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Jun 28, 2007 - In this work a procedure for easily selecting safe and productive operating conditions of homogeneous semibatch reactors in which exoth...
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Ind. Eng. Chem. Res. 2007, 46, 5333-5339

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Safe and Productive Operation of Homogeneous Semibatch Reactors Involving Autocatalytic Reactions with Arbitrary Reaction Order Francesco Maestri and Renato Rota* Dipartimento di Chimica, Materiali e Ingegneria Chimica “G. Natta”, Politecnico di Milano, Via Mancinelli 7, 20131 Milano, Italy

In this work a procedure for easily selecting safe and productive operating conditions of homogeneous semibatch reactors in which exothermic autocatalytic reactions are performed is developed and presented. Such a procedure is based on the boundary and temperature diagram method previously proposed in the literature for nonautocatalytic reactions. In the autocatalytic case an additional parameter appears in the mathematical model of the reactor as well as in the safety criterion, that is, the initial catalyst amount which is directly related to the coreactant accumulation in the system. A number of diagrams for identifying inherently safe operating conditions is provided, together with some rules of thumb for their use, allowing for easily selecting safe and productive operating conditions of homogeneous semibatch reactors in which exothermic reactions with autocatalytic behavior occur. Introduction The operating conditions of an indirectly cooled semibatch reactor (SBR) in which an exothermic reaction is performed can be considered safe if they correspond to a sufficiently low coreactant accumulation, so that the cooling system can control the heat evolved by the chemical reaction. Under such conditions the reaction temperature rise can also be limited, making the process safe and productive. The prevention of runaway phenomena in batch and semibatch reactors is a very common problem in the chemical industry that has been deeply investigated in the process safety literature of the past 30 years from the well-known accident of 1976 in Seveso, Italy. However, for a practical solution of the problem every scaleup procedure of such processes from the laboratory or pilot to the industrial scale must be simple and general at the same time. Such goals are often not easy to fit together, taking into account that especially in the fine chemical and pharmaceutical industries a wide range of products in relatively small amounts are produced and that a detailed mathematical model of the single process is often not available. The methods presented in the literature for the selection of safe and productive operating conditions of SBRs in which exothermic reactions are performed are all intended to minimize the coreactant accumulation in the system. On this basis Hugo et al.1-3 first developed a semiempirical criterion for homogeneous SBRs that has been then extended to heterogeneous (liquid-liquid) reactors by Steensma and Westerterp4-6 and Westerterp and Molga.7-9 These authors provided a more rigorous definition of coreactant accumulation in an SBR through the introduction of a target temperaturetime profile, to which the actual one can be compared to classify the thermal behavior of the reactor itself: such information is then represented in a suitable dimensionless space through the boundary diagrams, which allow easy discrimination between safe and excessive accumulation operating conditions without solving the mathematical model of the reactor. An experimental validation of this safety criterion has been carried out by van Woezik and Westerterp,10,11 who analyzed * To whom correspondence should be addressed. Fax: +39 0223993180. E-mail: [email protected].

the nitric acid oxidation of 2-octanol to 2-octanone with further oxidation of the reaction product to unwanted carboxylic acids. Maestri and Rota12-14 analyzed the dependence of the conclusions drawn through the boundary diagrams on the whole set of estimated kinetic parameters for both homogeneous and heterogeneous semibatch reactors, proving that unjustified assumptions on the reaction kinetics cannot be accepted for a reliable application of the method: such a conclusion holds for both heterogeneous and homogeneous reaction systems and makes a kinetic investigation (usually performed through calorimetric techniques) of crucial importance. When a threshold temperature (in the following referred to as the maximum allowable temperature, MAT) must not be exceeded during normal or during upset reactor operation, limiting accumulation phenomena can be just a necessary and not a sufficient condition to accept a given set of operating conditions. Such a temperature limitation can typically arise from safety problems (when the reaction mass can undergo a strongly exothermic and/or gas-producing decomposition reaction) or productivity problems (when above a threshold temperature, one or more side reactions can lead to the production of excessive amounts of unwanted impurities).14 For these reasons, Maestri and Rota15 developed a new typology of diagrams (called temperature diagrams) to be coupled with the boundary diagrams: such diagrams allow to safe prediction of the maximum temperature increase with respect to the initial temperature to be expected for a given set of operating conditions. The method has been then validated by Maestri and Rota,16 who analyzed the industrial process for the nitration of N-(2-phenoxyphenyl)methanesulfonamide to N-(4-nitro-2-phenoxyphenyl)methanesulfonamide, identifying with a minimum number of calorimetric experiments a set of operating conditions characterized by a low nitric acid accumulation in the system, which is crucial for minimizing the amounts of unwanted byproducts. Very recently Molga et al.17 repeated (even only for the particular case of (1,1) reaction order kinetics) the general analysis for homogeneous SBRs previously presented by Maestri and Rota,14 finding the same results in terms of inherently safe region boundaries for (1,1) reaction order kinetics. The method of boundary and temperature diagrams as developed in the cited literature cannot be applied to autocata-

10.1021/ie070427f CCC: $37.00 © 2007 American Chemical Society Published on Web 06/28/2007

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lytic reaction systems.7-9 However, reaction systems in which one of the products behaves as a catalyst are commonly encountered in industrial practice.10,11 For these reasons in this work the method of boundary and temperature diagrams has been extended to homogeneous SBRs in which an exothermic reaction with autocatalytic behavior is performed. The safety criterion developed for autocatalytic systems is based on diagrams involving the initial catalyst amount as an additional parameter with respect to the nonautocatalytic case. Such diagrams allow end users dealing with homogeneous autocatalytic SBRs to easily select operating conditions characterized by both a low coreactant accumulation and peak reaction temperatures lower than a threshold value. Mathematical Model Developing a criterion for selecting safe and productive operating conditions of homogeneous SBRs in which an exothermic autocatalytic reaction occurs requires a preliminary discussion on how the system can be modeled. We assume that at time equal to zero the SBR is filled with a molar quantity nB,0 of component B at a temperature T0 and that the feed of a molar quantity nA,1 of the coreactant A is started at a constant molar rate FA,D. In the system, the following overall reaction occurs:

A + B f 2B + C

(1)

The rate of this reaction can be described through a power law type functional dependence of the form

r ) kn,mCnACmB

(2)

A number of reacting systems that exhibit an autocatalytic behavior can be described through the kinetic scheme in eq 1:18-20 an example is provided by nitric acid oxidation reactions of organic compounds.10,11 According to the reaction scheme in eq 1 and to the kinetic expression in eq 2, component B behaves as a catalyst, so that a certain amount of this component must be present at time equal to zero for the reaction to take place.18 The kinetic parameters appearing in eq 2 can be estimated through a fitting of the results of proper calorimetric experiments (e.g., ARC, Phi-TEC II, RC1). A simple and realistic mathematical model of the reactor can be derived on the basis of the same assumptions stated elsewhere in the literature for nonautocatalytic systems.14 The mass balance equation for the ith chemical species is

dni/dt ) Fi,D ( reffVr

(3)

where Fi,D is the molar feed rate of the ith component, which differs from zero for species A only and the ( signs hold if the ith component is a product or a reactant, respectively. Moreover, reff is the effective conversion rate that can be determined either by the chemical reaction or by the coreactant supply rate. The material balance eqs 3 are coupled with the initial conditions: ni(t ) 0) ) ni,0, where ni,0 differs from zero for B only. Defining now the relative molar amount of the ith component as ζi ) ni/nA,1 and performing suitable combinations of the mass balance eqs 3, the following integral expressions of the concentrations of A and B can be derived:

CA )

nA,1(ϑ - ζC) Vr,0 (1 + ϑ)

(4)

CB )

nA,1(ζB,0 + ζC) Vr,0(1 + ϑ)

(5)

where ϑ ) t/tD is the dimensionless time,  ) VD/Vr,0 is the relative volume increase at the end of the supply period, and Vr,0 is the initial reaction volume. The mass balance equation for product C can be rewritten as

dζC/dϑ ) (Da)fκn,m

(6)

where Da ) kn,m,RtD(CB,0/ζB,0)n+m-1 is the Damko¨hler number for (n,m) order reactions of the form shown in eq 1, containing the physical information about the chemical kinetics and the dosing time, f ) [(ϑ - ζC)n(ζB,0 + ζC)m]/(1 + ϑ)n+m-1 is the only factor containing the functional dependence on ϑ and ζC, and κn,m ) exp[γ(1 - 1/τ)] is a dimensionless reaction rate constant, that is, the ratio of the reaction rate constant evaluated at the actual temperature, T, to that evaluated at a reference temperature, TR, with γ ) E/(RTR) and τ ) T/TR. The energy balance equation for the reactor can be written in dimensionless form as14

(1 + ϑ)

dζC dτ ) ∆τad,0 dϑ dϑ [(Co)(1 + ϑ) + RH](τ - τeff cool) (7)

where ∆τad,0 is the adiabatic temperature rise, which contains the physical information about the reaction enthalpy, Co ) (UA0)tD/(F˜ mC ˜ P,mVD) is the cooling number accounting for the cooling efficiency, and RH ) FDC ˜ P,D/F˜ mC ˜ P,m is the ratio between the volumetric heat capacities of the dosing stream and the reacting mixture. Moreover, τeff cool ) [(Co)(1 + ϑ)τcool + RHτD]/[(Co)(1 + ϑ) + RH] is an effective cooling temperature that summarizes the enthalpic contributions of both the heat removal by the coolant and the dosing stream. The model equations provided above are valid up to the end of the supply period. It can be easily demonstrated that the same equations can be extended to ϑ > 1 by substituting everywhere ϑ ) 1 and by setting TD ) T in the definition of the effective cooling temperature. It can be noticed that the mathematical model for autocatalytic reaction systems contains nine dimensionless parameters (that is, Da, ζB,0, , n, m, γ, ∆τad,0, Co, and RH), whereas the same model for nonautocatalytic reaction systems consists of only eight dimensionless parameters (that is, νA(Da)RE, , n, m, γ, ∆τad,0, Co, and RH):14 in the autocatalytic case the additional parameter ζB,0 depends on the initial catalyst amount, which is not related to the dosed coreactant quantity by any stoichiometric constraint. Moreover, the initial catalyst amount influences the initial reactivity of the system and for this reason is directly related to the coreactant accumulation, as well as the initial reaction temperature. Such a behavior can be better observed in Figure 1, where the influence of ζB,0 on the temperaturetime profiles (resulting from the integration of eqs 6 and 7 with the corresponding initial conditions) is summarized. It can be clearly recognized that, as ζB,0 decreases, the average reaction rates in the first part of the dosing period are lower and the coreactant accumulation becomes higher, so thatsas represented in Figure 1sit is possible to move from excessive accumulation operating conditions (recognized through an exceeding of the target temperature profile, as discussed in the following) at the lowest ζB,0 value to quick onset, fair conversion, smooth temperature profile conditions (in the following referred to as QFS4-6) at the highest ζB,0 value. Dealing with an SBR in which

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Figure 1. Temperature-time profiles in a homogeneous SBR in which an exothermic autocatalytic reaction of the form shown in eq 1 is carried out. Da ) 3,  ) 0.4, γ ) 38, RH ) 1, ∆τad,0 ) 0.7, Co ) 5. Coolant and dosing stream temperatures equal to the initial reactor temperature. Influence of the relative catalyst amount ζB,0.

an exothermic autocatalytic reaction is performed, the initial catalyst amount is therefore a crucial parameter to select operating conditions characterized by both a low coreactant accumulation and peak reaction temperatures lower than a given threshold value. 3. Thermally Safe Operating Conditions The safe operation of an indirectly cooled SBR in which an exothermic reaction is performed corresponds to conditions of so low coreactant accumulation that the conversion rate is at the limit entirely determined by the supply rate of the coreactant itself.4-6 On this basis, a number of boundary diagrams have been calculated elsewhere in the literature4-6,7-9,12-14,17 for nonautocatalytic reaction systems, allowing selection of safe and productive operating conditions of an existing reactor as well as scaleup of a process without solving the mathematical model of the reacting system. For this reason, in the following the calculation procedure of the boundary diagrams will be just briefly summarized, focusing on the relevant differences between the nonautocatalytic and autocatalytic cases. In particular, as for nonautocatalytic reaction systems,4-6,7-9,12-14,17 a target temperature-time profile can be defined as well

τta ) τeff cool + 1.05

∆τad,0 [(Co)(1 + ϑ) + RH]

(8)

to which the actual one can be compared to classify the thermal behavior of an existing reactor.14 Moreover, a dimensionless space given by an exothermicity and a reactivity parameter

Ex )

∆τad,0

γ

τcool (Co + RH)

Ry )

2

(Da)κ(τcool) (Co + RH)

(9)

(10)

can also be derived for autocatalytic reaction systems, in which the aforementioned typologies of reactor thermal behavior can be represented (see Figure 2).

Figure 2. Boundary diagram for the identification of excessive accumulation operating conditions in a homogeneous SBR in which an exothermic autocatalytic reaction of the form shown in eq 1 is carried out.

As is evident, in the autocatalytic case the single boundary diagram is identified through a set of five dimensionless parameters (corresponding to Co, RH, n, m, and ζB,0), that is, through an additional parameter with respect to nonautocatalytic reaction systems, represented by the initial catalyst amount ζB,0. Through an approach similar to that presented elsewhere in the literature for nonautocatalytic systems,7,8,17 an easy representation of the results can be kept in the autocatalytic case by summarizing in single plots calculated for given n and m values the Ex,MIN and Ry,QFS parameters (defining the inherently safe region; see Figure 2) as a function of the initial catalyst amount ζB,0 for different values of the cooling number Co. The dependence of Ex,MIN and Ry,QFS on the RH parameter is instead negligible, as for nonautocatalytic systems.14 In this way, only inherently safe operating conditions can be identified (that is, operating conditions characterized either by Ry values higher than the minimum value of Ry, Ry,QFS, for which excessive accumulation cannot occur for any Ex value or by Ex values lower than the maximum value of Ex, Ex,MIN, for which excessive accumulation cannot occur for any Ry value) and the method remains reasonably easy to use. Let us discuss at first how to identify the upper boundary of the inherently safe region, that is, the Ry,QFS value for a given set of Co, n, m, and ζB,0 values. The curves in Figure 3 provide the Ry,QFS values vs ζB,0 for several values of the parameters n, m, and Co. These diagrams allow end users dealing with homogeneous SBRs, in which exothermic autocatalytic reactions of the form shown in eq 1 occur and operating in industrially relevant ranges of the parameters involved, to easily identify the Ry,QFS value without solving the mathematical model of the reactor. As can be observed from these figures, Ry,QFS decreases with ζB,0. Such a behavior is expected since the initial reactivity of the system is directly related to the initial catalyst amount: in particular, the lower the ζB,0, the less the initial reaction rate and hence the higher the coreactant accumulation. However, at fixed n and m values the dependence of Ry,QFS on ζB,0 is lower at low cooling numbers (more commonly encountered at the full plant scale7,8), since a lower Co value is equivalent to a lower heat transfer efficiency, resulting in a higher temperature increase with the reaction extent and finally in a higher reaction rate. This prevents coreactant accumulation even for low ζB,0 values. Moreover, it can be observed that the Ry,QFS vs ζB,0

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Figure 3. Inherently safe conditions of homogeneous SBRs in which an exothermic autocatalytic reaction of the form shown in eq 1 is carried out. Influence of the relative catalyst amount ζB,0 on the Ry,QFS parameter at different Co values. RH ) 1, 0.02 < Da < 20, 0.05 <  < 0.6, 30 < γ < 45, 0.1 < ∆τad,0 < 0.7. (A) n ) 0.75, m ) 1. (B) n ) 0.75, m ) 2. (C) n ) 1, m ) 0.75. (D) n ) 1, m ) 1. (E) n ) 1, m ) 2. (F) n ) 2, m ) 1.

curves at different Co values can intersect at higher ζB,0 and Co values. This means that at fixed n, m, and ζB,0 the Ry,QFS parameter as a function of Co can exhibit a maximum, as observed elsewhere in the literature for nonautocatalytic systems.7,8,14 Such a behavior arises from the fact that high Co values can physically correspond either to a high dosing time (which can limit the coreactant accumulation in the system) or to a high cooling efficiency (which can keep the initial reaction temperatures and hence the initial reaction rates low, thus favoring the coreactant accumulation). Once the Ry,QFS value has been identified for a given set of n, m, Co, and ζB,0 values, to identify the whole inherently safe region, it is necessary to estimate also its left boundary, that is, the Ex,MIN value. This is much simpler since Ex,MIN does not depend on ζB,0, n, and m at fixed Co, as can be seen from the raw data reported in Figure 4 for all the values of n, m, and ζB,0 considered in this work. Such a behavior is correct, since

Ex,MIN arises from conditions of sufficiently low reaction exothermicity that the thermal effects related to an excessive coreactant accumulation (that is, the exceeding of the target temperature) tend to disappear, independently of the entity of the accumulation itself and hence of ζB,0 as well as of the kinetic parameters. It follows that, once the Co value is given, Figure 4 allows for a quick estimation of the related Ex,MIN value. It can be further observed that, as for nonautocatalytic SBRs,14 at a fixed m value the use of a diagram calculated for a higher n value than that estimated from the experimental data leads to a higher Ry,QFS and hence to safe results. The same statement holds true if for a fixed n value a diagram calculated for a higher m value than that estimated from the experimental data is used. Such a rule of thumb is important for end users, who can use their own diagrams calculated for a limited set of typical n and m values. It is important to notice that the safety criteria developed for nonautocatalytic reaction systems cannot be extended to the

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Figure 4. Inherently safe conditions of homogeneous SBRs in which an exothermic autocatalytic reaction of the form shown in eq 1 is carried out. Influence of the cooling number Co on the Ex,MIN parameter. RH ) 1, 0.75 < n < 2, 0.75 < m < 2, 0.01 < ζB,0 < 0.05. 0.02 < Da < 20, 0.05 <  < 0.6, 30 < γ < 45, 0.1 < ∆τad,0 < 0.7.

autocatalytic ones. Taking as an example the criterion of Hugo et al.2 for (1,1) reaction order, all the operating conditions characterized by Ry values higher than 1 should correspond to an acceptable coreactant accumulation (that is, this criterion can be summarized in the statement Ry,QFS ) 1), as verified also through the boundary diagram method for homogeneous SBRs with nonautocatalytic behavior.14,17 However, in the autocatalytic case such a criterion does not hold true as can be clearly observed for instance from Figure 3D (reporting the Ry,QFS values for n ) m ) 1), where even at low cooling numberss where Ry,QFS is lower and less dependent on ζB,0sthe Ry,QFS values are much higher than 1. When an exothermic semibatch reaction is performed, limiting the coreactant accumulation below an arbitrary value can be just a necessary but not a sufficient condition to discriminate whether a given set of operating conditions can be adopted or not: the second process constraint is often given by a threshold temperature value (referred to as MAT) that must not be exceeded to prevent the triggering of an undesired event.15 Such events are typically related either to safety problems (when dangerous decomposition reactions causing a reactor overpressurization could be triggered) or to product quality problems (when side reactions producing unwanted impurities could take place even to a small extent). In such cases, if the initial reaction temperature is too high, a low coreactant accumulation can be reached, but the MAT can be exceeded. The boundary diagrams are a useful tool for preventing excessive coreactant accumulations, but they do not provide any direct information about the maximum peak temperature to be expected for a selected set of operating conditions. To solve such a problem for practice, Maestri and Rota15 introduced the temperature diagrams that, through the same set of parameters employed for the representation of the boundary diagrams, allow end users to safely estimate the expected peak reaction temperature. Following the same procedure described elsewhere for nonautocatalytic reaction systems,14,15 it is possible to calculate the temperature diagrams also for systems in which an autocatalytic reaction such as that shown in eq 1 occurs. Such diagrams, at fixed Co, RH, n, m, and ζB,0 values, provide the ratio ψ of the peak to the initial reaction temperature as a function of the exothermicity number Ex.

As previously discussed, when dealing with autocatalytic reaction systems, an additional parameter (that is, the initial catalyst amount ζB,0) must be taken into account. However, a compact representation of the safety criterion can still be obtained if only inherently safe operating conditions are considered. This leads to a considerable simplification of the representation and use of the temperature diagrams, also considering that, as observed for nonautocatalytic reaction systems,14 the temperature profiles for Ry values higher than Ry,QFS become substantially independent of Ry itself. This means that the peak reaction temperature for a given set of operating conditions can be estimated through the same temperature curve, provided that such operating conditions are characterized by an Ry value higher than Ry,QFS, which can be first estimated through the diagrams reported in Figure 3. Moreover, under QFS conditions the characteristic time of the process (that is, the characteristic time related to the overall conversion rate as well as to its enthalpic effects) is at the same time close to the dosing time and higher than the characteristic time for the reaction mass cooling.14 This means that the thermal behavior of the system is completely determined by the enthalpic contribution of the reaction and that the kinetic related parameters (that is, n, m, and ζB,0) as well as the heat transfer related parameters (that is, Co and RH) have a small influence on the system thermal behavior. Under such conditions, the ψ parameter, which is in general a function of Co, RH, n, m, ζB,0, and Ex, becomes a function of Ex only and the temperature rise curves tend to overlap into a single curve, as can be seen from the raw data represented in Figure 5 for all the conditions investigated in this work. The curve represented in Figure 5 can be used to estimate the maximum peak temperature for any combination of the Co, RH, n, m, and ζB,0 parameters into their specified range of variation, provided that the operating conditions are characterized by an Ry value higher than Ry,QFS. It should be noticed that under QFS conditions the maximum peak temperature must be lower than the target temperature. However, since the target temperature decreases with time (see eq 8 and Figure 1) and the time at which the peak temperature occurs is unknown, it is not possible to identify which is the value of the target temperature bounding the peak temperature. This problem could be faced noting that the peak reaction temperature as well as the local value of the target temperature (eq 8) must be lower than its initial value Tta,0; consequently, the initial value of the target temperature could be used to estimate the peak reaction temperature. From eqs 8 and 9 the following functional dependence of the initial target temperature Tta,0 on the exothermicity number (eq 9) can be derived:

τ0 τta,0 ) 1 + 1.05 Ex τ0 γ

(11)

According to the range of variation of the γ parameter assumed in the calculation of the diagrams and taking into account that at the industrial scale the initial temperature values fall reasonably in the range -20 to +100 °C (corresponding to τ0 ) 0.851.25, with TR ) 300 K), the quantity in eq 11 varies between 1 + 0.02Ex and 1 + 0.04Ex, as represented through the two straight lines plotted in Figure 5. It can be clearly seen that at low Ex values (for instance, Ex < 5) the initial target temperature can be a good approximation of the maximum peak reaction temperature as estimated through the temperature diagrams. The two values (the one estimated using the initial value of the target temperature and the one estimated using the temperature diagrams) are more or less close

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Figure 5. Inherently safe conditions of homogeneous SBRs in which an exothermic autocatalytic reaction of the form shown in eq 1 is carried out. Influence of the exothermicity number Ex on the maximum temperature rise ψ ) (Tmax/T0)max. RH ) 1, 2 < Co < 80, 0.75 < n < 2, 0.75 < m < 2, 0.01 < ζB,0 < 0.05, 0.02 < Da < 20, 0.05 <  < 0.6, 30 < γ < 45, 0.1 < ∆τad,0 < 0.7.

to each other, depending on the slope of the Tta,0 vs Ex straight line, which in turn depends on the particular operating conditions considered. However, at high Ex values (corresponding either to low Co values or to high ∆τad,0 values, as frequently encountered at the industrial scale for strongly exothermic reactions7,8), the use of the initial target temperature expression (eq 11) to estimate the peak temperature can lead to huge overestimations with respect to the value estimated through the temperature diagrams. This overestimation would lead to the selection of operating conditions that significantly lower the reactor productivity. This behavior can be explained considering that under QFS conditions (on the basis of which the target temperature is derived) the characteristic time of the process (which is close to the dosing time) is much higher than the characteristic time for the reaction mass cooling. Such a requirement is satisfied if the ratio of these two characteristic times evaluated at t f 0 (where the accumulation phenomena play a major role)

|

tP eff tcool tf0

)

(UA)0tD F˜ mC ˜ P,mVr,0

+ RH ) (Co + RH)

(12)

is much higher than 1. Since practical values of  are lower than 0.6 and RH is on the order of 1, the quantity in eq 12 can be much higher than 1 only at sufficiently high cooling number values. This is the reason why at too low Co values (corresponding to high Ex values) the prediction of the peak reaction temperature through the initial target temperature expression (eq 11) can become less accurate. However, under different operating conditions, the value of the initial target temperature (which can never be overcome by the real peak reaction temperature) can be a better estimation of the real peak temperature than the value provided by the temperature diagrams, as clearly shown by Figure 5, where a small region characterized by values of the initial target temperature lower than that provided by the temperature diagrams also exists. This leads to a simple rule of thumb for selecting safe and productive operating conditions of SBRs in which an exothermic

reaction of the form shown in eq 1 takes place. Once operating conditions belonging to the inherently safe region have been selected (that is, with Ry values higher than the Ry,QFS value estimated through the proper curve provided by Figure 3 and/ or with Ex values lower than the Ex,MIN value estimated through Figure 4), both the initial target temperature (through eq 8) and the ψ value (through Figure 5) have to be computed: the best estimation of the real peak temperature expected during normal SBR operation will be the lower of these two temperature values. Conclusions In this work the thermally safe operation of homogeneous SBRs in which exothermic reactions with autocatalytic behavior occur has been analyzed. In the autocatalytic case the thermal behavior of the reactor depends on an additional parameter with respect to nonautocatalytic systems, that is, the initial catalyst amount, directly related to the coreactant accumulation. Accounting also for this additional degree of freedom, a safety criterion based on the use of the boundary and temperature diagram method previously proposed in the literature for nonautocatalytic systems has been developed, providing diagrams that allow for selecting inherently safe and productive operating conditions, with a minimum experimental and calculation effort. For a given reacting system, once the kinetic parameters appearing in eq 2 have been estimated through simple calorimetric experiments, the MAT value is given by either safety or process constraints and the main technical parameters of the real size reactor where the process has to be carried out are known, a set of operating conditions can be selected (for which the related values of the parameters Ry and Ex can be easily computed), fulfilling the following constraints: (1) Either Ry > Ry,QFS (where Ry,QFS is given by Figure 3) or Ex < Ex,MIN (where Ex,MIN is given by Figure 4). (2) MAT > min(Tta,0, T0ψ), where Tta,0 is given by eq 8 and ψ by Figure 5 and T0 is the initial temperature value. These operating conditions will be both safe and productive. Finally, it should be mentioned that, in the industrial practice, the usual isoperibolic semibatch technique can be not the best way of performing autocatalytic

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reactions. An alternative could be to ignite the system by starting the process under adiabatic conditions to rapidly reach the target temperature: from this point, usual isoperibolic conditions can be adopted.

ζ ) ni/nA,1, relative molar amount of the ith component, dimensionless

Nomenclature

ν ) stoichiometric coefficient, dimensionless

A ) heat transfer area of the reactor (associated with the jacket and/or the coil), m2 C ) molar concentration, kmol/m3 Co ) (UA)0tD/(F˜ mC ˜ P,mVD) ) cooling number, dimensionless C ˜ P ) molar heat capacity, kJ/(kmol‚K) Da ) kn,m,RtD(CB,0/ζB,0)n+m-1 ) Damko¨hler number for (n,m) order reactions, dimensionless E ) activation energy, kJ/kmol Ex ) exothermicity number, dimensionless Ex,MIN ) maximum value of Ex for which excessive accumulation cannot occur for any Ry value, dimensionless f ) function of dimensionless time and relative molar amount of B in eq 6, dimensionless F ) molar feed rate, kmol/s ∆H ˜ ) reaction enthalpy, kJ/kmol kn,m ) reaction rate constant, m3(n+m-1)/(kmoln+m-1‚s) MAT ) maximum allowable temperature, K n ) number of moles, kmol r ) reaction rate, kmol/(m3‚s) R ) gas constant ) 8.314 kJ/(kmol‚K) RH ) heat capacity ratio, dimensionless Ry ) reactivity number, dimensionless Ry,QFS ) minimum value of Ry for which excessive accumulation cannot occur for any Ex value, dimensionless RE ) reactivity enhancement factor for nonautocatalytic reaction systems,12-14 dimensionless t ) time or characteristic time, s T ) temperature, K ∆Tad,0 ) -∆H ˜ r/F˜ mC ˜ P,m (CB,0/ζB,0) ) adiabatic temperature rise, K U ) overall heat transfer coefficient, kW/(m2‚K) V ) liquid volume, m3

F˜ ) molar density, kmol/m3

Subscripts and Superscripts A, B, C ) components A, B, and C ad ) adiabatic cool ) coolant D ) dosing stream or dosing time eff ) effective i ) ith component m ) order of reaction with respect to component B m ) reacting mixture max ) maximum value of a quantity or at the maximum value of a quantity n ) order of reaction with respect to component A P ) process r ) reaction R ) reference ta ) target 0 ) start of the semibatch period 1 ) end of the semibatch period Greek Symbols γ ) E/(RTR) ) dimensionless activation energy  ) relative volume increase at the end of the semibatch period, dimensionless

ϑ ) t/tD, dimensionless time κ ) k/kR, dimensionless reaction rate constant

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ReceiVed for reView March 25, 2007 ReVised manuscript receiVed May 14, 2007 Accepted May 23, 2007 IE070427F