On the divergence criterion for runaway detection Application to complex controlled systems

On the divergence criterion for runaway detection  Application to complex controlled systems
On the divergence criterion for runaway detection  Application to complex controlled systems

On the divergence criterion for runaway detection:Application to complex controlled systems

Sabrina Copelli a,Vincenzo Torretta a,Christian Pasturenzi b,Marco Derudi c,

Carlo Sala Cattaneo c,Renato Rota c,*

a Universitàdegli Studi dell’Insubria,Dipartimento di Scienza e Alta Tecnologia,via G.B.Vico46,21100Varese,Italy

b INNOVHUB,Divisione Stazione Sperimentale per i Combustibili,viale A.De Gasperi3,20097S.Donato M.se,Italy

c Politecnico di Milano,Dipartimento di Chimica,Materiali e Ingegneria Chimica“G.Natta”,via Mancinelli7,20131Milano,Italy

a r t i c l e i n f o

Article history:

Received5January2013 Received in revised form

9May2013

Accepted9May2013

Keywords:

Runaway reactions Divergence criterion

Safety

Marginal ignition Temperature control a b s t r a c t

It is well-known that,for certain values of the operative parameters in?uencing the dynamic behavior of a chemical reactor,a phenomenon known as thermal runaway(that is,a loss of the reactor temperature control)may arise.Such a situation can be really dangerous because above a certain threshold tem-perature value unwanted side reactions or,worse,decompositions of the reacting mixture may be triggered evolving high amounts of?ammable or toxic gases that can cause reactor pressurization and, eventually,its explosion.For this reason,since the beginning of the previous century a number of studies concerning the prediction of the so called runaway boundaries has been carried out.In this work,a modi?ed version of the divergence criterion for runaway detection,originally developed by Zaldívar and co-workers,is presented.Such a modi?ed divergence criterion is capable of treating whatever type of complex controlled reacting system(taking into account not only temperature control but also dosing strategies)and its reliability has been demonstrated for isoperibolic semibatch reactors using literature experimental data concerning the nitration of4-Chlorobenzotri?uoride in mixed acids and the nitric acid oxidation of2-octanol to2-octanone and further carboxylic acids.

ó2013Elsevier Ltd.All rights reserved.

1.Introduction

Fast and highly exothermic chemical processes are very dif?cult to be optimized and scaled-up from laboratory to full plant scale because of the possible triggering of a reactor temperature loss of control,occurring whenever the rate of heat evolution becomes greater than the rate of heat removal provided by the installed cooling equipment.Such an unwanted phenomenon,called“ther-mal explosion”or“runaway”,can take place into every type of reactor(both continuous and discontinuous),operated in whatever temperature control mode(isothermal,isoperibolic,polytropic, etc.)and fed by the most disparate dosing strategies(linear,ramp, etc.),because of a great number of operating and human factors, such as:control system failure,stirrer breakdown,refrigerating system inef?ciency,maintenance,reactants loading errors,dead times enlarging or shortening,etc.

In?ne chemical and pharmaceutical industries,potentially runaway reactions are usually carried out in semibatch reactors (SBRs),where one or more reactants(called co-reactants)are dosed on an already loaded mixture in order to control the rate of heat generation by the dosing rate.However,if the process is operated under high accumulation conditions(that is,a large amount of unreacted co-reactants accumulates in the reactor during the dosing period),the desired reaction thermal control may be lost and the reactor temperature may increase up to values at which sec-ondary undesired reactions or,worse,decompositions of the reacting mixture can be triggered.Such reactions,apart from a reduction of the selectivity with respect to the desired product,may lead to a further and ultimate system thermal loss of control,since they are usually much more fast and exothermic than the desired reaction.Moreover,if a decomposition event arises because of the system thermal loss of control,the consequent release of incoerc-ible gases will pressurize the reactor leading to its physical explo-sion if the venting system cannot deal with such a phenomenon.

According to statistics,in a typical European country more than 100runaways are expected to occur annually(Benuzzi&Zaldivar, 1991).However,despite only a little amount of these accidents hurts the workers or the inhabitants of the neighborhood of the damaged factory,when strong runaways occur the consequences can be really serious.This motivates the great amount of work that has been done on runaway phenomena in exothermic chemical batch and semibatch reactors during the previous century and, particularly,in the last thirty years.As a matter of fact,many

*Corresponding author.Fax:t390223993180.

E-mail address:renato.rota@polimi.it(R.

Rota).Contents lists available at SciVerse ScienceDirect

Journal of Loss Prevention in the Process Industries jou rn al homepage:

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0950-4230/$e see front matteró2013Elsevier Ltd.All rights reserved.

https://www.360docs.net/doc/291969171.html,/10.1016/j.jlp.2013.05.004

Journal of Loss Prevention in the Process Industries28(2014)92e100

different studies on the detection of the so-called“runaway boundary”or“marginal ignition line”(that is,the system operating parameters in correspondence of which runaway phenomena are triggered)have been conducted(Alós,Nomen,Sempere,Strozzi,& Zaldívar,1998;Bosch,Strozzi,Zbilut,&Zaldívar,2004;Hugo& Steinbach,1985;Maestri&Rota,2005a,2005b;Morbidelli& Varma,1982,1985,1988;Steensma&Westerterp,1988,1990, 1991;Steinbach,1999;Strozzi,Zaldívar,Kronberg,&Westerterp, 1999;Varma,Morbidelli,&Wu,1999;van Woezik&Westerterp, 2000,2001;Wu,Morbidelli,&Varma,1998a,1998b;Zaldívar, Bosch,Strozzi,&Zbilut,2005;Zaldívar et al.,2003).

The?rst rigorous approach for isoperibolic SBRs was developed by Hugo and Steinbach(1985).Their work,based on homogeneous systems,introduced the accumulation criterion for the analysis of a SBR thermal behavior:co-reactant accumulation into the system, which arises from a not negligible characteristic time of the chemical reaction with respect to that one of the co-reactant sup-ply,must be kept at suf?ciently low values to avoid the reactor thermal loss of control(runaway).If co-reactant accumulates into the reactor,the system switches from semibatch to batch-like conditions and the cooling system might not control the heat evolution anymore.According to this criterion,operating condi-tions characterized by a suf?ciently low co-reactant accumulation are considered not only safe but also productive.

Steensma and Westerterp(1988,1990,1991)extended the re-sults obtained by Hugo and Steinbach(1985)to heterogeneous (liquid e liquid)SBRs,introducing the concept of target temperature and producing the so-called boundary diagrams(BDs)for single reactions of(1,1)reaction order kinetics.Particularly,such diagrams are generated in a suitable dimensionless space(i.e.,Reactivity,R y, versus Exothermicity,E x)that allows end users to easily discriminate between safe(referred to as“Quick onset,Fair conversion,Smooth temperature pro?le”,QFS)and excessive accumulation operating conditions without solving the mathematical model of the reactor.

Maestri and Rota(2005a,2005b,2006),analyzing the role that estimated kinetic parameters play on conclusions drawn through boundary diagrams calculated for reaction orders different from (1,1),prove that unjusti?ed assumptions on the reaction kinetics cannot be accepted for a reliable application of the BDs method: such a conclusion has been found to be true for both heterogeneous and homogeneous reaction systems(with or without autocatalytic behavior),and it makes a kinetic investigation(usually performed through calorimetric techniques)of great importance.Moreover, van Woezik and Westerterp(2000,2001)tried to extend the con-cepts introduced by Steensma and Westerterp(1988,1990,1991) for single reaction systems to the case of consecutive reactions with autocatalytic behavior,studying both theoretically and experimentally the nitric acid oxidation of2-octanol to2-octanone (?rst reaction)with further oxidation of the desired product to carboxylic acids(second reaction).These studies proposed to use boundary diagrams generated for single reactions to characterize the thermal behavior of such a system.Such an approach was re?ned by Copelli,Derudi,and Rota(2010b),who developed a rigorous method to generate boundary diagrams for consecutive reactions with autocatalytic behavior and used the same experi-mental data provided by van Woezik and Westerterp to validate it.

For what concern the identi?cation of the marginal ignition line only,the?rst general criterion able to handle multiple reactions systems and different reactor typologies was the generalized para-metric sensitivity criterion(Morbidelli and Varma,1982,1985,1988; Varma et al.,1999;Wu et al.,1998a,1998b).This criterion,with the shortcoming of being implicit(that is,it is not able to establish if the system operates under runaway conditions only by evaluating its constitutive parameters),identi?es the runaway boundary looking for the maximum of a suitable objective sensitivity coef?cient (Varma&Morbidelli,1997):the normalized sensitivity coef?cient of the maximum reactor temperature with respect to any operating parameter,s T

MAX;

f?f=T MAX$v T MAX=v f.The criterion shows the advantage of being applied both theoretically(through the use of a mathematical model describing the analyzed reacting system)and experimentally(through the monitoring of the maximum reactor temperature evolution by varying any system operating parameter).

Actually,the most general criterion for the detection of the runaway boundary,able to operate with multiple reactions, different temperature control modes and reactor typologies(e.g., batch reactor(BR),continuous stirred tank reactor(CSTR),plug-?ow reactor(PFR),etc.),is the divergence criterion(Bosch et al., 2004;Strozzi et al.,1999;Zaldívar et al.,2003,2005).This crite-rion,originally developed by Zaldívar and co-workers,states that if the system of ordinary differential equations(ODEs)that describes the analyzed process exhibits positive divergence(that is the trace of the corresponding Jacobian matrix),the synthesis is operating under runaway conditions.Such a criterion shows the relevant advantage of being usable both on-line(the divergence can be easily reconstructed,at any time during the process,through temperature measures provided by four thermocouples located into the reactor)and off-line(the runaway boundary is individu-ated by keeping constant the values of all the constitutive param-eters except one and,then,by setting the divergence expression with respect to the variable model parameter to zero).Moreover, since a fast reaction that does not produce or consume heat can generate a strong volume contraction in the N-dimensional state space that is not related to runaway phenomena,the term related to the extent-of-reaction of such a reaction must be disregarded into the divergence calculation in order to obtain predictions in agreement with the experimental evidences(Zaldívar et al.,2003).

In this work,it has been shown that,when controlled reacting systems are involved,also other state variables(apart from the extent-of-reaction related to a fast reaction that does not produce or consume heat)can generate a strong volume contraction in the N-dimensional state space that is not related to runaway phenomena,therefore leading the divergence criterion to fail in predicting runaway phe-nomena.As a consequence,a modi?ed version of the divergence criterion for runaway detection has been developed.Such a modi?ed version is capable of treating whatever type of controlled reacting system(taking into account not only temperature control but also dosing strategies)and its reliability has been demonstrated for iso-peribolic semibatch reactors using literature experimental data (Maestri et al.,2009a,2009b;van Woezik&Westerterp,2000,2001).

2.Modi?ed divergence criterion

The divergence criterion was originally developed by Zaldívar and co-workers(Strozzi et al.,1999)and it is based on the concepts of state space(that is,the geometric space where the set of all possible states of a dynamical system e namely,the solution or trajectory e is represented)and phase portraits(that is,the rep-resentation of the trajectories of a dynamical system in the phase space when an initial condition or a constitutive model parameter is varied in a suitable range)as key instruments that can be used for the complete description of whatever reacting system thermal behavior.As the matter of fact,the dynamics of a reacting system can be always represented by a system of ordinary differential equations(ODEs),that suitably describes material and energy balances,temperature and dosing control strategies,mixing rules, etc.in the general form:d x i=d t?F i?xetT .Particularly,to each state of the reacting system corresponds a unique point onto the state space,which is called state point,and the system state time evo-lution is represented as a motion along a trajectory.According to the Liouville’s theorem(Zaldívar et al.,2003)there is a relation

S.Copelli et al./Journal of Loss Prevention in the Process Industries28(2014)92e10093

between the state space volume of a N-dimensional dynamical system and its divergence.This relation can be expressed as:

V et T?V e0T$exp 2

4

Z t

di v f F ?x et T g d t 3

5

(1)

where:

di v f F ?x et T g ?

X N i ?1

v F i ?x et T v x i

(2)

Hence for a dynamical system the rate of change of an in ?ni-tesimal volume V (t ),following an orbit x et T,is given by the diver-gence of the ?ow,which is locally equivalent to the trace of the

F ?x et T Jacobian.Even though for dissipative systems,as chemical reactors are,the divergence unavoidably decreases as time ap-proaches N ,it has been observed (Strozzi et al.,1999)that when the system is operated under potentially runaway conditions it exhibits,during a certain period of time,a state space volume expansion.This means that trajectories originating from nearby starting points will diverge and such a feature can be correlated with the parametric sensitivity of the reactor temperature with respect to the system constitutive parameters or initial conditions.Therefore,taking into account such a correlation between para-metric sensitivity and the divergence of nearby trajectories,Strozzi et al.(1999)de ?ned a general criterion for runaway detection.Particularly,a reactor is operating under runaway conditions when the divergence of the ODEs system describing its thermal behavior becomes positive on a segment of the reaction path.

di v f F ?x et T gi 0(3)

This criterion is so general that can be easily applied to both controlled (there are suitable equations expressing temperature control strategy,e.g.isothermal)and uncontrolled (typically,when the reactor operates in the isoperibolic temperature control mode,coolant temperature is assumed to be constant for all process dura-tion)systems independently on the reactor type (CSTR,PFR,BR,SBR).

Nevertheless,such a general criterion needs to be subjected to a single constraint in order to be in accordance with both theoretical and experimental evidences on runaway detection.This constraint takes into account the presence of a fast reaction that does not

produce directly heat (Zaldívar et al.,2003).For describing the kinetics of the reacting systems of interest,usually the minimum number of global reactions (NR)is utilized;therefore,thanks to the stoichio-metric constraint,the NC number of moles of each species can be computed from the NR extent-of-reactions,which are easily related to the conversion of some reactants or to the yield of some products/intermediates.Consequently,the NC material balances can be recast in NR balance equations for the NR extent-of-reactions,one of these being that of the fast reaction that does not produce or consume heat.Particularly,such an extent-of-reaction can generate a strong volume contraction (synthesized in a no runaway condition ac-cording to the divergence criterion)in the N-dimensional state space arising by considering that extent-of-reaction in the diver-gence computation,but a global expansion (runaway)in the (N-1)-dimensional space arising by not considering that extent-of-reaction into the divergence computation.Therefore,as a general constraint for the runaway criterion based on divergence,all con-tributions arising from extent-of-reactions that are not related to heat evolutions must be disregarded into the divergence calcula-tion.Such an expedient is necessary in order to avoid evaluation errors due to the presence of strongly negative terms (the diver-gence is a sum with sign)that are not related to the runaway phe-nomenon and could lead to underpredict the chance of runaway.As will be shown in the following “Results ”section,this is a particular case of a more general behavior of the divergence crite-rion:when controlled reacting systems are involved,also other state variables (apart from the extent-of-reaction related to a fast reaction that does not produce or consume heat)can generate a strong volume contraction in the complete N-dimensional state space that is not related to runaway phenomena,therefore leading the divergence criterion to fail in predicting runaway phenomena.As a consequence,a modi ?ed version of the divergence criterion has been developed.Particularly,a more general constraint has been proposed based on the consideration that whatever type of runaway phenomenon is originated by a reactor temperature loss of control,only those terms of the F ?x et T trace that are related to the rate of heat evolution should be considered into the divergence calculation.Referring for the sake of illustration to a controlled isoperibolic semibatch reactor where NR reactions are carried out in homoge-neous phase,the following system of ODEs can be written to describe its thermal behavior:

S.Copelli et al./Journal of Loss Prevention in the Process Industries 28(2014)92e 100

94

where m dos is the dosing mass (kg),f dos is the dosing stream function,T cool,IN is the inlet coolant temperature (K),T cool is the actual coolant temperature (K),T cool,set is the set-point tempera-ture of the jacket (K),T is the reactor temperature (K),K pT is the proportional gain of the temperature controller (e ),K iT is the reset time of the temperature controller (s),V is the total liquid reacting volume (m 3),f r is a function accounting for mixing rules (that can assume a number of different time,temperature and compositions dependent expressions),UA is the global heat transfer coef ?cient for the cooling system (W/K),z j is the con-version of the j -th reaction (e ),f z is the conversion function,T dos is the dosing stream temperature (K),T ext is the ambient tem-perature (K),UA ext is the global heat transfer coef ?cient for the ambient (W/K),r j is the j -th reaction rate (kmol/(m 3s))according

to,e.g.,Arrhenius Law and D ~H rxn ;j is the j -th reaction enthalpy (J/

kmol).The meaning of other symbols is reported in the “Nomenclature ”section.

In the ODEs system (4),?rst and second equation express the dosing policy (e.g.constant feeding rate)and the reactor temper-ature control mode (in this case,isoperibolic),respectively;the third equation is the global material balance onto the reacting mass;the forth equation synthesizes the mixing rule for density;the ?fth one expresses the energy balance for the cooling jacket;the NR equations synthesized in the sixth one represent the bal-ances for the NR extent-of reactions;the last equation is the energy balance for the reactor.

Looking at the last equation of system (4),it is possible to observe that the term related to heat evolution (power released)is expressed by the following equation:

_Q rxn ?

X NR j ?1

r j $

àD ~H rxn ;j $V

(5)

If a single reaction of the type:

A t

B /

C (6)

is assumed to occur,the corresponding reaction rate can be expressed as follows:

r ?k N $exp

àE att

R $T

$f ez ;m dos T(7)

where k N is the pre-exponential factor and E att is the activation energy.

Therefore,analyzing all the dependent variables involved into Equation (5),it is possible to state that only temperature (T ),vol-ume (V ),extent-of-reaction (that is,conversion z )and dosing policy (m dos )affect the rate of heat evolution.As a consequence,all terms of the F ?x et T trace related to these state variables should be necessary and suf ?cient to characterize the system thermal behavior (that is,runaway).It is important to note that in this modi ?ed divergence criterion all the terms of F ?x et T trace corre-sponding to temperature control strategy,mixing rules and jacket energy balance are automatically excluded from the divergence calculation.As discussed in the following section,such a difference with respect to the original divergence criterion is effective to correctly detect runaway boundaries for isoperibolic semibatch reactors where one or more exothermic reactions take place.3.Results

In this section,results concerning the application of both the original divergence criterion for runaway detection and the modi ?ed version are presented with reference to the aromatic

nitration of 4-Chlorobenzotri ?uoride in mixed acids and the nitric acid oxidation of 2-octanol to 2-octanone and further carboxylic acids.For these systems,no underprediction of the marginal ignition line due to the presence of fast but no exothermic reactions (that can induce strong volume contractions)is expected for the original version of the diver-gence criterion (that is,no corrections in the divergence calculation must be done)because they involve only exothermic reactions.

3.1.Single non autocatalytic reaction

Since the aromatic nitration of 4-Chlorobenzotri ?uoride has been extensively discussed elsewhere (Copelli,Derudi,&Rota,2011;Maestri et al.,2009a ,2009b ),it will be only brie ?y summa-rized in the following.The following single non autocatalytic re-action represents the kinetic scheme considered into the 4-Chloro-3-nitrobenzotri ?uoride synthesis:

A t

B /

C t

D (8)

where A represents the species 4-Chlorobenzotri ?uoride,B is the nitronium ion (responsible for the nitration reaction),C is the species 4-Chloro-3-nitrobenzotri ?uoride (desired product)and D is water.

According to such a kinetic scheme,the reaction rate can be expressed as follows:

r ?k N $exp

àE att

RT

$?A c $?B c (9)

The dynamics of the isoperibolic SBR can be modeled through

the following ODEs system expressing,in dimensionless form:isoperibolic temperature PI control loop (?rst equation),energy balances on both jacket (second equation)and reactor (third equation),extent-of-reaction balance in terms of material balance for the desired product C (forth equation),global material

balance

https://www.360docs.net/doc/291969171.html,parison between original and modi ?ed approach for runaway detection based on the divergence criterion.a)Maximum value of the divergence,and b)sign (positive,t,or negative,à)of the divergence as a function of set point coolant tem-perature for the aromatic nitration of 4-Chlorobenzotri ?uoride in mixed acids.

S.Copelli et al./Journal of Loss Prevention in the Process Industries 28(2014)92e 10095

(?fth equation),mixing rule (sixth equation)and dosing policy (seventh equation).

In the time range 0 w <1:

with initial conditions:

s cool ;IN ?s cool ;IN ;0;s cool ?s cool ;0;s ?s 0;z ?0;v ?r ?1;v dos ?0(11)

Some of the equations above changes for w !1as follows:

8>><>>:d r ?0d v dos ?0

(12)

Referring to the ODEs system (10)and (12),it is possible to calculate the divergence according to both the original divergence criterion for runaway detection (div orig )and its corresponding modi ?ed version (div mod ).

di v orig ?

X 7i ?1

v F i ex Tv x i

(13)

di v mod ?

v F 3ex Tv x 3tv F 4ex Tv x 4tv F 5ex Tv x 5tv F 7ex T

v x 7

(14)

As it can be easily observed,the following terms are always

equal to zero and,therefore,they do not contribute to the diver-gence magnitude.

v F 1ex T1?v F 6ex T6?v F 7ex T

7

?0(15)

As a consequence,the only difference between div orig and div mod calculation is the computation of the term v F 2ex T=v x 2,which is associated to the jacket energy balance.Such a term is expressed by the following equation:

v F 2ex T

v x 2

?àq àNTU $v (16)

Observing Equation (16)is easy to notice that its contribution to the divergence is always negative and its magnitude depends on the constitutive process parameters.

After this preliminary observation,runaway boundaries can be identi ?ed according to the two different criteria.To generate the marginal ignition line,it is necessary to keep constant all consti-tutive model parameters and initial conditions except one,which

will be varied in a suitably wide range.In this case,for the sake of example,the set point coolant temperature has been chosen as the non-constant parameter.To each value of T cool ,set corresponds a unique value of the maximum of the divergence that can be computed with the two different criteria (div orig ,MAX ,computed with Equation (13),or div mod ,MAX ,computed with Equation (14);these are the absolute maxima detected during a single simulation at a current T cool ,set value).If such a divergence value is positive,the system is operating under runaway conditions during an interval of its reaction path;if it is negative,the system is always operating under safe conditions;if it is equal to zero,a boundary between safe and runaway conditions (?rst boundary)or runaway and QFS conditions (second boundary)is detected.

Fig.1,which has been computed using the recipe reported in Table 1,shows the comparison between the results obtained with the original divergence criterion and the modi ?ed one.Particularly,in Fig.1a,the maximum divergence value vs.time is reported for both the original (div orig ,MAX )and the modi ?ed criterion (div mod ,-MAX ),while in Fig.1b the sign of the maximum divergence is rep-resented.We can note that using the original divergence criterion no runaway boundary is detected in the analyzed range of T cool ,set (that is,20 C T cool ;set 50 C).However,it is known that around 25e 30 C a runaway arises,exhibiting an adiabatic temperature rise up to 70 C (Copelli,Derudi,&Rota,2011;Maestri et al.,2009a ,2009b ).

On the contrary,from the same Fig.1we can see that the modi ?ed divergence criterion identi ?es a runaway boundary around 25 C,in agreement with the experimental and theoretical behavior previously identi ?ed.Moreover,a QFS boundary around 43 C is also identi ?ed;such a value is slightly larger than that computed using other methods for QFS detection (Alós et al.,1998;Copelli,Derudi,&Rota,2011),which predict a QFS boundary at about 38 C.However,this is a well-known feature of the diver-gence criterion,which usually overpredicts QFS boundary (Strozzi et al.,1999).

The failure of the original divergence criterion in predicting the runaway boundary should be ascribed to the strongly negative magnitude of the term v F 2ex T=v x 2,which is associated to the jacket energy balance.Such a term is not directly linked to the runaway

S.Copelli et al./Journal of Loss Prevention in the Process Industries 28(2014)92e 100

96

behavior;therefore,analogously to what was previously found for a fast reaction that does not produce or consume heat(Zaldívar et al., 2003),it must be disregarded into the divergence calculation to avoid a strong but not meaningful volume contraction in the phase space and to obtain consequently predictions in agreement with the experimental evidences.

3.2.Consecutive reactions with autocatalytic behavior

Since even the nitric acid oxidation of2-octanol to2-octanone and further carboxylic acids has been extensively discussed else-where(Copelli,Derudi,&Rota,2010a;van Woezik&Westerterp, 2000,2001),it will be only brie?y summarized in the following. Particularly,the following consecutive reactions with autocatalytic behavior represent the kinetic scheme considered into the2-octanone synthesis:

1TAtB/2BtC

2TBtC/D

(17)

where A is2-octanol,B is nitrosonium ion(NOt,the species responsible for the oxidation of species A and C and also for the autocatalytic behavior of reaction1),C is2-octanone(product of reaction1)and D is a mixture of carboxylic acids(products of re-action2).

According to such a kinetic scheme,the reaction rates can be expressed as follows:1Tr1?k N;1$exp

à

E att;1

RT

$exp

à

àm H;1$H

á

$?A c$?B c

2Tr2?k N;2$exp

à

E att;2

$exp

à

àm H;2$H

á

$?C c$?B c

(18)

where m H,1and m H,2are the Hammett’s reaction rate coef?cients and H is the Hammett’s acidity function.

The dynamics of the isoperibolic SBR can be modeled through the following ODEs system expressing,in dimensionless form: isoperibolic temperature PI control loop(?rst equation),energy balances on both jacket(second equation)and reactor(third equation),extent-of-reactions balances in terms of C and D yields (forth and?fth equations),global material balance(sixth equation), mixing rule(seventh equation)and dosing policy(equation eight).

In the time range0w<1:

with initial conditions:

s cool;IN?s cool;IN;0;s cool?s cool;0;s?s0;z C?z D?0;

v?r?1;v dos?0(20) Some of the equations above changes for w!1as follows:

8

>><

>>:

d r

d w

?0

d v dos

d w

?0

(21)

Referring to the ODEs system(19)and(21),it is possible to calculate the divergence according to both the original divergence criterion for runaway detection(div orig)and its corresponding modi?ed version(div mod).

Table1

Process recipe,reactor and cooling system for the aromatic nitration of4-

Chlorobenzotri?uoride in mixed acids.

Initial load Reactor and cooling system

30[g]HNO3Reactor nominal volume 1.0?10à3[m3]

335[g]H2SO4Jacket nominal volume 5.0?10à4[m3]

Dosed stream Coolant?ow rate 5.0?10à4[m3/s]

83[g]4-Chlorobenzotri?uoride UA0 1.21[W/K]

S.Copelli et al./Journal of Loss Prevention in the Process Industries28(2014)92e10097

di v orig ?

X 8i ?1

v F i ex Tv x i

(22)

di v mod ?

v F 3ex Tv x 3tv F 4ex Tv x 4tv F 5ex Tv x 5tv F 6ex Tv x 6tv F 8ex T

v x 8

(23)

As it can be easily observed,the following terms are always

equal to zero and,therefore,they do not contribute to the diver-gence magnitude.

v F 1ex Tv x 1?v F 7ex Tv x 7?v F 8ex T

v x 8

?0(24)

As previously observed for a single reaction,the only difference between div orig and div mod calculation is the term v F 2ex T=v x 2.

Looking for runaway boundaries considering as non-constant parameter the set point coolant temperature,the results reported in Fig.2(which has been generated using the recipe reported in Table 2)have been obtained.Particularly,in Fig.2a,the maximum divergence vs.time evolution is reported for both the original (div orig ,MAX )and the modi ?ed divergence criterion (div mod ,MAX ),while in Fig.2b the sign of the maximum divergence is represented.We can note that using the original divergence criterion a unique runaway boundary at T cool ,set ?8 C,is detected.Such a value cor-responds to that at which the modi ?ed divergence criterion iden-ti ?es a second runaway boundary.In fact,the modi ?ed divergence criterion is able to identify also another thermal runaway effect,of small magnitude (as it can be seen from Fig.2a),occurring at a lower set point coolant temperatures (T cool ,set ?à9 C).This ?rst runaway boundary has been previously identi ?ed both experi-mentally and theoretically (Copelli et al.,2010a ;van Woezik &Westerterp,2001)and it is a key point in the safe process optimi-zation for what concerns the maximization of the 2-octanone production.

Similarly to that found in the ?rst case investigated,the failure of the original divergence criterion in predicting the runaway

boundary should be ascribed to the strongly negative magnitude of the term v F 2ex T=v x 2,which is associated to the jacket energy bal-ance without being directly linked to the runaway behavior.Therefore,it must be disregarded into the divergence calculation to avoid a strong but not meaningful volume contraction in the phase space and to obtain consequently predictions in agreement with the experimental evidences.4.Conclusion

In this work a modi ?ed version of the divergence criterion has been developed.Such a modi ?ed divergence criterion is able to deal with whatever type of complex controlled reacting system and its reliability has been demonstrated for isoperibolic semibatch re-actors using literature experimental data concerning the nitration of 4-Chlorobenzotri ?uoride in mixed acids and the nitric acid oxidation of 2-octanol to 2-octanone and further carboxylic acids.It has been found that where the original approach fails to detect a runaway boundary,the modi ?ed divergence criterion is capable of predicting reliably such a boundary.Acknowledgment

The authors wish to express their profound admiration to Pro-fessor J.M.Zaldívar inasmuch he should be taken as an example of a complete scientist and pioneer of mathematics applied to chemical reactors safety.During his career,professor Zaldívar explored many different ?elds,such as Neural Networks (Tomasin,Zaldivar,Gutierrez,Galvan,&Strozzi,2000;Zaldivar,Hernandez,&Panetsos,1992),Chemistry (Zaldivar,Alós,Hernández,Molga,&Westerterp,1995a ,1995b )and Chemical Engineering,without neglecting his collaboration in works on both neurotoxicity (Scelfo et al.,2012)and,as member of the JRC at Ispra,water and air pollution,with particular interest in PCB contaminations in northern Italy (Castro-Jiménez et al.,2009;Dueri,Castro-Jiménez,&Zaldívar,2009).

Professor Zaldívar should be considered a pioneer of reactor safety because of his works on the divergence criterion which,as stated in this paper,is a simple but powerful tool to estimate runaway conditions in chemical reactors with enough precision to be used in chemical industries.

Finally,it is worth mentioning that starting from the pioneering Zaldívar ’s works,applied mathematics has been increasingly used in thermal safety and optimization criteria studies opening new research ?elds (e.g.,Copelli,Derudi,Lunghi,Pasturenzi,&Rota,2011;Copelli et al.2010a ;Copelli,Derudi,&Rota,2011;Copelli et al.,2012;Copelli,Derudi,Sempere,et al.,2011).References

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criteria for boundary safe conditions in semi-batch processes:simulated anal-ysis and experimental results.Chemical Engineering and Processing,37,405e 421.Benuzzi,A.,&Zaldivar,J.M.(1991).Safety of chemical reactors and storage tanks .

Dordrecht:Kluwer Academic Publishers

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https://www.360docs.net/doc/291969171.html,parison between original and modi ?ed approach for runaway detection based on the divergence criterion.a)Maximum value of the divergence,and b)sign (positive,t,or negative,à)of the divergence as a function of set point coolant tem-perature for the nitric acid oxidation of 2-octanol to 2-octanone and further mixture of carboxylic acids.

Table 2

Process recipe,reactor and cooling system for the nitric acid oxidation of 2-octanol to 2-octanone and further carboxylic acids.Initial load

Reactor and cooling system 377[g]HNO 3Reactor nominal volume 2.0?10à3[m 3]251[g]H 2O Jacket nominal volume 1.0?10à3[m 3]Dosed stream Coolant ?ow rate 5.0?10à4[m 3/s]164[g]2-octanol

UA 0

4.3

[W/K]

S.Copelli et al./Journal of Loss Prevention in the Process Industries 28(2014)92e 100

98

Bosch,J.,Strozzi,F.,Zbilut,J.P.,&Zaldívar,J.M.(2004).On-line runaway detection in isoperibolic batch and semibatch reactors using the divergence criterion.

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Castro-Jiménez,J.,Dueri,S.,Eisenreich,S.J.,Mariani,G.,Skejo,H.,Umlauf,G.,et al.

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Copelli,S.,Derudi,M.,Lunghi,A.,Pasturenzi,C.,&Rota,R.(2011).Experimental design of topological curves to safely optimize highly exothermic complex reacting systems.Industrial&Engineering Chemistry Research,50,9910e9917. Copelli,S.,Derudi,M.,&Rota,R.(2010a).Topological criteria to safely optimize hazardous chemical processes involving consecutive reactions.Industrial& Engineering Chemical Research,49,4583e4593.

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Emulsion polymerization of vinyl acetate:safe optimization of a hazardous complex process.Journal of Hazardous Materials,192,8e17.

Dueri,S.,Castro-Jiménez,J.,&Zaldívar,J.M.(2009).Modelling the in?uence of thermal strati?cation and complete mixing on the distribution and?uxes of polychlorinated biphenyls in the water column of Ispra Bay(Lake Maggiore).

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Maestri,F.,Copelli,S.,Rota,R.,Gigante,L.,Lunghi,A.,&Cardillo,P.(2009a).Simple procedure for optimally scaling-up?ne chemical processes.I.Practical tools.

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Simple procedure for optimal scale-up of?ne chemical processes.II.

Nitration of4-chlorobenzotri?uoride.Industrial&Engineering Chemical Research,48,1316e1324.

Maestri,F.,&Rota,R.(2005a).Thermally safe operation of liquid-liquid semibatch reactors.Part I:single kinetically controlled reactions with arbitrary reaction order.Chemical Engineering Science,60,3309e3322.

Maestri,F.,&Rota,R.(2005b).Thermally safe operation of liquid e liquid semibatch reactors.Part II:single diffusion controlled reactions with arbitrary reaction order.Chemical Engineering Science,60,5590e5602.

Maestri, F.,&Rota,R.(2006).Safe and productive operation of homogeneous semibatch reactors.I:development of a general procedure.Industrial&Engi-neering Chemical Research,45,8002e8013.

Morbidelli,M.,&Varma,A.(1982).Parametric sensitivity and runaway in tubular reactors.American Institute of Chemical Engineers.AIChE Journal,28,705e713. Morbidelli,M.,&Varma,A.(1985).On parametric sensitivity and runaway criteria of pseudo-homogeneous tubular reactors.Chemical Engineering Science,40, 2165e2168.

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Nomenclature

A:Area,[m2],Dosed reactant

B:Loaded reactant

C:Desired product

b c p:Speci?c heat capacity,[J/(kg K)]

D:Byproduct or mixture of byproducts

div:?

P N

i?1

v F i?xetT =v x i,Divergence operator

Da:?k N$expeàE att=RT rifT$?B 0$t dos,Damk?ler number for the aromatic nitration of 4-Chlorobenzotri?uoride,[e]

Da1:?k N;1$expeàE att;1=RT rifT$?A dos$t dos,Damk?ler number for the?rst reaction of the nitric acid oxidation of2-octanol,[e]

Da2:?k N;2$expeàE att;2=RT rifT$?A dos$t dos,Damk?ler number for the second reaction of the nitric acid oxidation of2-octanol,[e]

E att:Activation energy,[kJ/kmol]

E x:Exothermicity number,[e]

F:Jacobian matrix

f:Generic function,[e]or with dimension

H:Hammett’s acidity function,[e]

D~H rxn:j-th reaction enthalpy,[J/kmol]

K iT:?150,Reset time,[s]

K pT:?6,Proportional gain,[e]

k N:Pre-exponential factor,[m3s/kmol]

i:Index of equations in a generic ordinary differential equation system,[e]

j:Index of reaction,[e]

m:Mass,[kg]

m H,j:Hammett’s reaction rate coef?cient for the j-th reaction,[e]

N:Dimension of the state space,[e]

NC:Number of species,[e]

NR:Number of reactions,[e]

n

!

:Number of moles vector,[kmol]

NTU:?eUAT0$t dos=b r cool$b c p;cool$V cool,number of transfer units,[e]

PI:Proportional e Integral control action

R:gas constant?8314,[J/(kmol K)]

r:Reaction rate,[kmol/(m3s)]

RE:?m A,Reactivity Enhancement Factor?Distribution coef?cient for species A,[e] RE1:?m A,Reactivity Enhancement Factor?Distribution coef?cient for species A,[e] RE2:?m C,Reactivity Enhancement Factor?Distribution coef?cient for species C,[e] R H:?b c p;dos=b c p;mix,heat capacity ratio,[e]

R y:Reactivity number,[e]

s:Normalized parametric sensitivity coef?cient,[e]

St:?eUAT0$t dos=b r0$b c p;mix$V0,number,[e]

St ext:?eUAText$t dos=b r0$b c p;mix$V0,number referred to ambient dispersion and evaporation losses,[e]

t:time,[s]

T:Temperature,[K]

U:overall heat transfer coef?cient,[W/(m2K)]

V:volume,[m3]state space volume

_V:Volumetric?ow rate,[m3/s]

v:?V/V0,Dimensionless reactor volume,[e]

v dos:?V dos=V TOT;dos,Dimensionless dosing volume,[e]

x:Dependent variables vector(for an ODEs system)

Subscripts and superscripts

ad:Adiabatic

c:Referred to as the continuous aqueous phase

cool:Referred to as the coolant or the external jacket

dos:Dosing stream or dosing time

ext:Referred to as the external heat losses

IN:Referred to as an inlet stream

j:Referred to as the j-th reaction

MAX:Maximum value of a quantity or at the maximum value of a quantity

mix:Referred to as the reacting mixture

S.Copelli et al./Journal of Loss Prevention in the Process Industries28(2014)92e10099

mod:Referred to as the modi?ed divergence criterion

orig:Referred to as the original divergence criterion

QFS:“Quick onset”,“Fair conversion”,“Smooth temperature pro?le”

rif:Referred to as reference conditions

set:Referred to as the desired set point

0:Start of the dosing period

1:Referred to as the?rst reaction

2:Referred to as the second reaction

Greek symbols

D s ad;j:?eàD~H rxn;jT$n A;TOT;dos=b r0$V0$b c p;mix$T rif,dimensionless adiabatic tempera-

ture rise of the j-th reaction,[e]

g:?E att=RT

rif

,dimensionless activation energy,[e]

3:?m TOT;dos=m0,relative mass increase at the end of the semibatch period,[e] z:dimensionless concentration or conversion of a generic species,[e]k j:?expeg

j

$e1à1=sTT,dimensionless j-th reaction kinetic constant,[e]

r:?b r=b r

Dimensionless reacting mixture density,[e]

b r:Reacting mixture density,[kg/m3]

s:?T/T rif,dimensionless temperature,[e]

q:?_V

cool

$t dos=V cool,dimensionless coolant contact frequency into the jacket,[e] w:?t/t dos,dimensionless time,[e]

f:Generic constitutive model parameter

4:Generic dimensionless function,[e]?

v dosàzT=3r

dos

,v dos

!

$e1àzT,

Concentration function for the aromatic nitration of4-Chlorobenzotri?uoride,

[e]

41:?ev dosàz Càz DT$ez B;0tz CT=v dos$expeàm H;1$HT,Concentration function for the second reaction of the nitric acid oxidation of2-octanol,[e]

42:?z C$ez B;0tz CT=v dos$expeàm H;2$HT,Concentration function for the second re-action of the nitric acid oxidation of2-octanol,[e]

S.Copelli et al./Journal of Loss Prevention in the Process Industries28(2014)92e100 100

初中英语语法知识点地点副词

初中英语语法知识点:地点副词 初中英语语法知识点:地点副词 表示地点的副词和表示位置关系的副词统称为地点副词。表示地点的:here, there, home, upstairs, downstairs, anywhere, everywhere, nowhere, somewhere, abroad, elsewhere等。地点副词常放在动词后面,如果是及物动词,一般就放在宾语后面。如:I remember having seen him somewhere. 副词可以用作表语,主要是地点副词,但时间副词和其他副词有时也可以用作表语。常见的这类副词有:表示地点的:here, there, home, upstairs, downstairs, anywhere, everywhere, nowhere, somewhere, abroad, elsewhere等。表示位置关系的:above, below, down, up, out, in, across, back, along, over, round, around, away, near, off, on, inside, outside, past等。在表示位置关系的副词中,有些副词也可用作介词(如:above, over, beyond, around, below, down, up, in, along, near, off, on, past等),在没有宾语时就是副词,有宾语时就是介词,如: Come in, please. (副词) They live in the next room. (介词) Let's take along. (副词) Let's walk along this street. (介词) She looked around. (副词) They sat around the table. (介词) Let's go on with the work...(副词) What subject will you speak on? (介词) 地点副词在句中的位置地点副词常放在动词后面,如果是及物动词,一般就放在宾语后面。如:I remember having seen him somewhere. Wuxia films are popular in China. 地点副词和时间副词并列使用时,一般要把地点副词放在时间副词之前。如: We had a meeting here yesterday. He did the work carefully here yesterday. 如果地点状语很长时,也可以放在时间状语之后。如: He was born in 1940 in a small village at the foot of Mount Tai。地点副词常可以用作表语副词可以用作表语,主要是地点副词,时间副词和其他副词有时也可以用作表语。如: They are inside. 他们在里面。 How long will she be away? 她要离开多久? When will you be back? 你什么时候回来? You haven't been around much. 你很少到这边来。 He'll be round in

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中考英语语法考点重点解析 中考英语语法考点重点分析(一) 中考是一种比较激烈的选拔性考试,它承担着为高级中学挑选优秀学生的任务,所以它一定要有必要的难度和区分度,我们在认真分析了近五年上海中考原题语法和词汇部分后(20xx年至20xx年),下面扼要的总结了些上海中考的重点和考点,帮助广大学生熟悉考试的具体要求和重点,为学生们备战中考提供必要的帮助。 初中生因为刚涉及英语学习,语法方面还没有过多学复合句等问题,中考的考核绝大多数是以词法为主,20xx年到20xx年语法单项选择15道题,涉及句法的题寥寥,一般在2道左右。20xx年单项选择增加到20道考核句法的题也没有超过两道。这体现了中考英语语法考核重在考察各种词法,强调最基础的学习和积累,为以后的高中学习打下牢固的基本功。 冠词和代词注意的问题: 例:The scientists from United States live in Ninth street.(20xx 上海中考题) A. the……the B. /……the C. /……/ D. the……/ 解析:这是考察冠词的一道典型试题,需要强调的是学生要牢牢记住一些冠词的特殊用法,如普通名词构成专有名词一定要用the 如:the Great wall长城 the Shanghai Museum 上海博物馆 the New Oriental School 新东方学校。另外注意零冠词的用法:街道、广场、

公园的前面不用任何冠词。所以这道题是选择D。 例:Liu xiang and Yao ming are world-famous sports stars.____ of them have set a good example to us .(20xx年上海中考题) A. all B. neither C. both D. none 例:There are many new high-rises on ___ side of Huaihai Road .What a magnificent view!(20xx年上海中考题) A. either B. neither C. both D. all 解析:代词部分尤其是不定代词部分历来是考试的重点。Both 是指两者的全肯定,是说都怎么样,而all是说全部都,这是在指三个人或者以上;none是指三个人或者以上都不怎么样,是否定的概念,neither是说两者的都不怎么样,也是否定的概念;最重要的是either这个单词,它表示两者都怎么样和both一样是肯定的,但是只说一个或者是任何一个。就上面这道题来讲,第一题说刘翔和姚明都给我们树立了榜样,两者的都肯定,所以选择C. 后一题是说淮海路两旁都是高楼大厦,按道理来讲应该是选择both,但是注意side 是一边,这是个单数,所以是说任意一边都是高楼大厦,选择A. 动词方面需要注意的问题: 一、近意动词的辨析选择 例:The VIPs from 21 countries will ___the APEC in Shanghai this autumn.(20xx年上海中考题) A. hold B. take part in C. join D. attend

初中英语语法大全知识点总结

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6、副词(adv.):修饰动词、形容词或其他副词,说明时间、地点、 程度等。如:now, very, here, often, quietly, slowly. 7、冠词(art..):用在名词前,帮助说明名词。如:a, an, the. 8、介词(prep.):表示它后面的名词或代词与其他句子成分的关系。如in, on, from, above, behind. 9、连词(conj.):用来连接词、短语或句子。如and, but, before . 10、感叹词(interj..)表示喜、怒、哀、乐等感情。如:oh, well, hi, hello. 2、句子成分:英语句子成分分为七种:主语、谓语、宾语、定语、 状语、表语、宾语补足语。 1、主语是句子所要说的人或事物,回答是“谁”或者“什么”。通 常用名词或代词担任。如:I’m Miss Green.(我是格林小 姐) 2、谓语动词说明主语的动作或状态,回答“做(什么)”。主要由动词 担任。如:Jack cleans the room every day. (杰克每天打 扫房间) 3、表语在系动词之后,说明主语的身份或特征,回答是“什么”或者 “怎么样”。通常由名词、代词或形容词担任。如:My name is Ping ping .(我的名字叫萍萍) 4、宾语表示及物动词的对象或结果,回答做的是“什么”。通常由

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10、感叹词(interj..)表示喜、怒、哀、乐等感情。如:oh, well, hi, hello. 2、句子成分:英语句子成分分为七种:主语、谓语、宾语、定语、状语、表语、宾语补足语。 1、主语是句子所要说的人或事物,回答是“谁”或者“什么”。通常用名词或代词担任。如:I’m Miss Green.(我是格林小姐) 2、谓语动词说明主语的动作或状态,回答“做(什么)”。主要由动词担任。如:Jack cleans the room every day. (杰克每天打扫房间) 3、表语在系动词之后,说明主语的身份或特征,回答是“什么”或者“怎么样”。通常由名词、代词或形容词担任。如:My name is Ping ping .(我的名字叫萍萍) 4、宾语表示及物动词的对象或结果,回答做的是“什么”。通常由名词或代词担任。 如:He can spell the word.(他能拼这个词) 有些及物动词带有两个宾语,一个指物,一个指人。指物的叫直接宾语,指人的叫间接宾语。间接宾语一般放在直接宾语的前面。如:He wrote me a letter . (他给我写了一封信) 有时可把介词to或for加在间接宾语前构成短语,放在直接宾语后面,来强调间接宾语。如:He wrote a letter to me . (他给我写了一封信) 5、定语修饰名词或代词,通常由形容词、代词、数词等担任。如: Shanghai is a big city .(上海是个大城市)

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