D]in = 5.82 mol L–1) using various methods: (left) the MV sensitivity criterion; (middle) loss-of-stabil-ity div-J criterion; (right) loss-of-stability div-SZ criterion (nominal conditions correspond to 323 K, [P]o = 0.72 mol L–1,batch time = 150 min)

F i g . 5 – Evaluation of the critical inlet conditions (Qc = critical feed flow rate of diketene solution) for the semi-batch reactor atless severe inlet conditions ([D]in = 4.0 mol L–1) using various methods: (left) the MV sensitivity criterion; (middle)loss-of-stability div-J criterion; (right) loss-of-stability div-SZ criterion (operating conditions correspond to 323 K,[P]o = 0.72 mol L–1, batch time tf = 150 min)

by using various sensitivity and divergence cri-teria.

Application of the MV criterion starts by eval-uating the runaway boundaries in the Q Ta� plane,by keeping nominal states for all other parameters.Thus, one evaluates the absolute sensitivity of thetemperature peak s T Q( ; ),max repeatedly done forvarious Q-levels under nominal conditions, and fora certain fixed value of � j aT� . The resultedcurve (Fig. 4-left) leads to retaining the critical Qc

value corresponding to the occurrence of the maxi-mum of | ( ; )| .maxs T Q The same rule is applied fordifferent Ta values, and the determined critical val-ues Qc are then represented in a Q Ta� plane, thusresulting the runaway critical curve Q Tc a( ). Therunaway boundary (the solid curve in Fig. 7 – left)divides the Q Ta� plane into two regions, corre-

sponding to a safe (below the critical curve) or anunsafe operation (above the critical curve) of the re-actor. As expected, the critical fed flow rate Qc de-creases as the operating severity increases, that isfor high Ta temperatures.

The procedure is repeated by choosing anotheroperating parameter � j (e.g. cD in, , cP ,0 , or T0),under nominal states for all other parameters, andderiving the corresponding runaway boundariesQc j( )� in separate planes (see Fig. 8-left, and Fig.9-left). As the obtained Q Tc ( )0 is roughly parallelto the T0 abscissa, it results that the initial batchtemperature has little influence on the critical Qc

over the investigated parameter domain (this plot isnot presented here).

When identifying the safe/unsafe operating re-gion in the parametric space, the parameter uncer-


F i g . 6 – Variation of the maximum of Lyapunov numbers (div-LY criterion) with the inlet feed flow rate of diketene solution atnominal conditions, and [D]in = 4.0 mol L–1 (left), or [D]in = 5.82 mol L–1 (right)

F i g . 7 – Runaway boundaries in the [Q vs.Ta] plane at nominal conditions predicted by MV sensitivity method (left) and div-J cri-terion (right). The confidence band (- - -) corresponds to the random deviations in the range of �Ta = � 3 K.

tainty can be accounted for by evaluating the confi-dence region of the runaway boundaries when theset point parameters present random variationsof type � ��j j" . Consequently, when derivingQc j( )� curve, by alternatively considering the pa-rameters at lower or upper bounds, the lower andupper bounds of the critical conditions Q Qc c" �can thus be obtained (Figs. 7–9 left side, dottedcurves). The indicated confidence band in the pa-rametric plane corresponds to a 100 % confi-dence level if parameters are uniformly distrib-uted, or to a lower confidence level for normaldistributed parameters depending on the distri-bution characteristics (i.e. a 68 % confidence le-vel for �� j j

� , a 95 % confidence level for�� j j

�2 , etc.). The approximate Qc variance (i.e.� Qc

2 ) can be estimated by using the error propagation

formula for the assumed uncorrelated parameters,

��

��

�

�Q

c

jjc j

Q2

2

2��

�

��

�� .6,28 Such an uncertainty in the

safety limits must be considered when determiningthe optimal operating policy of the SBR, usually bytaking the maximum sensitivities as constraints, andkeeping the solution inside the random variation re-gion of parameters that never intersects the con-straint boundaries.

The same procedure of deriving the runawayboundaries Qc j( )� in separate planes can be ap-plied by using another runaway criteria, for in-stance the div-J criterion (3). By retaining the esti-mated critical value Qc under nominal conditionsfor Ta = 323 K (Fig. 4 centre), the same rule is re-


F i g . 8 – Runaway boundaries in the [Q vs. cD,in] plane at nominal conditions predicted by MV sensitivity method (left) and div-Jcriterion (right). The confidence band (- - -) corresponds to the random deviations in the range of �cD in, = � 0.5 mol L–1.

F i g . 9 – Runaway boundaries in the [Q vs. cP,0] plane at nominal conditions predicted by MV sensitivity method (left) and div-Jcriterion (right). The confidence band (- - -) corresponds to the random deviations in the range of �cP ,0 = � 0.05 mol L–1.

peated for different Ta values, and the determinedcritical values Qc are then represented in a Q Ta�plane, thus resulting the critical curve Q Tc a( )(Fig. 7-right). The procedure is repeated by choos-ing another operating parameter �j (e.g. c cD in P, ,, 0 ),and the derived runaway boundaries Qc j( )� areplotted in separate planes (Fig. 8-right, and Fig.9-right, solid curves). The confidence band of thecritical conditions Q Qc c" � can be obtained onthe same way as for the MV-criterion (dottedcurves).

The comparison of the safety limits predictedby the MV-sensitivity and div-J criteria, presentedin Figs. 7–9, re-confirms the tendency of somediv-J methods to be more conservative, by predict-ing lower critical Qc values and slightly wider con-fidence bands. Generally, the confidence regionsize, for a certain confidence level, depends not

only on the parameter uncertainty but also on themodel non-linearity and used method of estimation.

To complete the sensitivity analysis, investiga-tion of QFS region existence is performed in theoperating space. QFS area is characterized by aquasi-insensitive operation at temperatures evenhigher than the runaway limit. In the first step, onepredicts the evolution of the maximum temperatureTmax and of the time tmax to reach the maximum peakas function of one of the most influential operatingparameter, that is the inlet flow rate Q. The results,plotted in Figs. 10–11 for jacket temperatures �a of50 °C (nominal) and 70 °C respectively, indicatedifferent conclusions as those obtained for singlereaction case, that is: i) a continuous increase of themaximum temperature, with no peak (left plots);ii) multiple local minima in the t Qmax � plots, athigher or lower values of tmax; iii) multiple local


F i g . 1 0 – Predicted maximum temperature Tmax of the SBR (left) and time to reach the maximum temperature tmax (centre) asfunction of the inlet flow rate for �a = 50 °C. (right) T tmax max� curve. Values of the other parameters as in Table 1.

F i g . 1 1 – Predicted maximum temperature Tmax of the SBR (left) and time to reach the maximum temperature tmax (centre) asfunction of the inlet flow rate for �a = 70 °C. (right) T tmax max� curve. Values of the other parameters as in Table 1.

minima in the T tmax max� plots. The complex suc-cessive-parallel reaction pathway, including exo-thermic reactions of very different enthalpy andtime constants, which are successively ‘ignited’ atdifferent temperatures, can explain such behaviourof the SBR. While mild and moderate operatingconditions lead to a quite insensitive tempera-ture-over-time profile (Fig. 12 left, for Q Qc# =0.0084 L min–1), higher feeding rates Q Qc� of theco-reactant inherently lead to its accumulation inthe reactor. Consequently, the high-level generatedheat induces an oscillatory behaviour of the temper-ature, which exhibits ever-growing amplitudes as

the feeding rate is increasing. The successive exo-thermic polymerisation reactions hinder stabilisa-tion of the temperature at super-critical conditionsand a QFS operation becomes improbable.

To check the QFS location, one follows thestandard procedure to build-up a boundary diagramin the Q Ta� plane. Following the MV-criterion,one evaluates the s T Q Q( ; )max � curves for differ-ent jacket temperatures Ta (Fig. 12 centre), andevery time determining the critical value Qc forrunaway. Separate plot of Q Tc a� in Fig. 13(left) splits the parametric plane in runaway andsafe operating regions. Further, one evaluates the


F i g . 1 2 – (left) Temperature evolution during the batch time for constant fed flow rates ofQ = 0.0020 L min–1 (1), Q = 0.0040 L min–1 (2), Q = 0.0060 L min–1 (3), Q (critical) =0.0083 L min–1 (4), Q = 0.0090 L min–1 (5), Q = 0.0140 L min–1 (6). Sensitivity of the reactormaximum temperature (centre) and the time-to-maximum-temperature (right) vs. the fed flowrate for various jacket temperatures of 313 K (1), 323 K (2), 333 K (3), 343 K (4), 353 K (5),363 K (6). Values of the other parameters as in Table 1.

F i g . 1 3 – (left) Boundary diagram for the thermal behaviour of SBR. Plots indicate the runaway boundary (––), and the inferiorlimit of QFS (- - -). (right) Maximum temperature dependence on the inlet flow rate and jacket temperature.

sensitivity function of tmax vs. Q, and then plots thes t Q Q( ; )max � curves for different values of thecooling agent temperature Ta (Fig. 12, right). Theresult is a quite oscillatory behaviour of s t Q( ; )max

in connection to the multiple local minima of thet Qmax � plot. Consequently, application of theQFS-criterion (9) lead not to only one but to multi-ple starting points QQFS of the QFS region, follow-ing the characteristics of the multi-reactions fromthe process. If one retains the lowest QQFS value forevery checked jacket temperature, the resultingboundary line of QFS region is displayed in Fig. 13(left). It must be mentioned that such inferior limitof QFS is located in the sub-critical region, belowthe runaway boundary. Indeed, the operating pointsin the QFS area are characterized by a quite flattemperature profile, with acceptable low oscilla-tions near the runaway limit (Fig. 12, left) and aquite fair conversion. For more severe conditions,that is for �a > 70–80 oC, the runaway curve corre-sponds to higher feeding rates Q, while the non-ig-nition area of flat temperature profile (not presentedhere) is much larger.

Such a sensitivity analysis is however incom-plete as long as the technological constraints are notaccounted for (see Table 1, down). In the presentcase, higher temperature regimes and longer batchtimes must usually be avoided in order to preventformation of secondary products in high amounts(such as DHA, Dn, G), while higher feeding ratesare avoided in order to prevent accumulation ofthe co-reactant D. Experimental observations andrough calculations indicate 70 °C as being the max-imum admissible temperature in the SBR, whilehigher temperatures lead to multiple negativeeffects (dangerous exothermic side-reactions,by-product formation [DHA]f > 0.15 mol L–1, sol-vent excessive vaporization and pressure increase,not included in the present model). By plotting thefeasible operating region (� max < 70 °C) in the sameQ Ta� plane (Fig. 13 right), and superposing itover the boundary diagram (Fig. 13 left), the feasi-ble portions of the non-ignition and QFS regionsare thus detected, to be of further use in locating thebest SBR set-point. Such a combined analysis re-veals that non-ignition region at high Ta levels ishowever non-feasible due to the mentioned draw-backs. Consequently, the feasibility of the problemsolution is decisive for the safe operation of theSBR (even if many-times sub-optimal).

Conclusions

Despite being computationally intensive, themodel-based evaluation of runaway boundaries ofthe operating region for an industrial reactor re-

mains a crucial issue in all design, operation andoptimal control steps. Particularly, the operation as-sociated with inherent random parameter fluctua-tions around the set point, and/or operation in ahigher productivity region in the vicinity of thesafety limits require a precise assessment of the run-away/critical conditions. From this point of view,both div-methods (e.g. div-SZ), based on detectionof loss of stability conditions, and parametric sensi-tivity (e.g. MV) methods can offer fair predictions,being quite similar and strongly connected.8

The method’s predictions are more accuratewhen the checked operating region is a sensitiveone, localized by preliminary model-based simula-tions. According to the parameter uncertainty char-acteristics, additional application of the same rulescan provide the confidence region of the safety lim-its. Such information can be used together with thestate variable sensitivities as constraints when de-riving the optimal operating conditions of the SBRby means of a certain optimization criterion.21,29

As the stability analysis precedes the analysisof the system’s sensitivity to perturbations,19 a com-bined application of div- and sensitivity methodscan offer better confidence in the estimated operat-ing boundaries. Besides, both div- and sensitivitymethods have their own value. While sensitive div-met-hods can detect early changes in the process charac-teristics, being recommended for on-line detectionof the runaway initiation, more accurate runaway-cri-teria can specify the distance in the parameter spacefrom the running/set point to the safety limits.

For a complex kinetic model, the application ofsensitivity and div-runaway criteria is a fairly com-putational task. To save time, it is preferable to re-duce the analysis to only the most influential pa-rameters and initial/inlet conditions of the reactor.Also, a good choice of the method of analysis, andthe correct interpretation of results are importantsteps in evaluating the critical conditions. While so-me method variants offer more conservative predic-tions of the critical conditions (e.g. sensitivity met-hods based on overall kinetics, or div-J method),others offer more accurate predictions (e.g. MV anddiv-SZ). Periodic determinations of the safe operat-ing limits for an SBR exhibiting a high thermal sen-sitivity are also necessary when variations in thecatalyst or raw-material characteristics are recorded.

An extended sensitivity analysis can revealpossible QFS regions at super-critical / severe oper-ating conditions characterized by a quasi-stableSBR behaviour due to quasi-instantaneous reac-tions. Thermally insensitive regions, together withtechnological constraints, have to be further ac-counted for establishing the SBR set-point fromboth economic and safe operation perspective.


ACKNOWLEDGMENT

This work was supported by CNCSIS – UEFISCSU,project number PNII – IDEI 1543/2008–2011“A nonlinear approach to conceptual design andsafe operation of chemical processes”.

N o t a t i o n s

Ar � heat exchange surface of the reactor measuredinside the reactor, m2

A � Arrhenius frequency coefficient, L mol–1 s–1, s–1

B T E RTad� � / ( )02 � reaction violence index21

cj � component j concentration, mol L–1

cp � specific heat capacity, J kg–1 K–1

dr � reactor inner diameter, m

Da v r cA a D A� �( ) ( )/ ,� 0 � Damköhler number for SBR

E � activation energy, J mol–1

Q � fed flow rate (liquid), L s–1

g � model function vector

G � Green’s function matrix

( )��H � reaction enthalpy, J mol–1

J g x� � �/ � system Jacobian

k � rate constants, L mol–1 s–1, s–1

Lmax � maximum of square root of JTJ eigenvalues

LYmax� maximum of Lyapunov stability numbers

q � orthogonalized vector of x

R � universal gas constant, J mol–1 K–1

r � chemical reaction rate, mol L–1 s–1

s x( ; )� � absolute sensitivity, � ��x t( )/

S x( ; )� � normalized sensitivity, ( / ) ( ; ),* *� �x s x or� � �ln ( ( ))/ ln ( )x t

St U A V cr D p� ( )/ ( )� � � Stanton number for SBR

t � time, s

T � thermodynamic temperature, K

� �T H c cad j p� �( ) / ( ),0 � � temperature rise under adi-abatic conditions, K

U � overall heat transfer coefficient, W m–2 K–1

V � liquid (reactor) volume, m3

x � state variable vector

G r e e k s

�i � Lyapunov stability numbers

� � finite difference

� � Kronecker delta function, or small perturbation

�x � perturbation of the fiducial trajectory

� � operating parameter

�j � eigenvalues of a matrix

�~

j � Lyapunov exponents

vj � stoichiometric coefficient of species j

� � liquid phase density, kg m–3

� � standard deviation, or relative sensitivity toler-ance

� � time constant, s

�D � time of addition of co-reactant D, s

� � temperature, °C

I n d e x

a � cooling agent

ad � adiabatic

c � critical

f � final

in � inlet

max � maximum

min � minimum

o � initial

$ � average value

A b b r e v i a t i o n s

D � diketene

DHA� dehydroacetic acid

G � secondary product

GM � geometry-based methods

MV � Morbidelli-Varma criterion

P � pyrrole

PAA � 2-acetoacetyl pyrrole

PSA � sensitivity-based methods

Py � pyridine

Re(·) � real part

SBA � stretching-based method

SBR � semi-batch reactor

SZ � Strozzi & Zaldivar

Trace(·) � trace of a matrix

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