temperature diagrams for preventing decomposition or side reactions in liquid–liquid semibatch...

Chemical Engineering Science 61 (2006) 3068–3078www.elsevier.com/locate/ces

Temperature diagrams for preventing decomposition or side reactions inliquid–liquid semibatch reactors

Francesco Maestri, Renato Rota∗

Politecnico di Milano, Dip. di Chimica, Materiali e Ingegneria Chimica “G. Natta” via Mancinelli 7 - 20131 Milano, Italy

Received 20 July 2005; received in revised form 18 November 2005; accepted 22 November 2005Available online 24 January 2006

Abstract

The operation of an indirectly cooled semibatch reactor in which an exothermic reaction occurs is usually considered safe if the characteristictime of the coreactant dosing is much higher than the characteristic times of all the other phenomena involved (chemical reaction and masstransfer), so that the conversion rate is controlled by the coreactant supply itself. Such operating conditions imply a small accumulation of thecoreactant in the system and are characterized by a temperature evolution which quickly approaches a target temperature and remains close toit throughout the dosing period, at the end of which the conversion is almost complete.

The so-called boundary diagrams are useful tools for identifying safe operating conditions without solving the mathematical model of thereactor. However, avoiding accumulation phenomena can be not sufficient for classifying a set of operating conditions as thermally safe whenthe maximum temperature reached by the system under normal operation exceeds a maximum allowable temperature (which can be relatedeither to safety problems, when dangerous decomposition reactions can be triggered, or to productivity problems, when side reactions cansignificantly lower the product yield above a given threshold temperature).

In this work the boundary diagrams for the prevention of excessive accumulation conditions in liquid–liquid semibatch reactors are coupledwith new diagrams, called temperature diagrams. These new diagrams, involving the same dimensionless parameters used for the representationof the boundary diagrams, allow determining—for a given set of operating conditions—the maximum temperature increase with respect to theinitial reactor temperature which can be expected to occur during normal operation. This information can be compared with the maximumallowable temperature for the reacting mixture. Then the operating conditions can be verified through the boundary diagrams in order to rejectconditions of excessive coreactant accumulation.

Several temperature diagrams are provided for various kinetically or diffusion controlled reactions with different reaction orders and theiruse together with a general procedure for calculating them is presented.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Semibatch reactors; Temperature diagrams; Boundary diagrams; Runaway; Safety; Scale-up

1. Introduction

Runaway phenomena in chemical reactors have been thor-oughly analyzed in the process safety literature of the last 30years. It is well known that a runaway reaction is a consequenceof the thermal loss of control of a reacting system in which anexothermic reaction occurs. This situation, also called thermalexplosion, is primarily responsible for an increase of the rateof the desired reaction and can also lead to the triggering of

∗ Corresponding author. Tel.: +39 0223993 154; fax: +39 0223993 180.E-mail address: [email protected] (R. Rota).

0009-2509/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.11.055

consecutive decomposition reactions of one or more compo-nents of the reacting mixture.

In order to evaluate the consequences of a runaway, the maindata required are the rate of heat evolution due to chemicalreactions and the heat transfer efficiency of the reactor. In par-ticular, it is necessary to know reaction enthalpy and heat ca-pacity of the reacting mixture, adiabatic temperature rise underprocess conditions, boiling point of the reacting mixture, tem-perature range in which dangerous decomposition reactions canbe triggered and their reaction enthalpy, amount and rate of gasevolution, effect of operational errors and impurities. Througha combination of these process information, Stoessel (1993)classified exothermic reaction processes into five classes as a

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F. Maestri, R. Rota / Chemical Engineering Science 61 (2006) 3068–3078 3069

function of the relative ranking of: process temperature, max-imum temperature that can be achieved by synthesis reactionas a consequence of cooling system failure (MTSR), boilingpoint of the solvent, maximum allowable temperature (MAT)to avoid decomposition reactions taking place. Situations char-acterized by MTSR values higher than the MAT values mustbe regarded as critical from a safety point of view: for in-stance, a less exothermic reaction system with a relativelylow decomposition temperature can be much more dangerousthan a more exothermic one with a very high decompositiontemperature.

It is important to stress that similar conclusions arise alsowhen a product decomposition or a side reaction which is notcritical for safety (since it is neither very exothermic nor pro-duces large amounts of gases) can take place above a thresh-old temperature value: such situations are very frequent in thefine chemical and pharmaceutical industries, where several pro-cesses involve products which can chemically degrade abovean experimentally determined temperature. Also in this case theMAT value must not be reached since the plant productivitywould be compromised. This means that, regardless the typol-ogy of problem we are dealing with (safety or productivity),one of the constraints we have to fulfill in the production plantis often summarized through a threshold temperature value thatcannot be exceeded neither during the normal reactor opera-tion nor under upset operating conditions, such as those arisingfrom a cooling system failure.

As thoroughly discussed in the literature (c.f. Steinbach,1999), if the coreactant accumulation in a semibatch reactor(SBR) increases above a critical value, a situation can arise inwhich, as the reaction ignites, the heat removal rate cannot bal-ance anymore the enthalpic contribution due to the chemical re-action: this results in a temperature jump which, depending onits entity, can exceed the MAT value. The most frequent causesof accumulation phenomena in SBRs are wrong assumptionson the reaction kinetics (which can be also associated to theuninvestigated presence of impurities), too high rates of core-actant supply, too low initial reaction temperatures, insufficientmixing or cooling system efficiency.

Among all the causes, we will focus on the too low initialtemperature, which primarily leads to a high coreactant accu-mulation in the system and, in most cases, to the thermal lossof control of the system itself as the reaction ignites. The re-sulting temperature jump can then reach the thermal range forthe triggering of the undesired decomposition event. For thisreason, in semibatch conditions a higher initial temperature isoften safer than a lower one in order to avoid accumulationphenomena and temperature jumps. However, this can be a nec-essary condition to prevent the exceeding of the MAT value(note that avoiding accumulation of the coreactant usually al-lows to avoid thermal runaways even when the cooling systemfails provided the coreactant supply is suddenly stopped), butnot a sufficient condition. If the initial temperature is too high,a situation can arise in which the coreactant accumulation isconfined below critical values but the initial temperature itselfis too close to the MAT value for avoiding the overshooting ofthe threshold temperature.

1.6

1.5

1.4

1.3

1.2

1.1

11 2 3 4 5 6 7 8 9 10 11

Exothermicity number "Ex"

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Rea

ctiv

ity n

umbe

r "R

y"�

= (

Tm

ax/T

0)m

ax

Ex=

Ex,

MIN

Ex=

Ex,

MIN

Ry=0.2

Ry=0.05

Ry=0.3Ry=0.4

Ry=Ry,QFS

Ry=10Ry,QFS

A

1 2 3 4 5 6 7 8 9 10 11Exothermicity number "Ex"

EXCESSIVE ACCUMULATION REGION

marginal ignition conditions

Tcool increasing

"QFS" conditions

INHERENTLY SAFE REGION

Ry=Ry,QFS

no ignition

A

(A)

(B)

Fig. 1. (A) temperature and (B) boundary diagram.

For the identification of low accumulation operating condi-tions in liquid–liquid SBRs in which exothermic reactions oc-cur, Steensma and Westerterp (1988, 1990, 1991) firstly devel-oped the so-called boundary diagrams method. Such diagramsprovide, in a suitable dimensionless space defined through an“exothermicity number” (Ex) and a “reactivity number” (Ry ,see Fig. 1(B)), a representation of the regions in which exces-sive accumulation and inherently safe operating conditions areexpected.

van Woezik and Westerterp (2000, 2001) extended thismethod to the case of multiple (consecutive) reactions, an-alyzing both theoretically and experimentally the nitric acidoxidation of 2-octanol to 2-octanone with further oxidation ofthe reaction products to unwanted carboxylic acids. However,this approach requires the knowledge of the kinetics of both thesynthesis and the decomposition (or side) reactions: becauseof money and time constraints this information is usually notavailable (at least for the unwanted reaction) and is lumped ina MAT value.

3070 F. Maestri, R. Rota / Chemical Engineering Science 61 (2006) 3068–3078

Recently the original works of Steensma and Westereterpon slow liquid–liquid reaction systems were improved byWesterterp and Molga (2004a,b), who extended the bound-ary diagrams for slow (1,1) reaction orders to the full rangeof cooling numbers of interest in the industrial practice andprovided some insights on how to practically recover the in-formation required by the boundary diagrams method and howto extend it to multiple reactions.

Moreover, Maestri and Rota (2005a,b) demonstrated the keyrole that the reaction kinetics can play in determining the shapeand location of the boundary diagrams for both slow and fastreaction regimes and provided a general procedure for calcu-lating such diagrams.

Thanks to these previous works, for many practical systemsthe problem of avoiding excessive coreactant accumulation ina semibatch reactor can be easily faced.

However, as previously mentioned, when dealing with anexothermic semibatch reaction system in which an unwantedreaction can be triggered above a threshold temperature the useof the boundary diagrams does not in general allow to classifyas acceptable a selected set of operating conditions. In fact, evenif operating conditions belonging to the excessive accumulationregion of the boundary diagram are normally characterized by ahigher reaction temperature rise than the so-called quick onset-fair conversion-smooth temperature profile (QFS) conditions(Steensma and Westerterp, 1990), it must be stressed that fora higher initial temperature even a lower temperature rise cancause the maximum reaction temperature to exceed the MATvalue.

As a consequence, it would be useful to find a way to in-tegrate the information that can be deduced from the bound-ary diagrams with another information, that is, the maximumtemperature value that can be expected under normal operatingconditions. Finding such a way (which has to be as simple asthe use of the boundary diagrams to be effective) is the mainaim of the present work.

As discussed in the following, the boundary diagrams forthe identification of excessive accumulation and inherently safeoperating conditions in liquid–liquid SBRs can be coupled witha new kind of diagrams, which have been called temperaturediagrams. Such diagrams, involving the same set of dimension-less parameters employed for the representation of the bound-ary diagrams, provide for a given set of operating conditionsthe highest ratio of the maximum to the initial reactor temper-ature that can be expected to occur. This was the missed in-formation to be compared with the MAT value. The procedurefor building the temperature diagrams is discussed and a num-ber of such diagrams for various operating conditions are pro-vided to allow end users to easily identify operating conditionscharacterized by both a sufficiently low coreactant accumula-tion and a maximum temperature value under normal operationconditions lower than the MAT value.

2. Mathematical model

Developing diagrams for the simple and reliable predictionof the maximum temperature attained in a SBR in which an

exothermic liquid–liquid reaction of the form

�AA + �BB → C + �DD (1)

occurs requires a preliminary discussion on how the processcan be modeled through the statement of a few reasonableassumptions.

In the following we assume that the microkinetic rate of reac-tion (1) can be expressed through a power-law type functionaldependence

r = kn,mCnACm

B (2)

derived from a data fitting of RC1 experiments performed inthe kinetically controlled regime.

The following model assumptions can be reasonably stated:

(1) reaction mass is perfectly macromixed,(2) influence of chemical reaction on the volume of the single

phase is negligible,(3) no phase inversions occur,(4) solubility of species A (the dosed coreactant) and C in the

continuous phase, c, and of components B (the componentinitially charged in the reactor) and D in the dispersedphase, d, is small (in other words, species A and C arealmost all in phase d, whereas species B and D are presentalmost only in phase c),

(5) chemical reaction takes place only in one of the two liquidphases: this situation is very common in many industrialprocesses (such as nitrations and oxidations), in which thecatalyst (typically a strong acid) is present only in onephase,

(6) heat effects are associated to the chemical reaction only,(7) at the beginning, the reaction mass is at the mean coolant

temperature, Tcool, which is assumed to remain constant forthe whole duration of the process (i.e., the reactor operatesunder isoperibolic conditions).

Since the equations constituting the mathematical model underthese assumptions have been thoroughly discussed elsewhere(Steensma and Westerterp, 1988, 1990, 1991; Maestri and Rota,2005a,b), they are only briefly summarized in the following.

The mass balance equation for the component B can be writ-ten in dimensionless form as (Steensma and Westerterp, 1988)

d�B

dϑ= �BtD

nB,0reffVr , (3)

where the effective conversion rate, reff (that depends on themicrokinetic rate expression (2) and on the phase in which thereaction occurs) can be in general determined by the role ofeither chemical reaction, mass transfer or coreactant supply.

Depending on the kinetic regime, Eq. (3) can be rearrangedin the form

d�B

dϑ= �A Da REslow,c/d fslow,c/d �n,m (4)


Table 1Expressions of the reactivity enhancement factor, RE, and of the function, f , for slow and fast reactions taking place in the dispersed or continuous phase.(Maestri and Rota, 2005a,b)

Reaction in the dispersed phase, d Reaction in the continuous phase, c

REslow,c/d

(�B

�A

)1−n

mmB

(�B

�A

)1−n

mnA

REfast,c/d6

db,0m

(m+1)/2B

C(1−n−m)/2B,0

(�B

�A

)1−n/2[

2DLd,B

(m + 1)kn,m,Rd

]1/26

db,0m

(n+1)/2A

C(1−n−m)/2B,0

(�B

�A

)(1−n)/2[2DLc,A

(n + 1)kn,m,Rc

]1/2

fslow,c/d(ϑ − �B)n(1 − �B)m

(�ϑ)n−1(ϑ − �B)n(1 − �B)m

(�ϑ)n

ffast,c/d(ϑ − �B)n/2(1 − �B)(m+1)/2(�ϑ)1−n/2

1 + 2.5�ϑ/(1 + �ϑ)

(ϑ − �B)(n+1)/2(1 − �B)m/2(�ϑ)(1−n)/2

1 + 2.5�ϑ/(1 + �ϑ)

for the kinetically controlled regime (Maestri and Rota, 2005a)and in the form

d�B

dϑ= �A Da REfast,c/d ffast,c/d �1/2

n,m (5)

for the diffusion controlled regime (Maestri and Rota, 2005b).In this equation Da=kn,m,RtDCn+m−1

B,0 is the Damköhler num-ber for a reaction with (n, m) reaction orders, which containsthe information on the dosing time, while �n,m=exp[�(1−1/�)]is the dimensionless reaction rate constant, that is, the ratio ofthe reaction rate constant to the same quantity evaluated at areference temperature, TR; � = E/(RT R) is the dimensionlessactivation energy and � = T/TR is the dimensionless tempera-ture.

The expressions for the calculation of the reactivity enhance-ment factor, RE, and for the function, f , have been derivedelsewhere (Maestri and Rota, 2005a,b) and they are summa-rized in Table 1.

The energy balance equation for the reactor yields (Steensmaand Westerterp, 1988)

(1 + RH �ϑ)d�

dϑ= ��ad,0

d�B

dϑ− [U∗ Da(1 + �ϑ) + �RH ](� − �eff

cool), (6)

where d�B/dϑ is the only factor that depends on the ki-netic regime (slow or fast reactions) and T eff

cool = (U∗ Da(1 +�ϑ)Tcool + �RH TD)/(U∗ Da(1 + �ϑ) + �RH ) is an effectivecooling temperature which takes into account the effects ofboth the heat removal by the coolant and the sensible heat ofthe dosing stream.

These equations are valid for 0 < ϑ�1, that is up to the endof the dosing period. It can be easily demonstrated that the sameequations can be extended to ϑ > 1 by substituting everywherethe (ϑ − �B)/ϑ or (ϑ − �B) terms with (1 − �B), the �ϑ termswith � and by setting TD ≡ T in the definition of the effectivecooling temperature.

In order to identify an excessive accumulation region thepredictions of this model can be lumped using two dimension-less parameters, called exothermicity (Ex) and reactivity (Ry)

number. They are defined as the ratios of an exothermicity anda reactivity factor to a cooling factor, whose expressions can

be derived from the mass and energy balance equations pre-viously reported. Exothermicity and reactivity numbers can becalculated through the following equations (Maestri and Rota,2005a,b):

Ex = �

�2cool

��ad,0

U∗ Da + �RH

= �

�2cool

��ad,0

�(Co + RH ), (7)

Ry = �A Da RE �(�cool)

U∗ Da + �RH

= �A Da RE �(�cool)

�(Co + RH )(8)

for the kinetically controlled (or slow reaction) regime and

Ex = �

2�2cool

��ad,0

U∗ Da + �RH

= �

2�2cool

��ad,0

�(Co + RH ), (9)

Ry = �A Da RE �1/2(�cool)

U∗ Da + �RH

= �A Da RE �1/2(�cool)

�(Co + RH )(10)

for the diffusion controlled (or fast reaction) regime.A boundary diagram is shown, for the sake of example, in

Fig. 1(B). The excessive accumulation region (EAR) is boundedby the continuous line and is surrounded by the no ignition re-gion (for low Ry values), the QFS region (for high Ry values)and by an inherently safe region. The latter is characterized ei-ther by Ex values lower than the minimum Ex value of theEAR boundary, Ex,MIN, or by Ry values larger than the max-imum Ry value of the EAR boundary, Ry,QFS . More detailson the development and use of the boundary diagrams can befound elsewhere in the literature (c.f. Westerterp and Molga,2004a,b; Maestri and Rota, 2005a).

3. Thermally safe operating conditions

Following a procedure similar to that employed for the cal-culation of the boundary diagrams and using the same dimen-sionless parameters introduced for their representation (i.e., theexothermicity number, Ex , and the reactivity number, Ry), it ispossible to build new diagrams that, for every (Ex, Ry) couple,provide the maximum value of the Tmax/T0 ratio which canbe expected for the chosen set of operating conditions. Suchdiagrams, an example of which is represented in Fig. 1(A) to-gether with the related boundary diagram reported in Fig. 1(B),will be referred to in the following as temperature diagrams.


1.6

1.5

1.4

1.3

1.2

1.1

11 2 3 4 5 6 7 8 9 10 11


1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

01 2 3 4 5 6 7 8 9 10 11


Rea

ctiv

ity n

umbe

r "R

y"�

=(T

max

/T0)

max

Ex=

Ex,

MIN

Ex=

Ex,

MIN

Ry=0.05

Ry=0.1

Ry=0.2

Ry=0.3

Ry=0.4

Ry=Ry,QFS

Ry=Ry,QFS

Ry=10Ry,QFS

(A)

(B)

Fig. 2. Slow reaction regime. Reaction in the dispersed phase. Co = 10,RH = 1, n = 1, m = 1. 0.025 < �A Da RE < 14, 0.3 < �< 0.55, 32 < �< 42,0.29 <��ad,0 < 0.7: (A) temperature diagram and (B) boundary diagram.

Before discussing the development of such diagrams it isworthwhile illustrating their use. Given a set of operating con-ditions represented by the point A on the boundary diagramof Fig. 1(B), we know from such a diagram that no danger-ous coreactant accumulation is expected since point A is out-side the EAR. Moving to the related temperature diagram ofFig. 1(A), the same point can be identified from the valuesof Ex (in this case equal to 4) and Ry (in this case equal toRy,QFS�0.78). From the temperature diagram the value ofthe parameter � = (Tmax/T0)max�1.13 can be read. Knowingthe initial reaction temperature T0, it can be concluded that themaximum temperature under normal operation will not exceed1.13 T0: this is the value to be compared with the MAT value.

In the following a general procedure for building the temper-ature diagrams is discussed and a number of such diagrams areprovided for different combinations of reaction phase, kineticregime and Co, RH , n and m values.

1.45

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

10 1 2 3 4 5 6


1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 1 2 3 4 5 6


Rea

ctiv

ity n

umbe

r "R

y"�

=(T

max

/T0)

max

Ex=

Ex,

MIN

Ex=

Ex,

MIN

Ry=0.05

Ry=0.2

Ry=0.4

Ry=0.6

Ry=Ry,QFS

Ry=Ry,QFS

Ry=10Ry,QFS

(A)

(B)


In the mass and energy balance equations (4)–(6) or—analo-gously—in expressions (7)–(10) of the exothermicity and reac-tivity numbers, 8 dimensionless parameters appear: �A Da RE,�, �, RH , ��ad,0, Co, n, and m. Assigning a value to each of theseparameters in compliance with its accepted range and lettingthe coolant temperature as a variable parameter, the functionaldependence between the exothermicity and reactivity numbers,Ry = �(Ex), is univocally identified. Eqs. (7) and (8) or (9)and (10) are the parametric form of this functional dependence(for the slow and the fast reaction regime, respectively), sincethey provide, for each value of the coolant temperature, a cou-ple of related values of Ry and Ex as shown, for the sake ofexample, in Fig. 1(B). As discussed elsewhere (Maestri andRota, 2005a), from Eqs. (7) and (8) or (9) and (10), it is evi-dent that the single Ry = �(Ex) line (i.e., a line representingthe functional dependence between Ry and Ex) on the singleboundary diagram (i.e., for Co, RH , n, m values assigned and


1.6

1.5

1.4

1.3

1.2

1.1

12 4 6 8 10 12 14 16 18 20 22



0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

02 4 6 8 10 12 14 16 18 20 22

Rea

ctiv

ity n

umbe

r "R

y"�

=(T

max

/T0)

max

Ry=0.05

Ry=0.2

Ry=0.4

Ry=Ry,QFS

Ry=10Ry,QFS

Ex=

Ex,

MIN

Ex=

Ex,

MIN

Ry=Ry,QFS

(A)

(B)


constant) is identified through the values of the following threedimensionless groups:

��ad,0

�(Co + RH )= const,

�A Da RE

�(Co + RH )= const,

� = const

(11)

that must be also constant along the single Ry = �(Ex) line.As previously mentioned, the number of parameters that

must be assigned to solve the mathematical model of thereactor at the current coolant temperature, as previously men-tioned, is equal to eight. Since 4 of them must be constantwhen a single boundary diagram is considered, the numberof independent parameters is equal to 4. These must also ful-fill the three constraints (11). It follows that the number ofsets of parameters which generate the same Ry = �(Ex) lineon a given boundary diagram is equal to ∞1. These sets of

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

10 1 2 3 4 5 6 7 8 9 10

Exothemicity number "E"

Ry=Ry,QFS

Ry=0.2

Ry=0.05

Ry=0.1

Ry=0.3

Ry=0.4

Ry=10Ry,QFS

Ry=Ry,QFS

Ex=

Ex,

MIN

Ex=

Ex,

MIN

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 1 2 3 4 5 6 7 8 9 10


�=

(Tm

ax/T

0)m

axR

eact

ivity

num

ber

"Ry"

(A)

(B)

Fig. 5. Slow reaction regime. Reaction in the dispersed phase. Co = 10,RH =2.5, n=1, m=1. 0.025 < �A Da RE < 14, 0.3 < �< 0.55, 32 < �< 42,0.29 <��ad,0 < 0.7: (A) temperature diagram and (B) boundary diagram.

parameters can be generated from each member of the singlefamily multiplying by the same factor K the parameters �A Da

RE, �, ��ad,0 and U∗Da, while keeping the remaining 4 param-eters (i.e., RH , �, n, m) constant. It should be noticed that thisallows to keep Co = U∗Da/� constant as well as to fulfillconstraints (11).

Selecting a value of the coolant temperature means to selecta point on the boundary diagram (i.e., a couple of Ex and Ry

values), where it is possible to solve ∞1 systems of the massand energy balance equations (4) or (5) and (6) with the properinitial conditions, being the factor K the parameter that gener-ates the ∞1 model equations. For each K value (that allows toobtain values of the parameters �A Da RE, �, � and ��ad,0 in-side proper ranges) the model equations can be solved. There-fore, the dimensionless temperature T/T0 as a function of thedimensionless time ϑ can be computed and its maximum valueTmax/T0 can be identified. Scanning locally (i.e., at the currentvalues of Ex and Ry) the range of variation of the K parameter,


1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

10 2 4 6 8 10 12


Ex=

Ex,

MIN

Ex=

Ex,

MIN

Rea

ctiv

ity n

umbe

r "R

y"�

=(T

max

/T0)

max

Ry=0.05

Ry=0.1

Ry=0.2

Ry=0.3Ry=0.4

Ry=Ry,QFS

Ry=10Ry,QFS

Ry=Ry,QFS

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 2 4 6 8 10 12


(A)

(B)

Fig. 6. Slow reaction regime. Reaction in the dispersed phase. Co = 10,RH =0.4, n=1, m=1. 0.025 < �A Da RE < 14, 0.3 < �< 0.55, 32 < �< 42,0.29 <��ad,0 < 0.7: (A) temperature diagram and (B) boundary diagram.

it is possible to compute a set of Tmax/T0 values whose maxi-mum value, �= (Tmax/T0)max, is the maximum expected ratioof the reaction temperature to the initial reactor temperatureat the current Ex and Ry values on the considered boundarydiagram. In other words, a K value able to locally maximizethe ratio of the maximum to the initial reactor temperature isselected.

By repeating the same procedure on several Ry=�(Ex) linesat a constant Ry value a functional dependence �=(Ex) at thecurrent Ry value can be generated. In Fig. 1(A) such functionaldependences for different Ry values are represented. Note thatRy varies in a range selected on the basis of the extension andlocation of the related boundary diagram, i.e., the boundarydiagram calculated for the same reaction phase, kinetic regimeand for the same Co, RH , n, m values. When a set of operatingconditions for a liquid–liquid SBR is selected (from which avalue of the Ex and Ry parameters can be computed, according

1.6

1.5

1.4

1.3

1.2

1.1

10 2 4 6 10 128

Exothemicity number "Ex"

0.3

0.25

0.2

0.15

0.1

0.05

00 2 4 6 8 10 12


Rea

ctiv

ity n

umbe

r "R

y"

� =

(Tm

ax/T

0)m

ax

Ex=

Ex,

MIN

Ry=Ry,QFS

Ry=Ry,QFS

Ry=0.15

Ry=0.1

Ry=0.05

Ry=0.01

Ry=10Ry,QFS

Ex=

Ex,

MIN

(A)

(B)


to the relations (7) and (8) or (9) and (10)), the temperaturediagrams can be used to check whether

�T0 < MAT. (12)

If check (12) is satisfied, two situations may be possible:

(a) the calculated (Ex , Ry) point belongs to the Ex < Ex,MINzone of the temperature diagram (see Fig. 1(A)) or tothe Ex > Ex,MIN zone with Ry �Ry,QFS . In this case, theselected operating conditions can be accepted since theycannot lead to the triggering of unwanted reactions andthey belong to the inherently safe region of the boundarydiagram where no excessive coreactant accumulation canoccur,

(b) the calculated (Ex, Ry) point belongs to the Ex > Ex,MINzone of the temperature diagram (see Fig. 1(A)) withRy < Ry,QFS . In this case, it is necessary to refer tothe related boundary diagram to check whether the


1.6

1.5

1.4

1.3

1.2

1.1

10 2 4 6 108 12


2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 2 4 6 8 10 12


Rea

ctiv

ity n

umbe

r "R

y"

Ex=

Ex,

MIN

Ex=

Ex,

MIN

Ry=0.05

Ry=0.2

Ry=0.4

Ry=0.6Ry=0.8

Ry=Ry,QFS

Ry=10Ry,QFS

Ry=Ry,QFS

�=

(Tm

ax/T

0)m

ax

(A)

(B)

Fig. 8. Slow reaction regime. Reaction in the dispersed phase. Co = 10,RH =1, n=0.5, m=1. 0.025 < �A Da RE < 14, 0.3 < �< 0.55, 32 < �< 42,0.29 <��ad,0 < 0.7: (A) temperature diagram and (B) boundary diagram.

aforementioned point belongs to the safe region, to the ex-cessive accumulation region or to the no ignition region. Itmust in fact be emphasized that even operating conditionswhich satisfy check (12) cannot be considered thermallysafe if they imply an excessive coreactant accumulationbecause such operating conditions are typically character-ized by sudden temperature and conversion jumps that areresponsible for a bad control of the reacting system andcan become very dangerous in case of cooling system fail-ures. Analogously, no ignition operating conditions mustbe avoided, since they imply low reactor productivity.

In Figs. 2–12 several couples of temperature and boundarydiagrams are reported for different combinations of the reac-tion phase, the kinetic regime and the Co, RH , n and m val-ues. In particular, the case of kinetically controlled reactionstaking place in the dispersed phase has been analyzed since itcorresponds to the most critical typology of liquid–liquid SBR

1.6

1.5

1.4

1.3

1.2

1.1

10 2 4 6 8 10 12


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 2 4 6 8 10 12


Rea

ctiv

ity n

umbe

r "R

y"�

=(T

max

/T0)

max

Ex=

Ex,

MIN

Ex=

Ex,

MIN

Ry=0.05

Ry=0.1

Ry=0.2

Ry=0.3Ry=0.4

Ry=Ry,QFS

Ry=Ry,QFS

Ry=10Ry,QFS

(A)

(B)


from the safety point of view (Westerterp and Molga, 2004a,b).These diagrams allow for a simple and fast analysis of theoperating conditions of a SBR from the thermal safety pointof view. However, it is worthwhile mentioning that the proce-dures presented in this work for the temperature diagrams andin Maestri and Rota (2005a) for the boundary diagrams allowto build the temperature and boundary diagrams for any com-bination of reaction phase, kinetic regime and Co, RH , n andm values.

The ranges of Ex and Ry to which the single temperaturediagram is extended are strictly related to the location and ex-tension of the coupled boundary diagram, so that, for a givenreaction phase and kinetic regime, the same sensitivity of theaforementioned ranges with respect to the Co, RH , n and m

parameters can be expected for both boundary and temperaturediagrams (c.f. Maestri and Rota, 2005a,b).

It is interesting to note that on the single temperature diagramat Ry values lower than Ry,QFS the � vs. Ex curves typically


1.6

1.5

1.4

1.3

1.2

1.1

10 2 4 6 8 10 12

Ex=

Ex,

MIN

Ry=0.05

Ry=0.1

Ry=0.2

Ry=0.3Ry=0.4

Ry,Ry,QFS

Ry,10Ry,QFS


�=

(Tm

ax/T

0)m

ax

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 2 4 6 8 10 12


Rea

ctiv

ity n

umbe

r "R

y"

Ex=

Ex,

MIN

Ry=Ry,QFS

(A)

(B)

Fig. 10. Slow reaction regime. Reaction in the dispersed phase. Co = 10,RH =1, n=1, m=0.5. 0.025 < �A Da RE < 14, 0.3 < �< 0.55, 32 < �< 42,0.29 <��ad,0 < 0.7: (A) temperature diagram and (B) boundary diagram.

exhibit a maximum. Such a behavior is expected since whilethe � vs. Ex curve at Ry = Ry,QFS is representative of situ-ations in which the maximum reactor temperature is less thanor equal to the local target temperature (Steensma and West-erterp, 1988), the same curves at Ry < Ry,QFS and in the Ex

range corresponding to the EAR are representative of situationsin which the maximum reactor temperature can be higher thanthe local target value. The entity of such a temperature exceed-ing is normally much higher than the variations of the targettemperature itself with Ry (and hence with the related initialtemperature values).

As it can be observed from the temperature diagrams repre-sented in Figs. 2–12, for Ry values much lower than Ry,QFS themaximum � values may be of the order of 1.5; this means thatif the initial reaction temperature is 300 K the maximum tem-perature attained by the reacting mixture can reach 450 K, tem-perature which is well above the normal boiling point of almostall the solvents normally employed in the industrial practice.

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

10 1 2 3 4 5 6


� =

(T

max

/T0)

max

Ex=

Ex,

MIN

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5 6


Rel

ativ

ity n

umbe

r "R

y"

Ex=

Ex,

MIN

Ry=Ry,QFS

(A)

(B)

Ry=0.4

Ry=0.2

Ry=0.6

Ry=0.8

Ry=1.0

Ry=1.2Ry=1.4

Ry=Ry,QFS

Ry=10Ry,QFS

Fig. 11. Fast reaction regime. Reaction in the dispersed phase. Co = 10,RH = 1, n = 1, m = 1. 0.025 < �A Da RE < 14, 0.3 < �< 0.55, 32 < �< 42,0.29 <��ad,0 < 0.7: (A) temperature diagram and (B) boundary diagram.

On the other hand, for Ry values greater than or equal toRy,QFS , the maximum � values corresponding to the exploredranges of the model parameters are of the order of 1.2: thisanalogously means that, under inherently safe conditions, if theinitial reactor temperature is equal to 300 K the maximum tem-perature will not be higher than 360 K regardless the value ofthe Ex parameter. However, much lower values of the tempera-ture increase are expected for lower values of Ex . In particular,for Ex < Ex,MIN and always referring to the investigated rangesof parameters, a maximum value of � ≡ 1.1 is expected.

4. Conclusions

Semibatch reactors in which exothermic reactions takeplace should never be operated under conditions of exces-sive coreactant accumulation because, as the desired reactionignites, a thermal loss of control of the system may oc-cur which can cause the temperature to raise so much that


1.45

1..4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

10 2 4 6 8 10 12


� =

(T

max

/T0)

max

0.04

0.035

0.03

0.025

0.02

0.015

0.010 2 4 6 8 10 12


Rea

ctiv

ity n

umbe

r "R

y"

Ex=

Ex,

MIN

Ry=0.01

Ry=0.015

Ry=0.02Ry=0.025

Ry=Ry,QFS

Ry=Ry,QFS

Ry=10Ry,QFS

Ex=

Ex,

MIN

(A)

(B)

Fig. 12. Slow reaction regime. Reaction in the continuous phase. Co = 10,RH = 1, n = 1, m = 1. 0.025 < �A Da RE < 14, 0.3 < �< 0.55, 32 < �< 42,0.29 <��ad,0 < 0.7: (A) temperature diagram and (B) boundary diagram.

decomposition reactions of one or more components of the re-acting mixture can be triggered. Such events, which can be de-tected through calorimetric experiments, are normally accom-panied by a much higher exothermicity and by the evolutionof gases, which are responsible for an increase of the systempressure at rates that can be dangerous.

However, even when dangerous accumulation is avoided, themaximum temperature reached by the system during normaloperation can overcome the maximum allowable temperature(MAT), which can be related either to safety problems (exother-mic or gas producing decomposition reactions) or to produc-tivity problems (unwanted side or consecutive reactions).

While the boundary diagrams are a powerful tool for iden-tifying excessive accumulation operating conditions in SBRs,they do not provide direct information about the temperatureincrease that can be expected to occur in the reacting system.

In this work the temperature diagrams have been introduced.Such diagrams, using the same dimensionless parameters in-volved in the representation of the boundary diagrams, allow de-termining the aforementioned maximum temperature increase

without solving the mathematical model of the reactor and pro-viding the possibility to extend the laboratory information toscaled-up systems.

The two typologies of diagrams (i.e., temperature and bound-ary diagrams) are strictly related from the point of view oftheir use: once a set of operating conditions for a liquid–liquidSBR has been selected, the temperature diagrams must be usedfirstly to check whether the temperature increase can reach theMAT value. Then, for conditions involving Ex > Ex,MIN andRy < Ry,QFS , the boundary diagrams must be used to rejectexcessive accumulation or no-ignition operating conditions.

Notation

A heat transfer area of the reactor (associated to thejacket and/or the coil), m2

C molar concentration, kmol/m3

Co U∗Da/�, cooling number, dimensionlessC̄P molar heat capacity, kJ/(kmol K)db,0 db at hold up of the dispersed phase approaching

zero, mD diffusivity, m2/sDa =Kn,m,RtDCn+m−1

B,0 , Damköhler number for(n, m) order reactions, dimensionless

E activation energy, kJ/kmolEx exothermicity number, Eqs. (7) and (9), dimen-

sionlessf function of the dimensionless time and conver-

sion of B in Eqs. (4) and (5), dimensionlesskn,m reaction rate constant, m3(n+m−1)/(kmoln+m−1 s)K multiplying factor of the �A Da RE, �, ��ad,0 and

U∗Da parameters, dimensionlessm equilibrium distribution coefficient (mA =

CA,c/CA,d ; mB = CB,d/CB,c, dimensionlessMAT maximum allowable temperature, KMTSR maximum temperature reached due to the syn-

thesis reaction, Kn number of moles, kmolr reaction rate referred to the total liquid volume,

kmol/(m3 s)R gas constant = 8.314 kJ/(kmol K)

RE reactivity enhancement factor in Eqs. (4) and (5),dimensionless

RH heat capacity ratio, dimensionlessRy reactivity number, Eqs. (8) and (10), dimension-

lesst time, sT temperature, KU overall heat transfer coefficient, kW/(m2 K)

U∗Da (UA)0tD/(̃cC̃P,cVc) modified Stanton number,dimensionless

V liquid volume, m3

Subscripts and Superscripts

ad adiabaticA, B, C, D components A, B, C and D


b in the dispersed phase drop diameter db

c continuous phasecool coolantd dispersed phaseD dosing stream or dosing timeeff effectivefast fast reaction regimeH in the heat capacity ratio RH

L in the liquid phasem order of reaction respect to component B

max maximum value of a quantity or at the maximumvalue of a quantity

MIN in Ex,MINn order of reaction respect to component A

QFS inRy,QFS

r reactionR referenceslow slow reaction regimex in the exothermicity number Ex

y in the reactivity number Ry

0 start of the semi batch period

Greek letters

� E/(RT R), dimensionless activation energy, dimen-sionless

�H̃ reaction enthalpy, kJ/kmol�Tad,0 adiabatic temperature rise, K� relative volume increase at the end of the semibatch

period, dimensionless� molar conversionϑ t/tD , dimensionless time, dimensionless� k/kR , dimensionless reaction rate constant, dimen-

sionless� stoichiometric coefficient, dimensionless̃ molar density, kmol/m3

� T/TR , dimensionless temperature, dimensionless� functional dependence between Ry and Ex

� (Tmax/T0)max, maximum dimensionless tempera-ture rise, dimensionless

functional dependence between � and Ex at as-signed Ry

References

Maestri, F., Rota, R., 2005a. Thermally safe operation of liquid–liquidsemibatch reactors. Part I: single kinetically controlled reactions witharbitrary reaction order. Chemical Engineering Science 60, 3309–3322.

Maestri, F., Rota, R., 2005b. Thermally safe operation of liquid–liquidsemibatch reactors. Part II: single diffusion controlled reactions witharbitrary reaction order. Chemical Engineering Science 60, 5590–5602.

Steensma, M., Westerterp, K.R., 1988. Thermally safe operation of a cooledsemi-batch reactor. Slow liquid–liquid reactions. Chemical EngineeringScience 43 (8), 2125–2132.

Steensma, M., Westerterp, K.R., 1990. Thermally safe operation of asemibatch reactor for liquid–liquid reactions. Slow reactions. Industrialand Engineering Chemistry Research 29, 1259–1270.

Steensma, M., Westerterp, K.R., 1991. Thermally safe operation of asemibatch reactor for liquid–liquid reactions. Fast reactions. ChemicalEngineering Technology. 14, 367–375.

Steinbach, J., 1999. Safety assessment for chemical processes. Wiley–VCH,Weinheim.

Stoessel, F., 1993. What is your thermal risk? Chemical Engineering Progress,68–75.

van Woezik, B.A.A., Westerterp, K.R., 2000. The nitric acid oxidation of2-octanol. A model reaction for multiple heterogeneous liquid–liquidreactions. Chemical Engineering Process 39, 521–537.

van Woezik, B.A.A., Westerterp, K.R., 2001. Runaway behavior and thermallysafe operation of multiple liquid–liquid reactions in the semibatch reactor.The nitric acid oxidation of 2-octanol. Chemical Engineering Process 41,59–77.

Westerterp, K.R., Molga, E.J., 2004a. No more runaways in fine chemicalreactors. Industrial and Engineering Chemistry Research 43 (16),4585–4594.

Westerterp, K.R., Molga, E.J., 2004b. Runaway prevention in liquid–liquidsemibatch reactors. Inzynieria Chemiczna i Procesowa 25 (3/4),2041–2050.

temperature diagrams for preventing decomposition or side reactions in liquid–liquid semibatch...

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