Topological Criterion To Safely Optimize Hazardous Chemical

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Ind. Eng. Chem. Res. 2011, 50, 1588–1598

Topological Criterion To Safely Optimize Hazardous Chemical Processes Involving Arbitrary Kinetic Schemes Sabrina Copelli, Marco Derudi, and Renato Rota* Politecnico di Milano, Dip. di Chimica, Materiali e Ingegneria Chimica “G. Natta”, Via Mancinelli 7, 20131 Milano, Italy

Safe and productive operating conditions for complex exothermic chemical processes are difficult to obtain because of the strong interaction between selectivity and safety constraints. In this work a generalization of the topological criterion to nonautocatalytic, chain, and autocatalytic catalyzed reaction schemes is presented. Such a criterion, useful for isoperibolic semibatch processes, is able to detect the so-called QFS boundary by performing a topological analysis of a suitable reduced phase portrait of the system of ordinary differential equations describing the process. Moreover, a safe and general procedure aimed to obtain the optimum values of both the dosing time and the initial reactor/coolant temperature, based on such a criterion, has been proposed and validated through the use of literature experimental data and laboratory tests concerning the case studies analyzed. 1. Introduction In pharmaceutical and fine chemical industries, fast and exothermic reactions are often carried out in semibatch reactors (SBRs) in order to better control the heat evolution during the synthesis. In fact, for such processes, a phenomenon known as “thermal runaway” (namely, a loss of the reactor temperature control) may be triggered whenever the rate of heat removal becomes lower than the rate of heat production. During the last 25 years, a considerable number of studies on the detection of the runaway boundaries has been perfomed.1-16 Usually, such studies dealt with isoperibolic semibatch reactors involving a single synthesis reaction,6-13 but two of them were able to handle complex kinetic schemes, different reactor typologies, and temperature control modes: the generalized parametric sensitivity criterion1 and the divergence criterion.1-5 However, from a practical viewpoint, the desired goal is to define a set of operating parameters (e.g., initial reactor temperature and dosing time) for which the maximum reactor productivity is attained, maintaining safe conditions. Knowing runaway boundaries allows avoiding the selection of potentially hazardous operating conditions, but it does not permit one to optimize the analyzed process. The first approach accounting for both optimization and safety aspects was based on the construction of boundary diagrams (BDs).6-9 These diagrams are generated in a dimensionless space reactivity (Ry) vs exothermicity (Ex) and are able to identify both the runaway region (RW) and the so-called “quick onset, fair conversion, smooth temperature profile” region (QFS).7 In particular, in the QFS region the process exhibits an almost linear conversion to the desired product and a high cooling efficiency; therefore, this region is considered the most safe and productive for an isoperibolic SBR. However, the definition of the boundary that divides the runaway region from the QFS one is arbitrary.10 Moreover, it is true that an isoperibolic SBR operating in the QFS region is safe from the dosed reactant accumulation viewpoint, but nothing is known about neither the thermal stability of the system (since side reaction kinetics is not contained in the mathematical model * To whom correspondence should be addressed. Fax: +39 0223993180. E-mail: [email protected].

that generates the BDs) nor the optimum set of operating parameters in such a region. To deal with the first problem, the so-called temperature diagrams (TDs) have been proposed.11 Such diagrams allow comparing the expected maximum reactor temperature with the system maximum allowable temperature, MAT, that corresponds to a temperature above which the rate of undesired reactions has been experimentally proved to be not negligible. If the maximum reactor temperature is above the MAT, the selected operating parameters are not suitable and must be rejected. The second problem (optimization) is more complex to deal with because it requires an unambiguous definition of QFS conditions. Such an issue has been discussed by Alo`s et al.,10 who proposed, for isoperibolic SB systems involving a single reaction, a definition of QFS using the objective function “time at which the maximum reactor temperature, ϑ(TMAX), occurs”. In particular, the QFS is individuated, for a given value of the dosing time, by the coolant temperature, Tcool,QFS, at which such an objective function exhibits a local minimum. In this work the topological criterion, previously proposed for systems involving two autocatalytic reactions to identify the QFS conditions without referring to any arbitrary definitions or functions,17 is extended to arbitrary kinetic schemes and validated through literature experimental data and laboratory tests. It has been found that such a theory, based on the analysis of the topology of a suitable reduced system phase portrait (PP), can be used successfully to optimize a strongly exothermic isoperibolic SB process involving whatever complex kinetic scheme. 2. Chemical Topology Theory In order to model a generic isoperibolic semibatch process, it is required to write mass and energy balance equations.12,13,18 Such balances, which are summarized in dimensionless form in eqs 1, constitute a system of nonlinear ordinary differential equations (ODEs) having as independent variable the dimensionless time, ϑ, and as dependent variables the dimensionless reactor temperature, τ, and all the conversions, ζi, required to characterize completely the system kinetics

10.1021/ie102014w  2011 American Chemical Society Published on Web 01/06/2011

{

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 dζj ) dϑ

(

NR

j)1

ζ, φ1 ϑ, b

(

Table 1. Process Recipe, Reactor Cooling System, Thermochemical, and Kinetic Parameters for the Analyzed Slow Heterogeneous Reaction Occurring in the Continuous Phase: Aromatic Nitration of 4-Chlorobenzotrifluoride in Mixed Acids

∑ Da · RE · f (ϑ, bζ) · κ (τ) j

)

j

b dτ dζ ) · dϑ dϑ

b dζ φ3 ϑ, b ζ, dϑ

)

j

j

NR

∑ ∆τ j)1

(

( )

ad,j · φ2

)

b dζ + dϑ

j ) 1, 2, ..., N

recipe and reactor cooling system

b dζ eff - φ4 ϑ, b ζ, · (τ - τcool ) dϑ

initial load and dosed stream

cooling system

30 g of HNO3 (66% w/w) 335 g of H2SO4 (98% w/w) 83 g of 4-Cl-BTF

external jacket siliconic oil UA0 ) 1.21 [W/K]

(1)

The solution of the equations in eq 1 is constituted by a series b(ϑ),τ(ϑ)), representing of dependent variables (N + 1)uples (ζ a one-dimensional variety, called trajectory, whose evolution can be appreciated by reporting such (N + 1)uples onto the phase space (PS) correspondening to eq 1. By varying one system constitutive parameter (e.g., dosing time, global heat transfer coefficient, etc.) or initial condition (IC, e.g., initial reactor temperature, etc.) in a suitable range and then reporting the solutions obtained onto the corresponding PS, it is possible to build a diagram called a “phase portrait” (PP). It is necessary to mention that when a portrait is generated by varying whatever system parameter but IC the nomenclature “phase portrait” is not mathematically rigorous. In this contest it will be used, even if improperly, for the sake of convenience. Using the tools provided by topology, it is possible to analyze a phase portrait in order to detect runaway and QFS boundaries for the studied process. However, before conducting such a topological analysis, it is necessary to perform a reduction of the analyzed phase space, that is to eliminate all the axes corresponding to dependent variables that are not involved into the energy balance equation and, consequently, that do not contribute directly to the observed system thermal behavior. Such a phase space reduction has been already carried out by Zaldı´var and co-workers2-5 to explain how some dependent variables had to be disregarded into the divergence calculation. The phase space obtained after the reduction operation constitutes a new space that can be used to generate the desired reduced phase portrait (by varying whatever system parameter or IC). 3. Phase Portraits Projections and QFS Conditions In order to identify the QFS conditions for an isoperibolic SB process involving whatever arbitrary kinetic schemes, it is necessary to specify which product the QFS refers to. In fact, when more than one reaction proceeds a relevant selectivity problem may arise and more than one QFS may be detected, depending on which product one is interested in. In the following, three different kinetic schemes will be discussed: single nonautocatalytic, chain, and autocatalytic catalyzed reactions. For each kinetic scheme, it will be shown how to perform a topological analysis of the desired reduced phase portrait to detect QFS conditions and establish if a runaway phenomenon may be triggered. 3.1. Single Nonautocatalytic Reaction. The most general reaction scheme may be written as νAA + νBB f C + νDD

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(2)

where A is the dosed reactant, B is the initially charged reactant, C is the desired product, and D is the unwated product. The system can be either homogeneous (a single liquid phase) or heterogeneous (two immiscible liquid phases), and in the latter case, the reaction can occur either in the continuous or in the dispersed phase.

thermodynamic and kinetic parameters 3.228 × 1012 m3/(kmol s) 87 268 kJ/kmol 2.358 kmol/m3 150.1 kJ/mol 0.4387 35 0.4403 0.3038 0.01 95 °C

A Eatt [B]0 ∆Hrxn RH γ ∆τad ε mA MAT

Along the same line discussed elsewhere,12,13 eqs 1 are reduced to

{

dζ ) νA · Da · RE · f · κR dϑ dτ dζ ) ∆τad · (1 + β · RH · ε · ϑ) · dϑ dϑ [St · (1 + ε · ϑ) + ε · RH] · (τ - τeff cool)

IC

ζ(ϑ ) 0) ) 0,

(3)

τ(ϑ ) 0) ) τ0 ) τcool

where Da ) A · exp(-Eatt/RTrif) · [B]0n+m-1 · tdos is the Damko¨hler number, RE is the reactivity enhancement factor,12,13 f is the conversion function,12,13 κ ) exp(γ · (1 - 1/τ)) is the dimensionless kinetic constant, R is a factor that takes into account the operating reaction regime (slow, 1, or fast, 1/2), β is a factor accounting for the number of system phases (for one phase, β eff ) RHε · τdos + St · (1 + ) 1/RH, for two phases, β ) 1), and τcool εϑ) · τcool/(St · (1 + εϑ) + RHε) is the effective coolant temperature. The meaning of all the other terms is reported in the Nomenclature section. Observing the structure of eqs 3, it is possible to state that no phase space reduction is required because the number of exothermic reactions (one) equals the number of independent mass balance equations needed to characterize completely the kinetic scheme (one). A direct consequence of this aspect is that only one QFS may be defined: the QFS with respect to production of the desired species C. Using, as an example, model parameters reported in Table 1 and solving eqs 3, a trajectory, described at any time ϑ by a couple (ζ,τ), can be obtained. Such a variety, when it is reported onto its corresponding phase space, shows a characteristic upper-lower limited trend with one maximum located corresponding to the couple (ζ(τMAX),τMAX) and an attractor point in (1,1). Since the most relevant system constitutive parameter and initial condition is the mean coolant temperature, Tcool (which is related to the reactor temperature at the start of the dosing period, T0, by a numerical constant, ∆T0, associated with heat dissipations, reactor thermal inertia, etc. by the relation Tcool ) T0 + ∆T0), it is possible to generate a phase portrait by varying such a parameter, as shown in Figure 1. Even if the temperature axis has been referred to the coolant temperature (in order to make comparable all the trajectories on the same plot) and a coolant temperature axis has been inserted (in order to better

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Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Figure 1. Phase portrait for the single nonautocatalytic reaction system considered. Coolant temperature and dosing stream temperature vary between 273 and 353 K. The axes parameters are ζ, ψ ) T/Tcool, and Tcool. Model parameters are reported in Table 1.

is possible to notice that on increasing the coolant temperature, beyond point A the conversion corresponding to the maximum temperature starts to decrease while before such a point it was increasing. Such a trend inversion is referred to as “transition inversion” in C space and indicates physically that the system thermal loss of control shifts its occurrence from times larger to lower than the dosing period and reduces its magnitude. These phenomena may be explained by considering that a coolant temperature increase always leads to a lower coreactant accumulation (the reaction starts before a high degree of reactant accumulation exists) and, consequently, to a reduction of the runaway magnitude. For a further increase of the coolant temperature, another tendency inversion occurs. Beyond point B, the maximum dimensionless temperature diminishes (even if less and less) whereas the conversion restarts to increase. This inversion is referred to as “QFS inversion”. The coolant temperature at which such an inversion occurs, Tcool,QFS, is the boundary temperature between “runaway” conditions and QFS conditions for the reaction considered. It is worth mentioning that in the C space a sort of loop, originated by the intersection of the ζ(ψMAX) vs ψMAX curve with itself, may be detected. Such a behavior had been already observed by Steinbach,14 even if in a different dimensionless space, and it indicates the existence of a runaway region. Finally, a general observation arises from the analysis of the inversion points character: the transition inversion is individuated by a concavity of the topological curve toward the left while the QFS inversion is individuated by a concavity toward the right. 3.2. Chain Reactions. The most simple kinetic scheme involving chain reactions is that represented by solution homopolymerizations, which may be summarized by the following reactions

Figure 2. C space for the single nonautocatalytic reaction system considered: aromatic nitration of 4-chlorobenzotrifluoride in mixed acids. Point A represents the transition inversion and point B the QFS inversion. Coolant temperature varies from 273 to 353 K. Model parameters as in Table 1. Experimental runs (see Table 4) are compared with the correspondent theoretical ones by the means of arrows: (O) runaway runs, (b) QFS runs, (-) runaway region, (--) QFS region.

appreciate the evolution of all the trajectories) the obtained phase portrait does not show any detectable property. However, some interesting topological properties can be identified by projecting the maxima of all the trajectories onto a modified space ζ(ψMAX) vs ψMAX (in the following referred to as C space), as reported in Figure 2. The topological criterion utilizes the turning point of the curve shown in Figure 2, which is also referred to as topological curve, to identify the boundaries that separate the different thermal behavior regions obtainable for an isoperibolic semibatch reactor by varying the coolant temperature from low to high values: NO IGNITION (NO IGN, flat temperature profile characterized by low maximum temperature increase with respect to the initial reactor temperature and slow conversion variations during all the process duration), RUNAWAY (RW, sharp temperature profile characterized by high maximum temperature increase with respect to the initial reactor temperature and violent conversion variations either after or before the end of the dosing period), and QFS (smooth temperature profile characterized by medium maximum temperature increase with respect to the initial reactor temperature and almost linear conversion during the dosing period). In fact, a clear change in the thermal behavior of the system can be associated to any turning point, where an inversion of the curve tendency with respect to the conversion trend occurs. Observing Figure 2, it

kd

(1) Initiation: I2 98 2R kp

(2) Propagation: R + M 98 R

(4) ktd

(3) Termination (by disproportion): R + R 98 2P

where I2 is the initiator, R is the pseudoradical species (constituted by all the radical species regardless of their chain lengths), M is the monomer, and P is the dead polymer chain. In particular, the following assumptions can be stated: (1) the system is homogeneous (a solvent, present in large excess, is perfectly mixed with the monomer, radical, and polymer species); (2) only reactions 4 occur; (3) the terminal kinetic model has been assumed to describe the overall kinetics of the process (that is, the length of the active chains does not affect the kinetics of the process, particularly the propagation kinetics); (4) the only reaction developing heat is the propagation reaction; (5) the increase of the mixture viscosity affects the heat removal efficiency (such a phenomenon can be taken into account by using a suitable function of the monomer conversion, whose parameter, ω, must be experimentally determined); (6) the variation of the mixture density can be taken into account by introduction of a contraction factor δ ) 1 - FM/FP.


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Table 2. Process Recipe, Reactor Cooling System, Thermodynamic, And Kinetic Parameters for the Chain Reaction System Considered: Solution Homopolymerization of Butyl Acrylate in Ethyl Acetate Thermally Initiated by AIBN recipe and reactor cooling system initial load and dosed stream

cooling system

205 g of ethyl acetate 2 g of AIBN 120 g of butyl acrylate

external jacket siliconic oil UA0 ) 2.295 [W/K]

thermodynamic and kinetic parameters

{

1.33 × 1015 1/s 1.285 × 105 kJ/kmol 3.20 × 107 m3/(kmol s) 2.62 × 104 kJ/kmol 89.1 kJ/mol 3.70 × 109 m3/(kmol s) 1.33 × 104 kJ/kmol 1.021 0.8882 0.499 77 °C

Ai Ei Ap Ep ∆Hrxn,p Atd Etd RH ∆τad ε MAT

Figure 3. Phase portrait for the solution homopolymerization system considered. Dosing time varies between 30 and 10 800 s, and dosing stream temperature is 288 K. The axes parameters are ζP, ψ, and tdos. Model parameters are reported in Table 2.

Using these assumptions, mass and energy balances lead to dζI 1 ) Dai · (1 - ζI) · exp γi 1 dϑ τ dζR 1 ) 2Dai · (1 - ζI) · exp γi 1 dϑ τ

[ ( )] [ ( )] ζ 1 · exp[γ (1 - )] 2Da · [1 + εϑ(1 - δζ )] τ 2 R•

t

td

P

ζR•(ϑ - ζP) dζP 1 ) Dap · · exp γp 1 dϑ [1 + εϑ(1 - δ · ζP)] τ dζP dτ [1 + εϑ(1 - δζP)] · ) ∆τad,p dϑ dϑ eff (St[1 + εϑ(1 - δζP)] · (1 - ωζP) + RHε) · (τ - τcool )

IC

[ (

( )

)]

(5)

ζI(ϑ ) 0) ) ζR(ϑ ) 0) ) ζP(ϑ ) 0) ) 0, τ(ϑ ) 0) ) τ0 ) τcool - ∆τ0

where Dai ) f · Ai · exp(-γi) · tdos, Dat ) Atd · exp(-γtd) · [I]0 · tdos, and Dap ) Ap · exp(-γp) · [I]0 · tdos are the Damko¨hler numbers of the initiation, global termination, and propagation reactions while τeff cool ) (RHε · τdos + St[1 + εϑ · (1 - δζP)] · (1 - ωζP) · τcool)/ (St · [1 + εϑ · (1 - δζP)] · (1 - ωζP) + RHε) is the effective cooling temperature. The meaning of all the other terms is reported in the Nomenclature section. Observing the structure of eqs 5, a phase space reduction is required because the number of exothermic reactions (one, the propagation reaction) is lower than the number of independent mass balance equations needed to characterize completely the kinetic scheme (three). On the contrary, as there is only one exothermic reaction, one QFS may be defined: the QFS with respect to species P. Analogously to the single reaction case and using, as an example, model parameters reported in Table 2, the solution of eqs 5 generates a trajectory that can be represented onto a suitable phase space. In this case, such a trajectory cannot be visualized directly because it is defined by a 4-uple of dependent system variables, (ζI, ζR, ζP, τ). As previously discussed, a phase space reduction (from four to two dimensions) must be performed by taking into account only the dependent variables that are involved directly into the energy balance equation, namely, ζP and τ. After this reduction, the obtained reduced phase space is perfectly analogous to a single reaction phase space. Therefore, it is possible to generate the desired reduced phase portrait by varying one system constitutive parameter or

Figure 4. P space for the solution homopolymerization system considered: solution homopolymerization of butyl acrylate in ethyl acetate initiated by AIBN. Point A represents the transition inversion and point B the QFS inversion. Dosing time varies from 30 to 5400 s. Model parameters as in Table 2. Experimental runs (see Table 5) are compared with the correspondent theoretical ones by the means of arrows: (O) runaway runs, (b) QFS runs, (-) runaway region, (--) QFS region.

IC. Since polymerization reactions require a high product quality (e.g, peaked molecular weight distribution, high average molecular weight, little branching, etc.), which can be obtained only by limiting as much as possible temperature fluctuations, coolant temperatures are forced to vary in a narrow range. For this reason, coolant temperature is not a suitable parameter to generate the desired phase portrait because a generic PP can be constructed only if the generating parameter can be varied into a sufficiently wide range (conversely, the PP would be incomplete and, consequently, useless). Therefore, the dosing time, which is the second most important parameter involved in process optimization, has been chosen as the generating parameter to accomplish the same goal. The result is represented in Figure 3, where the main coordinated axes are the polymer conversion ζP, dimensionless reactor temperature τ, and dosing time. Along the same line discussed for single reactions, a topological analysis of the phase portrait can be performed by projecting all the trajectory maxima onto a modified space ζP(ψMAX) vs ψMAX called, in this case, P space. The resulting diagram is represented in Figure 4. Starting from low dosing time values and then increasing, it is possible to detect at first inversion point A (which has been referred to as the transition inversion) and then inversion point B (which has been referred to as the QFS with respect to the desired species P). It is worth noting that even if the curve in P space is quite different from the others reported in this work or elsewhere,17 the same character of the inversion points (representative of a system thermal behavior shift) is mantained regardless of the phase portrait generating parameter (Tcool or tdos): the transition inversions are individuated by a concavity of the projecting

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curve toward the left, and on the contrary, the QFS inversion is individuated by a concavity toward the right. Such a general rule may be applied to any isoperibolic SB process involving whatever complex kinetic scheme in order to detect the QFS conditions and, consequently, safely optimize the process itself. 3.3. Autocatalytic Catalyzed Reactions. The reaction scheme here considered can be written as K1, K2

(1) A + B 98 C + D

Table 3. Process Recipe, Reactor Cooling System, Thermodynamic, and Kinetic Parameters for the Autocatalytic Catalyzed Reaction System Considered: Acid Esterification between Propionic Anhydride and 2-Butanol Catalyzed by Sulfuric Acid recipe and reactor cooling system initial load and dosed stream

cooling system

511 g of 2-butanol 5 g of sulfuric acid 890 g of propionic anhydride

UA0 ) 5.110 [W/K]

(6) (2) K1 f K2 where A is the dosed reactant, B is the initially loaded reactant, C is the product responsible for the autocatalytic behavior of the system because it is directly involved in the kinetic expression of reaction 1, D is the desired product, and K1/K2 are two different types of catalyst. Mass and energy balance equations for the homogeneous case can be written as follows

[

(ϑ - ζ)(1 - ζ) + (1 + εϑ) (1 - ζK1) (ϑ - ζ)(1 - ζ) dζ · + ) κ1(τ) · k1,rif · X · [B]0 · [K1]0 · tdos (1 + εϑ) (1 + εϑ) dϑ ζK1 · (ϑ - ζ) κ2(τ) · k2,rif · X · [K1]0 · tdos (1 + εϑ) (1-ζK1) p4 · Trif dζK1 ζ (p1 · [K1]0 +p2 · [B]0 ) · (p3+ ) ) κ3(τ) · k3,rif · [B]0 · tdos · 10 · (1+εϑ) (1+εϑ) τ dϑ (1 - ζ)(1 - ζK1) κ0(τ) · k0,rif · X · [B]0 · tdos

(1 + εϑ) · IC

]

(1 + εϑ) dτ dζ eff ) ∆τad,0 · - (St · (1 + εϑ) + RHε) · (τ - τcool ) dϑ dϑ ϑ ) 0 ⇒ ζ ) ζK1 ) 0, τ ) τ0

(7) where κi ) exp (γi · (1 - 1/τ)) is the ith dimensionless kinetic eff constant and τcool ) RHε · τdos + St · (1 + εϑ) · τcool/(St · (1 + εϑ) + RHε) is the effective coolant temperature. The meaning of all the other terms is reported in the Nomenclature section. Observing the structure of eqs 7, a phase space reduction is required because the number of exothermic reactions (one, the first) is below the number of independent mass balance equations needed to characterize completely the kinetic scheme (two). However, since there is a single exothermic reaction, only one QFS may be defined: the QFS with respect to the desired product D. Using, as an example, model parameters reported in Table 3 and solving eqs 7, a trajectory, showing a maximum in correspondence of the triple (ζ(τMAX),ζK1(τMAX),τMAX) and an attractor point in (1,1,1) is obtained. As previously discussed, a phase space reduction (from three to two dimensions) must be performed by taking into account only the dependent variables that are involved directly in the energy balance equation, namely, ζ and τ. The obtained reduced phase space is perfectly analogous to than one of a single reaction phase space. Analogously to the case of a single reaction, it is possible to build the phase portrait reported in Figure 5 by varying the mean coolant temperature and referring to it the reactor temperature (in order to make comparable all the trajectories reported onto the reduced phase portrait). By projecting the maxima of the different trajectories shown in Figure 5 onto the plane reported in Figure 6, called D space (ζ(ψMAX) vs ψMAX), it is possible to notice that, even in this

thermodynamic and kinetic parameters A0 E0 A1 E1 A2 E2 A3 E3 p1 p2 p3 p4 [B]0 [K1]0 ∆Hrxn St RH ∆τad ε MAT

5.362e+7 m3/(kmol s) 80 479 kJ/kmol 2.807e+10 m6/(kmol2 s) 79 160 kJ/kmol 3.948e+10 m3/(kmol s) 69 975 kJ/kmol 1.403e+8 m3/(kmol s) 76 617 kJ/kmol 0.200 m3/kmol 0.032 m3/kmol -21.375 12 706 K-1 10.401 kmol/m3 0.062833 kmol/m3 63.0 kJ/mol 3.0778 1.1693 1.4213 1.3688 99 °C

case, two inversion points are detectable. The first one (labeled as point A and showing a concavity toward the left) is characterized by a trend inversion which implies the simultaneous reduction of both the maximum temperature and conversion when Tcool increases and can be referred to as the transition inversion in the D space. The second one (labeled as point B and showing a concavity toward the right) is characterized by a trend inversion which implies the decrease of the maximum dimensionless temperature and increase of the conversion when Tcool increases and can be referred to as the QFS of species D. The coolant temperature at which such an inversion occurs, Tcool,QFSD, is the boundary temperature between “runaway” conditions and QFS conditions for the desired species D. In this D space, the runaway loop originated by the “virtual” intersection of the ζ(ψMAX) vs ψMAX curve with itself is not detected because the investigated coolant temperature range was too narrow to generate it. Even in this case, the transition inversion is individuated by a concavity of the topological curve

Figure 5. Phase portrait for the single autocatalytic catalyzed reaction system considered. Coolant temperature and dosing stream temperature vary between 273 and 353 K. The axes parameters are ζ, ψ, and Tcool. Model parameters are reported in Table 3.


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The optimization procedure can be suitably complemented with a method useful to identify runaway boundaries, like the sensitivity one, as well as with dedicated experiments. While the first way would confirm that the operating conditions identified through the aforementioned optimization procedure are not beyond the runaway boundary of the whole system (therefore providing a cross validation of the results through two independent methods), the latter one would validate at a lab scale the optimization procedure itself for the specific case of interest. 5. Experimental Validation

Figure 6. D-space for the single autocatalytic catalyzed reaction system considered: acid esterification between propionic anhydride and 2-butanol catalyzed by sulfuric acid. Point A represents the transition inversion and point B the QFS inversion. Coolant temperature varies from 273 to 353 K. Model parameters as in Table 3. Experimental runs (see Table 6) are compared with the correspondent theoretical ones by the means of arrows: (9) no ignition, (O) runaway, (b) QFS, (- · -) no ignition region, (-) runaway region, and (--) QFS region.

toward the left while the QFS inversion is individuated by a concavity toward the right. 4. Optimization Procedure From the proposed general topological criterion for QFS detection, a procedure aimed to optimize dosing time (tdos) and mean coolant temperature (Tcool) arises straightforward along the same lines previously proposed.17 In the following, the desired product will be considered the generic species X: such a species may be an intermediate or a final product. In particular, for any isoperibolic SB process involving whatever complex kinetic scheme, tdos should be minimal to obtain the maximum productivity of species X fulfilling the safety constraints. The general optimization procedure can be summarized in the following steps, which are reported in Figure 7. (1) Search for microkinetic and thermochemical system parameters, reactor, and cooling equipment characteristics and definition of a suitable coolant temperatures [Tcool,MIN;Tcool,MAX] and dosing times [tdos,MIN;tdos,MAX] range. (2) Identification of the QFSX coolant temperature (Tcool,QFSX) in the X space for tdos ) tdos,n and Tcool ∈ [Tcool,MIN;Tcool,MAX] (analogously, we can identify QFSX dosing time (tdos,QFSX) in the X space for Tcool ) Tcool,n and tdos ∈ [tdos,MIN;tdos,MAX]). (3) At the QFSX conditions, maximum reactor temperature TMAX has to be checked with respect to the experimentally determined MAT. If TMAX > MAT, the dosing time should be increased (or the coolant temperature should be decreased) and step 2 repeated with this new value. (4) As a final check, conversion of species X at the end of the dosing time ζX,dos is required to be larger than a given threshold value ζX,MIN. If this is not true, the dosing time should be increased (or the coolant temperature should be decreased) and step 2 repeated with this new value. (5) Once Tcool and tdos values such that optimum QFSX conditions are attained and TMAX < MAT as well as ζX,dos > ζX,MIN, the time at which the maximum reasonable conversion of species X occurs has to be identified: such a time will be the stop time of the process. Practically, such a time is considered to be that corresponding to conversion variation reduces to 0.0001 min-1. The real stop time will be lower than this value as determined by a cost/benefit analysis.

5.1. Single Nonautocatalytic Reaction. As a first-case study involving a single nonautocatalytic reaction, an aromatic nitration by mixed acids has been analyzed. In the present case study, the species to be nitrated (that is, 4-chlorobenzotrifluoride, organic, or dispersed phase) is added to a mixture of sulfuric and nitric acids (aqueous or continuous phase) in which sulfuric acid acts both as a solvent and as a dehydrating agent vs nitric acid to form the nitronium ion, NO2+, which is the electrophilic species reacting with the aromatic ring. The system is heterogeneous liquid-liquid, and the overall reaction takes place in the continuous phase under a slow reaction regime. Since the reaction is carried out using a high excess of sulfuric acid and intense stirring (500 rpm), mass transfer phenomena occurring during synthesis are not limiting (slow reaction regime).20 As previously discussed, in this case only one QFS may be defined: the QFS with respect to the desired species C (4-chloro,3-nitrobenzotrifluoride). Since the most relevant model parameter, among all the system constitutive ones and initial conditions, is the coolant temperature, it is possible to generate the C space (ζ(ψMAX) vs ψMAX) necessary to optimize the process choosing Tcool as a portrait variable. The first step in the optimization procedure summarized in Figure 7 is the search for system microkinetics and thermodynamic parameters. Since this reacting system has been extensively investigated elsewhere,20 the microkinetic reaction rate equation (pre-exponential factor, A, and activation energy, Eatt) and all thermodinamic parameters (reaction enthalpy, ∆Hrxn, equilibrium distribution coefficient, m), necessary to generate the C space, are already known, as reported in Table 1. In order to experimentally validate the optimization procedure, the RC1 scale (Mettler Toledo, MP06, 1 L) has been selected. Process recipe (relative volume increase, ε, reactants number of moles, etc.) and reactor global heat transfer coefficient, (UA)0, at such a scale are also reported in Table 1. For this process the MAT parameter has been assumed equal to 95 °C for safety reasons,20 while the desired minimum conversion at the end of the dosing period, ζMIN, has been set equal to 0.85 to avoid an excessive degree of coreactant accumulation. Finally, according to physical constraints (no solidification of the reacting mixture must occur)20 and MAT value (no decomposition of the reacting mixture must occur), the following ranges of variation can be defined: Tcool ∈ [20-60] °C and tdos ∈ [180-1800] s. Moreover, a temperature step equal to 0.5 °C and a dosing time step equal to 60 s have been used into the iterating part of the optimization procedure. After these preliminary steps, starting from tdos,min ) 180 s, the optimum coolant temperature and dosing time can be searched for by performing the optimization procedure summarized in Figure 7 (case 1). The optimum Tcool-tdos couple

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Figure 7. Optimization procedure scheme.

has been found to be equal to 40-600 s, while the corresponding value of the maximum stop time, tstop, has been found equal to 3600 s. In order to validate the C space predictions for the analyzed system, a set of experimental isoperibolic RC1 runs has been carried out using the obtained optimum dosing time value, 600 s, and four different coolant temperature values into the selected range: 32, 35, 37, and 40 °C (the theoretical optimum coolant temperature), as shown in Figure 8 (where temperature and calorimentric conversion vs time curves have been reported). In particular, for each experimental run, maximum reactor temperature and calorimetric conversions corresponding to such a maximum and the end of the dosing time have been collected (see Table 4). Moreover, on the basis of such data and analysis of all the experimental temperature and calorimetric conversion profiles, each run has been classified into one of the three different thermal behavior classes available for an isoperibolic SB reactor7-10 whose operating variable to be optimized is the coolant temperature: no ignition, runaway, and QFS. Table 4 also reports the theoretical classification arising from analysis of the C space. A graphical comparison between experimental and theoretical results is shown in Figure 2. Each experimental run, which is

represented by a couple (ζMAX,exp,ψMAX,exp), has been located in the C space and compared with the corresponding theoretical one lying on the topological curve (where runaway and QFS regions are marked, respectively, with a continuous and a dashed line) by means of arrows. Little gaps between experimental and theoretical points are due to perturbations such as delays on reactant loadings or heat losses that may affect experimental temperature and calorimetric conversion profiles. It can be seen that each experimental run is located in the predicted thermal behavior region. In particular, runs 1 and 2, which show the typical sharp temperature profile (see Figure 2), are located in the runaway region, run 3, which exhibits hybrid temperature and conversion profiles, is located at the boundary between runaway and QFS conditions, and run 4, which clearly shows a smooth temperature profile and an almost linear conversion during the dosing period, is located into the QFS region. Moreover, from Table 4 we can also see that for run 4 (40-600 s) all constraints imposed by the optimization procedure are fulfilled so that its operating conditions can be considered, from both theoretical and experimental viewpoints, the most safe and productive for the analyzed process.


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Figure 8. Experimental temperature and calorimetric conversion vs time profiles for the aromatic nitration of 4-chlorobenzotrifluoride in mixed acids (see Tables 1 and 4).

Figure 9. Experimental temperature and calorimetric conversion vs time profiles for the solution homopolymerization of butyl acrylate in ethyl acetate initiated by AIBN (see Tables 2 and 5).

Table 4. Comparison between the Experimental and Theoretical Results Obtained for the Aromatic Nitration of 4-Chlorobenzotrifluoride in Mixed Acids

Table 5. Comparison between the Experimental and Theoretical Results Obtained for the Solution Homopolymerization of Butyl Acrylate in Ethyl Acetate Initiated by AIBN

run

Tcool (°C)

TMAX,exp (°C)

ζMAX,exp

ζdos,exp

experimental classification

proposed criterion

run

tdos (s)

TMAX,exp (°C)

ζMAX,exp

ζdos,exp


proposed criterion

1 2 3 4

32 35 37 40

98.0 103.3 97.5 94.0

0.89 0.84 0.76 0.78

0.44 0.92 0.90 0.90

RW RW QFS QFS

RW RW QFS QFS

1 2 3 4

1200 1800 3300 3900

82.8 82.0 76.7 75.4

0.85 0.64 0.42 0.80

0.88 0.76 0.77 0.78

RW RW QFS QFS

RW RW QFS QFS

5.2. Chain Reactions. As a second case study, the solution homopolymerization of butyl acrylate (BA, g99%, SigmaAldrich) in ethyl acetate (EtOAc, CHROMASOLV Plus for HPLC, 99.9%, Sigma-Aldrich) initiated by 2,2′-dimethyl-2,2′azodipropiononitrile (AIBN, g98%, Fluka) has been investigated. In particular, the monomer (that is, butyl acrylate) is added to an initial reactor load constituted by ethyl acetate (which acts as a solvent) and dissolved AIBN (initiator). The system is liquid homogeneous. Conversely to the single reaction case, the most relevant system constitutive parameter is the dosing time. Also in this case, as the system has been extensively studied elsewhere,21,22 pre-exponential factors, activation energies, and propagation reaction enthalpy, necessary to generate the P space, have been taken from the literature, as reported in Table 2. In order to experimentally validate the optimization procedure, the RC1 scale (Mettler Toledo, MP06, 1 L) has been selected. Process recipe (relative volume increase, ε, reactants number of moles, etc.) and reactor global heat transfer coefficient, (UA)0, at such a scale are summarized in Table 2. For this process the MAT value has been assumed equal to 77 °C, that is, 5 °C below the temperature above which the reacting mixture has been experimetally observed to start boiling vigorously, while the desired minimum conversion at the end of the dosing period, ζP,MIN, has been set equal to 0.80. Finally, according to physical constraints and the MAT value the following ranges of variation can be defined: Tcool ) 70 °C

(the maximum allowable value considering that the reactor will be operated in the isoperibolic mode) and tdos ∈ [300, 5400] s. A dosing time step equal to 300 s has been chosen to be used into the iterating part of the optimization procedure. After these preliminary steps, starting to iterate from tdos,min ) 300 s, the optimum dosing time is searched for by performing the optimization procedure summarized in Figure 7 (case 2). The optimum Tcool-tdos couple has been found equal to 70-3900 s, while the corresponding value of the maximum stop time, tstop, has been found equal to 8400 s (where a conversion of about 97.0% is achieved). In order to validate the P space predictions for the analyzed system, a set of experimental isoperibolic RC1 runs has been carried out using the coolant temperature value of 70 °C and four different dosing time values: 20, 30, 55, and 65 min. All experimental temperature and calometric conversion vs time curves are reported in Figure 9. In particular, for each experimental run, maximum reactor temperature and calorimetric conversion corresponding to such a maximum and the end of the dosing time have been collected (see Table 5). Moreover, on the basis of such data and analysis of the experimental temperature and calorimetric conversion profiles, each run has been classified into one of the three different thermal behavior classes available for an isoperibolic SB reactor whose operating variable to be optimized is the dosing time: runaway (RW), QFS, and starving (STV), as shown in Table 5, where the

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theoretical classification resulting from the topological criterion is also reported. A graphical comparison between experimental and theoretical results is shown in Figure 4. Each experimental run, which is represented by a couple (ζMAX,exp,ψMAX,exp), has been located into the P space and compared with the corresponding theoretical one lying on the topological curve (where runaway and QFS regions are marked, respectively, with a continuous and a dashed line) by means of arrows. It can be seen that each experimental run is located in the predicted thermal behavior region. In particular, runs 1 and 2, which show the typical sharp temperature profile (see Figure 4), are located in the runaway region, while runs 3 and 4, which clearly show a smooth temperature profile and an almost linear conversion during the dosing period, are located in the QFS region. Moreover, from Table 5 we can also see that for run 4 (70-3900 s) all constraints imposed by the optimization procedure are fulfilled so that its operating conditions can be considered, from both theoretical and experimental viewpoints, the most safe and productive for the analyzed process. 5.3. Single Autocatalytic Catalyzed Reaction. The last case study refers to the esterification between 2-butanol and propionic anhydride catalyzed by sulfuric acid. The system is a homogeneous liquid, and the only reaction developing heat is the esterification one. In this case, a phase space reduction is required because the number of exothermic reactions (one) is lower than the number of independent mass balance equations needed to characterize completely the kinetic scheme (as previously discussed). However, since there is a single exothermic reaction, only one QFS may be defined: the QFS with respect to the desired species D. Analogously to the first case, the most relevant system constitutive parameter is the average coolant temperature. Therefore, it is possible to generate the D space (ζ(ψMAX) vs ψMAX) necessary to optimize the process using Tcool as a portrait variable. Also for this case study, the reacting system has been extensively investigated elsewhere.10,19,23 The microkinetic (preexponential factors and activation energies) and thermodinamic ˜ rxn, etc.) necessary to generate parameters (reaction enthalpy, ∆H the D space are summarized in Table 3. In order to validate experimentally the developed procedure, literature experimental data10,23 carried out at the RC1 scale (Mettler Toledo, AP01, 2 L) have been used. Process recipe (relative volume increase, ε, reactants number of moles, etc.) and reactor global heat transfer coefficient, (UA)0, at such a scale are also reported in Table 3. For this process the MAT value has been assumed equal to 85 °C for safety reasons. The desired minimum conversion at the end of the dosing period, ζMIN, has been set to 0.90 in order to avoid an excessive degree of coreactant accumulation. According to physical constraints10 and the MAT value the following ranges of variation can be defined: Tcool ∈ [0, 70] °C and tdos ∈ [300, 5400] s. Moreover, a temperature step equal to 1 °C and a dosing time step equal to 300 s have been used in the iterative part of the optimization procedure. After these preliminary steps, starting from tdos,min ) 300 s, the optimum coolant temperature and dosing time can be searched for by performing the optimization procedure summarized in Figure 7 (case 1). The optimum Tcool-tdos couple has been found equal to 50-3600 s, while the corresponding value of the final time, tstop, has been found equal to 3900 s (where a conversion of about 99.5% is achieved).

Table 6. Comparison between the Experimental and Theoretical Results Obtained for Acid Esterification between Propionic Anhydride and 2-Butanol Catalyzed by Sulfuric Acid run

Tcool (°C)

TMAX,exp (°C)


proposed criterion

1 2 3 4

11 25 50 60

14.1 94.9 80.0 78.8

NO IGN RW QFS QFS

NO IGN RW QFS QFS

In order to validate the D-space predictions for this system, the set of experimental isoperibolic RC1 runs reported in Table 6,10,23 which have been carried out at a dosing time of 3600 s and four different coolant temperature values into the seleced range 11, 25, 50, and 60 °C, has been used. Since no information about conversion with respect to the desired product D has been reported in the literature, the experimental thermal classification provided by Alo´s et al.10 and Strozzi et al.23 has been directly compared with the theoretical one arising from the topological curves generated into D space. Such a comparison has also been reported in Table 6. A graphical comparison between experimental results (represented by dots generated by solving eqs 7 using model parameters listed in Table 3 and Tcool,exp reported in Table 6) and theoretical classification arising from the topological curve, where no-ignition, runaway, and QFS regions are marked, respectively, with a dashed-dotted, a continuous, and a dashed line, is shown in Figure 6. It can be seen that each experimental run is located in the predicted thermal behavior region. In particular, run 1, which has been experimentally classified by Alo´s10 as a no ignition because of its low temperature increase (only 3 °C), is located into the no ignition region, run 2, which has been experimentally classified as a runaway10 because its extremely high temperature increase (70 °C) is located in the runaway region, while runs 3 and 4, which have been experimentally classified as QFS10 because of their moderate temperature increase (respectively, 30 and 18 °C), are located in the QFS region. Moreover, from Table 6 we can also see that for run 3 (50-3600 s) all constraints imposed by the optimization procedure are fulfilled so that its operating conditions can be considered, from both theoretical and experimental viewpoints, the most safe and productive for the analyzed process. 6. Conclusions In this work the chemical topology theory, previously used to optimize an isoperibolic SB process involving consecutive reactions,17 has been extended to whatever complex kinetic scheme. In particular, it has been shown that any inversion point detectable onto the X space is associated to a thermal behavior change that defines the character of the inversion point. Two different general characters have been individuated regardless of the generating phase portrait parameter: the transition character and QFS character. The first one refers to an inversion point corresponding to that where the topological curve shows a concavity toward the left, while the second one refers to an inversion point corresponding to that where the concavity is toward the right. Once the desired QFS conditions have been individuated, a suitable optimization procedure can be used to fully optimize the process. All theoretical predictions arising from the topological criterion and its related procedure have been then experimentally validated through a dedicated set of isoperibolic RC1 experiments and literature data. The good agreement between theoretical predictions and experimental results demonstrates that the


inversion points observable on the topological curve represent real thermal behavior boundaries that can be experimentally detected. Nomenclature A ) pre-exponential factor, m3/(kmol · s) or (1/s) B ) species B C ) species C, molar concentration, kmol/m3 c˜p ) molar heat capacity, kJ/(kmol · K) D ) species D Dai ) Damko¨hler number of the ith reaction Eatt ) activation energy, kJ/kmol fi ) dimensionless concentration function of the ith reaction ∆Hrxn ) reaction enthalpy, kJ/kmol krif,i ) Ai · exp(-Eatt,i/RTrif), kinetic constant at the reference temperature m ) reaction order with respect to species B, equilibrium distribution coefficient MAT ) maximum allowable temperature, °C or K n ) reaction order with respect to species A, number of moles of a species, mol R ) gas constant ) 8.314 kJ/(kmol · K) REi ) reactivity enhancement factor of the ith reaction12,13,17 RH ) (F˜ dos · c˜p,dos)/(F˜ 0/mix · c˜p,0/mix) heat capacity ratio St )((UA)0 · tdos)/(F˜ 0/mix · c˜p,0/mix · V0), Stanton number t ) time, s T ) temperature, K (UA)0 ) initial overall heat transfer coefficient, kW/(K) V ) volume, m3 Subscripts and Superscripts A ) referred to as the species A ad ) adiabatic att ) referred to as the activation energy B ) referred to as the species B C ) referred to as the species C cool ) coolant D ) referred to as the species D dos ) dosing stream or dosing time exp ) referred to as the experimental data i ) reaction number index, referred to as the initiation reaction j ) generic model constitutive parameter M ) referred to as the monomer species MAT ) at the maximum allowable temperature MAX ) maximum value of a quantity or at the maximum value of a quantity N ) dimension of a variety NR ) referred to as the total number of reactions in a process p ) referred to as the propagation reaction P ) referred to as the polymer species QFS ) “quick onset”, “fair conversion”, “smooth temperature profile” rif ) referred to as the reference conditions stop ) referred to as the stop time td ) referred to as the termination by disproportion reaction 0 ) start of the dosing period f ) referred to as a vector Greek Symbols R ) factor accounting for the rection regime β ) factor accounting for the number of phases γ ) Eatt/RTrif, dimensionless activation energy δ ) 1 - FM/FP, monomer/polymer contraction factor

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˜ rxn,i) · nA/M,dos)/(F˜ 0/mix · c˜p,0/mix · V0 · Trif), dimensionless ∆τad ) ((- ∆H adiabatic temperature rise of the ith reaction ζ ) dimensionless concentration or conversion ε ) (Vdos)/(V0), relative volume increase at the end of the semibatch period κ ) exp(γ(1 - 1/τ)), dimensionless kinetic constant ν ) stoichiometric coefficient F and F˜ ) density and molar density, kg/m3 and kmol/m3 ϑ ) (t)/(tdos), dimensionless time τ ) T/Trif, dimensionless temperature eff τcool ) effective dimensionless coolant temperature ψ ) T/Tcool b/dϑ φi ) generic functions of ϑ, b ζ, and dζ ω ) overall heat transfer coefficient contraction factor due to the viscosity increase at high monomer conversion

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ReceiVed for reView October 4, 2010 ReVised manuscript receiVed November 28, 2010 Accepted December 2, 2010 IE102014W

Topological Criterion To Safely Optimize Hazardous Chemical

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