Topological Criteria to Safely Optimize Hazardous Chemical

Ind. Eng. Chem. Res. 2010, 49, 4583–4593

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Topological Criteria to Safely Optimize Hazardous Chemical Processes Involving Consecutive Reactions 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 operating conditions, for strongly exothermic chemical systems involving multiple reactions, are particularly critical to be obtained, because of the complex interactions between selectivity and safety constraints. In this work, new criteria, which are useful for isoperibolic semibatch processes involving consecutive reactions and based on the topology of a particular phase space, are presented. Such criteria are able to detect both the runaway boundary and the so-called “quick onset, fair conversion, smooth temperature profile” (QFS) operating region by performing a topological analysis of the phase space of the system of ordinary differential equations (ODEs) that describe the analyzed process. Moreover, a safe optimization procedure, the objective of which is to obtain the optimum values both of the dosing time of the dosed reactant and the initial reactor temperature, based on such criteria, has been developed. Finally, such a set of optimum operating parameters has been validated through a comparison with experimental data from published literature. 1. Introduction In fine chemistry and pharmaceutical industries, fast and strongly exothermic reactions are often performed to produce many different intermediates, which must be further processed to obtain the desired final products. Such reactions, which are usually performed in discontinuous reactors, may trigger a phenomenon known as “thermal runaway”, as a consequence of a loss of the reactor temperature control when the rate of heat removal (through the refrigerating system) becomes lower than the rate of heat production of the synthesis reactions. To control the rate of heat generation, such reactions may be performed in semibatch reactors (SBRs) (which is one possible operating mode of a discontinuous reactor) where one or more reactants (called co-reactants) are dosed on an already-loaded reacting mixture. Consequently, the heat-release rate is controlled by the feeding rate of the co-reactants and, if any refrigerating system failure occurs during the dosing time, stopping the dosing stream can be enough to prevent a loss of reactor temperature control. However, if the process is performed under high co-reactant accumulation conditions (that is, a large amount of unreacted co-reactants accumulates in the reactor), the reactor temperature can increase up to temperature values at which secondary undesired reactions or decomposition reactions are triggered. Such reactions (apart from a reduction of the selectivity, with respect to the desired product) can lead to a thermal loss of control, since they are usually extremely fast and exothermic. Moreover, if a decomposition event is triggered because of the thermal loss of control, the consequent release of uncondensable gas may pressurize the reactor, leading to its venting through the installed emergency pressure relief systems and, possibly, to its physical explosion. The amount of co-reactant accumulation and the maximum temperature above which undesirable phenomena may be triggered are two fundamental information items needed to perform a safe optimization of a SB process involving fast and exothermic reactions. * To whom correspondence should be addressed: Fax: +39 0223993180. E-mail: [email protected].

During the last 25 years, different studies on the detection of the so-called “runaway boundary” or “marginal ignition line”sthat is, the system operating parameters where runaway phenomena are triggeredshave been conducted.1-19 Usually, studies whose objective is to detect runaway boundaries include isoperibolic semibatch reactors involving a single synthesis reaction. One criterion that is able to handle multiple reaction systems and different reactors (e.g., batch, continuously stirred tank reactor (CSTR), plug-flow reactor (PFR)) is the generalized parametric sensitivity criterion,1 with the shortcoming of being implicit; that is, it is not able to establish if the system operates under runaway conditions only by evaluating the constitutive model parameters. The criterion identifies the runaway boundary looking for the maximum of a suitable objective sensitivity coefficient, that is, the normalized sensitivity coefficient of the maximum temperature with respect to any operating parameter: sTmax,φj )

φj dTmax Tmax dφj

A different general criterion for the detection of runaway phenomena, able to operate with multiple reactions, different operating modes and reactor typologies, is the divergence criterion.2-5 Such a criterion states that if the system of ordinary differential equations (ODEs) that describes the analyzed process exhibits positive divergence, the process itself is operated under runaway conditions. Consequently, the runaway boundary is individuated by keeping constant the values of all the constitutive parameters except one (e.g, the coolant temperature) and then by setting the divergence expression to zero, with respect to the variable model parameter. However, from a practical viewpoint, the optimization of all (or some) operating parameters (e.g., coolant temperature and dosing time) is the most relevant aspect. Knowing the runaway boundary allows one to avoid the selection of potentially hazardous operating conditions, but it does not allow one to optimize the analyzed process. The first combined “safety-optimization” approach for isoperibolic SBRs was developed by Hugo and Steinbach,6 for homogeneous liquid reactors, and then by Steensma and

10.1021/ie901679q  2010 American Chemical Society Published on Web 04/14/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010

Figure 1. Boundary diagram for an isoperibolic semibatch process involving one reaction.14

Westerterp,7-9 for heterogeneous liquid-liquid reactors, with the construction of boundary diagrams (BDs). These diagrams are generated in a suitable dimensionless space (i.e, reactivity (Ry) versus exothermicity (Ex)) that is able to identify not only a runaway boundary but also a runaway region (see Figure 1). The area inside the continuous bold curve is the “runaway” or “excessive co-reactant accumulation” region; outside this region, there are the “no-ignition” region, for low-reactivity numbers, and the “inherently safe” region, for high-reactivity numbers. The last area, which is also referred to as the “quick onset, fair conversion, smooth temperature profile” (QFS) region,7 is characterized by an almost-linear conversion to the desired product, with respect to the dosing time, and by an almost pseudo-stationary reactor temperature profile, which means high cooling efficiency. However, note that the definition of the QFS region (and, consequently, of the boundary that divides such a region from the runaway region) is somehow arbitrary.10 Moreover, it is true that an isoperibolic SBR operating in the QFS region is safe from the co-reactant accumulation viewpoint (no runaway that is due to the desired reaction can occur) and it exhibits a high conversion degree at the end of the dosing time, but nothing is known about either the thermal stability of the system in such a region (since the mathematical model, which generates the BD, is based on the thermal effects associated with the synthesis reaction only) or the optimum set of operating parameters. To face the first problem, so-called “temperature diagrams” (TDs) have been proposed.11 Such diagrams allow upperbounding the maximum temperature increase as a function of some parameters that define the BDs (that is, Ex and Ry). The value of such a maximum temperature rise can be compared with the experimentally determined value of the maximum allowable temperature (MAT) for the considered system.11 If the maximum reactor temperature is above the MAT value, the operating parameters are not safe from the thermal stability viewpoint and they must be rejected. The second problem (optimization) is more complex to face. Using a generalized form of BDs and TDs, a simple procedure to estimate the minimum dosing time that guarantees both the effective maximum reactor temperature to be lower than the MAT and the system to be operated in the QFS region has been developed.12,13 However, such a procedure can be used only for isoperibolic SBRs that involve a single exothermic reaction

and it utilizes an arbitrary definition of optimum operating conditions (that is, the QFS one). The problem of having an arbitrary definition of the QFS conditions has been solved by Alo`s et al.,10 who proposed, for reacting systems involving a single reaction, a clear definition of QFS using a suitable objective function. In particular, the QFS of an isoperibolic SBR is individuated, for a given value of the dosing time, by the coolant temperature (Tcool,QFS), at which the objective function “time at which the maximum reactor temperature, ϑ(τjmax), occurs” exhibits a minimum. The intrinsic nature of such a criterion can be proved by calculating the parametric sensitivity of ϑ(τjmax), with respect to each system parameter (φj), sϑ(τjmax),φj, as a function of Tcool. In correspondence of Tcool,QFS, it may be observed that sϑ(τjmax),φj ) 0, regardless of the φj parameter considered. This means that the selected objective function exhibits an intrinsic minimum, regardless of the parameter considered. This criterion arises from an analysis of the target line used to generate BDs,14,15 which is an ideal temperature profile that an isoperibolic SBR exhibits when the co-reactant conversion is linear during the dosing time and all the heat developed by the desired reaction is readily removed by the cooling system. Such an ideal temperature profile is shown in Figure 2, together with a real SBR temperature profile. As it can be easily observed from Figure 2, the maximum of the ideal target temperature profile occurs at time zero. Such an observation explains the aforementioned choice of the objective function, ϑ(τjmax), for the definition of the QFS condition.10 Thus, in summary, for a multireaction system, it is not yet clear how to define the QFS conditions of an isoperibolic SBR without involving any arbitrary definition, as well as how to develop a suitable procedure to identify safe and productive operating conditions. In this work, a new criterion to identify QFS conditions, which is based on the topology of the phase space of the system analyzed (and, therefore, it is free from any arbitrary definition), has been developed. Moreover, this criterion has been used to develop a safe optimization procedure for isoperibolic semibatch (SB) processes involving consecutive reactions. 2. Mathematical Model To identify the QFS condition for an isoperibolic SB process involving consecutive reactions, it is necessary to specify the product to which the QFS condition refers. In fact, when more than one reaction proceeds, a relevant selectivity problem arises and more than one QFS condition may be detected, depending on the product in which one is interested. Among all the reacting systems involving consecutive reactions, in this work, the following reacting system (which has been thoroughly investigated experimentally in the literature16,17) has been considered, as a case study: the oxidation of 2-octanol to 2-octanone by means of nitric acid (60% w/w). The kinetic scheme of the process may be represented by the following two consecutive reactions: A + B f 2B + C

(1)

B+CfD

(2)

where A is 2-octanol, B is the nitrosonium ion (NO+, which is the species responsible for the oxidation of species A and C and also for the autocatalytic behavior of reaction 1), C is 2-octanone (a product of reaction 1), and D is a mixture of carboxylic acids (the products of reaction 2).


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Figure 2. Generic isoperibolic SBR temperature profile (dotted line) and the corresponding target temperature ideal profile (bold line).

This case study presents many peculiar aspects, which are of interest when the safe optimization of consecutive reactions systems is considered. As several oxidations of organic compounds in nitric acid, the reactions are fast, they are exothermic, and they exhibit a strong selectivity problem with respect to the intermediate species C (2-octanone). Particularly, the undesired oxidation of 2-octanone leads to the irreversible formation of a mixture of carboxylic acids (referred to as a pseudo-species D) constituted by two molar parts of esanoic acid and acetic acid with one molar part of eptanoic acid and formic acid.16 Such an oxidation (reaction 2) is much more exothermic than that of 2-octanol to 2-octanone (reaction 1) and therefore, besides the loss of selectivity with respect to 2-octanone (C), it releases a large amount of heat, increasing the reactor temperature toward the mixture thermal threshold. As a further peculiarity, the first reaction exhibits an autocatalytic behavior. Therefore, for such a system, it is crucial to perform a safe optimization that is able to both minimize the risk of a runaway phenomenon and maximize the productivity of the desired product. The mathematical model that describes this reacting system is based on some hypotheses, namely, the following: (i) the system is heterogeneous: two liquid phases exist: an “organic” (or “dispersed”) one and an “aqueous” (or “continuous”) one; the latter is present in large amounts, with respect to the organic phase; (ii) species A, C, and D are present almost only in the organic phase, whereas species B is only present in the aqueous phase; (iii) oxidations happen only in the aqueous phase (where the nitrosonium ion, which is the species responsible for the oxidation, is present); (iv) the reactions are performed in an isoperibolic (constant Tcool) semibatch reactor in which species A is dosed above an initial charge of species B and nitric acid; (v) no phase inversion may occur; and (vi) the microkinetic rate expressions are represented by the following modified power laws: n r1 ) k1(T)ψ1(C)CA,acq CmB,acq

(3)

r2 ) k2(T)ψ2(C)CpC,acqCqB,acq

(4)

where n, m, p, and q are the reaction orders; k1 and k2 are the kinetic constants of reactions 1 and 2 (depending on temperature); Ψ1(C) and Ψ2(C) are whatever functions of the concen-

trations of reactants, products, and catalytic substances involved into the system, as discussed elsewhere.18 Material and energy balances for such a SBR, in dimensionless form, lead to the following equations:16,17

{

{

(RHεϑ + 1)

dξC ) Da1RE1f1κ1 - Da2RE2f2κ2 dϑ dξD ) Da2RE2f2κ2 dϑ IC ξC(ϑ ) 0) ) ξD(ϑ ) 0) ) 0

(

)

(5)

( )

dξD dξC dξD dτ ) ∆τad,1 + + ∆τad,2 dϑ dϑ dϑ dϑ

[Coε(1 + εϑ) + RHε](τ - τeff cool) 0 e ϑ < 1 (RHε + 1)

(

)

( )

dξD dξC dξD dτ ) ∆τad,1 + + ∆τad,2 dϑ dϑ dϑ dϑ

[Coε(1 + ε) + RHε](τ - τeff cool) ϑ g 1 IC

τ(ϑ ) 0) ) τ0 (6)

The meanings of all the terms involved in these equations are reported in the Nomenclature section. 3. Phase Spaces and Topology: Definition of the QFS Condition For potentially runaway systems, safety aspects have a strong influence on the optimization of the entire process. For the 2-octanol-2-octanone system, the optimization problem is further complicated because (i) the desired product is an intermediate (selectivity problem); (ii) the undesired degradation of the intermediate (2-octanone) is extremely fast and exothermic and it can lead to decomposition of the reacting mixture; and (iii) the first reaction exhibits autocatalytic behavior. Particularly, the last aspect strongly affects the method to be used to identify the runaway boundary (or the marginal ignition line, which is the transition between the “no-ignition state” and the “runaway state”). The most robust available criteria that are able to address autocatalytic systems are the generalized parametric sensitivity criterion1 and the target line criterion.16,17 The latter is useful only when a target temperature profile can be defined. The other two aspects (namely, the selectivity problem and the exothermicity of reaction 2) directly affect the optimization procedure. To face such problems, it is important to understand that, in this case, two different QFS conditions, leading to two different optimization problems, can arise: the first one is related

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to the production of the intermediate C, and the second one is related to the production of the final product D. In fact, as discussed elsewhere, with reference to the use of the BD methods for multireaction systems,18 several different behaviors can be found, depending on both thermokinetic and operating parameters. The most general behavior is expected when the operating conditions allow the triggering of the two consecutive reactions at very different temperature values. Starting from a low coolant temperature, the system is basically not ignited and the maximum reactor temperature is very low, as the reactant conversions progress. As a consequence, the co-reactant accumulates into the reactor. However, the reactor temperature is so low, no safety problems are expected (no ignition conditions). By increasing the coolant temperature, the maximum value of the reactor temperature increases and the ignition of the first reaction occurs: the marginal ignition conditions for reaction 1 (MI1) are reached. Beyond this point, the operating conditions become usually unsafe, because there is an excessive accumulation of reactant A and reaction 1 is ignited. As a matter of fact, even if the reaction heat of the first reaction is much lower than that of the second one and, consequently, these conditions could be considered suitable from the safety point of view (because the more-exothermic reaction is not triggered yet), they are not suitable from a productivity point of view, because they are far from the optimal QFS conditions for the desired intermediate C. Further increases of the coolant temperature reduce the accumulation of co-reactant and, if the second reaction is not triggered, the reactor temperature profile attains QFS conditions for reaction 1 (QFS1). The maximum value of the reactor temperature increases again if the coolant temperature is increased, because the second reaction also becomes ignited, leading first to a thermal loss of control in reaction 1 and, suddenly after, to a thermal loss of control in reaction 2. In other words, a second marginal ignition of the first reaction (MI12) is attained just before the marginal ignition of the second reaction (MI2), where the thermal control of both reactions is lost. Increasing the coolant temperature again finally leads to QFS conditions for reaction 2 (QFS2). However, not all these operating conditions (namely, MI1, MI12, MI2, QFS1, and QFS2) can be attained in practice, depending on the actual thermokinetic parameters of the reacting system, as well as the operating parameters of the SBR. In this work, the most general case involving all the five boundaries is discussed. Particular cases where some of these boundaries overlap, leading to the disappearance of QFS1, are not of interest, because, in these cases, no optimization for C production is possible. The identification of safe operating conditions with respect to the production of either C or D requires the definition of the QFS conditions, avoiding arbitrary definitions, and it can be done by generating and analyzing the “phase space” of the system at different coolant temperatures (that is, by generating the so-called “phase portrait”). Actually, such a “phase portrait” is quite unusual, because (i) it is a parametric surface (with the surface-generating parameter being the coolant temperature (Tcool), which is equal to the initial reactor temperature (T0)) generated by an envelope of trajectories (with the single trajectory-generating parameter being the dimensionless time, θ); and (ii) the surface can be generated by varying Tcool in a suitable range while all the other parameters are kept constant. Moreover, temperature (which represents one of the coordinate

Figure 3. Phase portrait. Coolant temperature varies between 260 K and 325 K, and the dosing stream temperature is 293 K. The axes parameters are ξC, ξD, and jτ ) τ/τcool. All of the model parameters for the experimentally investigated system (eqs 1 and 2)16,17 are reported in Table 1. Table 1. Kinetic and Constitutive Model Parameters for the System Considereda

a

parameter

value

A1 Eatt,1/R mH0,1 ∆H1 A2 Eatt,2/R mH0,2 ∆H2 Co RH n)m)p)q ξB0 Da1RE1 Da2RE2 γ1 γ2 ∆τad,1 ∆τad,2 ε Ψ1(C) Ψ2(C)

2 × 10 m3/(kmol s) 11300 K 6.6 160 kJ/mol 1.67 × 1012 m3/(kmol s) 12000 K 2.2 520 kJ/mol 21 0.6 1 0.035 9.91 × 10-8 9.62 × 10-4 37.7 40 0.37 1.21 0.4 exp(-6.6H0) exp(-2.2H0) 7

Data taken from refs 16 and 17.

axes defining the “phase space”) is made dimensionless by means of the Tcool value used to generate each trajectory, τ)

τ T ) τcool Tcool

to make all of the trajectories on the same plot comparable. Figure 3 reports an example of such a “phase portrait”. All model parameters used to generate Figure 3 and the successive figures are reported in Table 1. The parametric surface reported in Figure 3 defines a socalled “topological manifold”. Topology studies the “base properties” of an n-dimensional space, namely, the properties that depend only on the fold of a manifold; distances, which are fundamental to study the geometry of a space, are ignored. Topology is very useful to analyze the behavior of a system of three ODEs, such as the problem here considered, which is represented by eqs 5 and 6, because it leads to a threedimensional (3-D) differential manifold that can be analyzed in a 3-D space through the correspondent “phase portrait”. Referring to Figure 3, this “phase portrait” does not show any detectable properties. However, some interesting topological properties can be identified by focusing on the maxima of the trajectories, with respect to jτ. To better visualize such maxima, they can be reported on the dimensionless temperature-dimen-


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Figure 4. C-space: maxima of trajectories with respect to jτ (τjmax), as a function of dimensionless C concentration, in correspondence to jτmax (ξC(τjmax)). Symbols indicate different coolant temperature values (from 260 K to 325 K); the difference of Tcool values between two near symbols is equal to 1 K. System parameters are as described in Table 1.

Figure 5. D-space: maxima of trajectories with respect to jτ (τjmax), as a function of dimensionless D concentration, in correspondence to jτmax (ξD(τjmax)). Symbols indicate different coolant temperature values (from 260 K to 325 K); the difference of Tcool values between two near symbols is equal to 1 K. System parameters are as described in Table 1.

sionless C concentration plane (τj vs ξC, hereafter referred to as C-space) and on the dimensionless temperature-dimensionless D concentration plane (τj vs ξD, hereafter referred to as D-space). The resulting diagrams are reported in Figures 4 and 5, respectively. On these diagrams, each symbol represents a different Tcool value, which increases by a 1 K step. This means that the Tcool difference between two close symbols is always the same (equal to 1 K) and allows for identifying discontinuities in the system behavior, arising from the same increase in Tcool. The proposed criterion for identifying QFS conditions utilizes the topological properties of the curves shown in Figures 4 and 5, particularly, the turning point of such curves. A clear change in the behavior of the system can be associated to any turning point, where an inversion of the curve tendency occurs, as will be discussed in the following. 3.1. QFS of Species D (QFS2). Observing Figure 5, it is possible to notice that, as Tcool starts to increase from the initial value of 260 K, a smooth increase of both jτmax and ξD(τjmax) follows. However, at a given Tcool value, a sharp transition between “no-ignition” reactor behavior (low maximum temperature and low D concentration, point C in the figure) and “runaway” reactor behavior (high maximum temperature and

D concentration, point D in the figure) is clearly detectable. During such a transition, following a 1 K increase in Tcool, the maximum dimensionless temperature increases from 1.08 to 1.55 and the corresponding D concentration increases from 0.11 to 0.59. Beyond point D, when the coolant temperature increases again, an inversion of the behavior of the maximum temperature and its relative D concentration occurs, because both variables start to decrease. Such an inversion will be referred to as the first inversion in the D-space, and it is characterized by a simultaneous reduction of both the maximum temperature and its relative D concentration when Tcool increases. This indicates that the thermal loss of control occurs when the dosing period is not finished yet: increasing the coolant temperature leads to a lower co-reactant accumulation and, consequently, to a reduction of the magnitude of the runaway phenomenon. For a further increase of the coolant temperature, another tendency inversion in the D-space occurs. Beyond point E, the maximum dimensionless temperature still diminishes (even if less and less) whereas the dimensionless D concentration increases. This inversion will be referred to as the second inversion in the D-space. The coolant temperature at which such an inversion occurs (Tcool,2°INV,D) is the boundary temperature between “runaway” conditions and QFS conditions for reaction 2. This can be easily understood by comparing the results of the proposed criterion with those of the generalized criterion for defining QFS conditions that was previously proposed by Alo`s at al.10 Such a criterion was developed for single reaction systems and identifies the QFS boundary as the Tcool value where the function ϑ(τjmax) vs Tcool shows a local minimum. For the system considered here (involving two consecutive reactions), the shape of the ϑ(τjmax) vs Tcool is shown in Figure 6. In this case, ϑ(τjmax) shows two minima, whose physical meaning is not straightforward. The same figure also reports the dimensionless D concentration, in correspondence of the maximum reactor temperatures, ξD(τjmax), as a function of Tcool: this last function shows a local minimum that corresponds with the turning point E (see Figure 5). Moreover, from Figure 6, it is possible to observe that the function ξD(τjmax) vs Tcool exhibits a local minimum that corresponds to the same coolant temperature at which the function ϑ(τjmax) vs Tcool exhibits a local minimum (Tcool,2°INV,D).

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Figure 6. Dimensionless D concentration (ξD(τjmax)) and time (ϑ(τjmax)), in correspondence of the maximum temperatures, as a function of the coolant temperature (Tcool). System operating parameters are as described in Table 1.

Figure 7. Objective parametric sensitivity coefficients of ξD(τjmax), with respect to the relevant system constitutive parameters (namely, Tcool, Co, γ1, ∆τad,1, γ2, ∆τad,2), as a function of coolant temperature (Tcool). System parameters as in Table 1.

This cross-validates the proposed criterion and that proposed by Alo`s at al.10 (suitably extended to multiple reactions systems), and it gives a clear interpretation of the physical meaning of the second minimum of the ϑ(τjmax) vs Tcool function: it is the boundary between runaway and QFS2 conditions. Moreover, the Tcool,2°INV,D value determined from the D-space agrees well with the experimentally determined QFS2 value.16,17 The intrinsic characteristic of the proposed criterion can be easily demonstrated by analyzing its dependence on the considered parameter (which is Tcool in the curve reported in Figure 5). As previously mentioned, at the turning point E (which identifies QFS2 boundary), the ξD(τjmax) vs Tcool function shows a local minimum. In other words, the sensitivity coefficient sξD(τjmax),Tcool is equal to zero, in correspondence to the turning point E. If the proposed criterion is independent from the considered parameter, all the sensitivity coefficients sξD(τjmax),φj should be equal to zero at the QFS2 boundary. The behavior of the sensitivity coefficients of ξD(τjmax), with respect to the relevant constitutive parameters of the system, as a function of Tcool, is shown in Figure 7. It is possible to observe that, at a coolant temperature equal to Tcool,2°INV,D, the dimensionless D concentration, in correspondence to the maximum reactor temperature, exhibits zero

Figure 8. D-space: maxima of trajectories with respect to jτ (τjmax), as a function of dimensionless D concentration, in correspondence to jτmax (ξD(τjmax)). Symbols indicate different coolant temperature values (from 260 K to 325 K); the differences in Tcool between two near symbols equal to 1 K. System parameters are as described in Table 1, except Co ) 65.

sensitivity, with respect to all of the considered parameters. This indicates a generalized behavior of the proposed criterion. Turning point E constitutes a potential optimum operating point to safely optimize the reactor productivity, with respect to species D. It is referred to as a “potential” optimum operating point, because also other parameters, besides the coolant temperature, must be optimized (in particular, the dosing time). It is worth mentioning that, in the D-space, a sort of loop, originated by the “virtual” intersection of the jτmaxvs ξD(τjmax) curve with itself (the uniqueness theorem ensures that no intersections may occur between two different trajectories in a phase space; because the D-space is just a two-dimensional (2D) projection of the 3-D phase space shown in Figure 3, it can show intersections that do not exist in the phase space) may be detected. Such a behavior has been already observed by Steinbach,19 in a different dimensionless space, for single reaction systems. Also, in agreement with Steinbach findings,19 in this case, the presence of a loop in the D-space indicates the existence of a runaway region that divides two safe operating regions (at least for that which concerns D-species, that is, reaction 2): MI2 and QFS2. If the D-space does not exhibit a loop, no runaway phenomena due to the production of species D (that is, to reaction 2) can occur in the range of coolant temperatures analyzed: the system operates in the so-called invariably safe conditions.16,17 Increasing the cooling number from 21 to 65, the loop in the D-space disappears, because the maximum temperature always increases as the coolant temperature increases (see Figure 8); these operating conditions are inherently safe, with respect to the triggering of reaction 2 and no QFS2 conditions can be identified in agreement with the experimental results.16,17 3.2. QFS of Species C (QFS1). As it can be observed from Figure 4, when increasing Tcool from the lowest value of 260 K, a smooth increase of both jτmax and ξC(τjmax) follows, up to the first turning point (point A in Figure 4), where ξC(τjmax) starts to decrease. Turning point A will be referred to as the first inversion in the C-space, and it is representative of the marginal ignition of the first reaction (MI1). However, no sharp transition arises around MI1, because the temperature smoothly increases at first and then smoothly decreases. A maximum in the sensitivity coefficient of jτmax, with respect to Tcool, can be found around the turning point A. However, since reaction 1 is much less exothermic than reaction 2, the boundary between no ignition and runaway conditions is not well-defined, as evidenced by the absence of a sharp peak of sjτmax,Tcool vs Tcool. This means that, beyond MI1, a real runaway is not expected, because the heat of reaction 1 is too low; however, an excessive


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Figure 9. Dimensionless C concentrations (ξC(τjmax)) and time (ϑ(τjmax)), in correspondence to the maximum temperatures, as a function of the coolant temperature (Tcool). System operating parameters are as given in Table 1.

accumulation of co-reactant leads to operating conditions beyond MI1 that are not suitable, at least from a productivity point of view, because they are far from the optimal QFS conditions for the desired intermediate C, even if the more-exothermic reaction 2 is not triggered. When continuing to increase the coolant temperature, a second inversion in the C-space occurs (point B). As previously discussed for the D-space, also, in this case, in correspondence to such a Tcool,2°INV,C value, the function ξC(τjmax) vs Tcool exhibits a local minimum (see Figure 9) that overlaps with the first local minimum exhibited by the function, ϑ(τjmax) vs Tcool. This cooling temperature value is the boundary between runaway (or, in this case, excessive accumulation conditions) and QFS conditions for reaction 1. Once again, this cross-validates the two criteria that have been investigated: the one proposed in this work and that previously proposed,10 suitably extended to multiple reactions systems. Therefore, the physical meaning of the first minimum in ϑ(τjmax) vs Tcool curve is a boundary between runaway and QFS1 conditions. As previously discussed for the QFS2 boundary in the D-space, the intrinsic characteristic of the proposed criterion for identifying QFS1 boundary can be investigated looking at the value of sξC(τjmax),φj in correspondence with the turning point B. Also, in this case, the sensitivity coefficients of ξC(τjmax), with respect to the relevant constitutive model parameters, as a function of Tcool, has been computed, as shown in Figure 10. As a consequence, in correspondence with Tcool,2°INV,C, all of the sensitivity coefficients are equal to zero, showing that ξC(τjmax) is zero-sensitive, with respect to any relevant constitutive model parameter. As previously discussed, this indicates a generalized behavior of the proposed criterion. Also, the turning point B constitutes a potential optimum operating point to safely optimize the reactor productivity, with respect to the intermediate C. Moreover, even in this case, it is possible to observe the occurrence of a runaway loop, which identifies the presence of a runaway (or excessive accumulation) region separating the safe (for what concerns C-species, that is reaction 1) operating regions: MI1 and QFS1. An increase of the cooling number from 21 to 65 leads to the disappearance of the runaway loop, as shown in Figure 11. These operating conditions have been previously referred to as invariably safe, with respect to reaction 2; that is, the QFS2 conditions cannot be identified. Nevertheless, a QFS1 condition can be detected even if the maximum reactor temperature values continues to increase smoothly when increasing the coolant temperature: the second inversion in the C-space is still present. By increasing the Tcool value beyond QFS1, a third inversion in the C-space can be evidenced (turning point C in Figure 4),

Figure 10. Objective parametric sensitivity coefficients of ξC(τjmax), with respect to the relevant system constitutive parameters (namely, Tcool, γ1, ∆τad,1, Co, ∆τad,2), as a function of coolant temperature (Tcool). System parameters are as given in Table 1.

Figure 11. C-space: maxima of trajectories with respect to jτ (τjmax), as a function of dimensionless C concentration in correspondence to jτmax (ξC(τjmax)). Symbols indicate different coolant temperature values (from 260 K to 325 K); the difference in Tcool values between two nearby symbols is equal to 1 K. System parameters are as given in Table 1, except Co ) 65.

leading to a sharp decrease of ξC(τjmax) and a sharp increase of jτmax (from 1.08 to 1.55), which is representative of runaway of the entire system. This inversion, which occurs at the same coolant temperature where a runaway ignition was experimentally detected,17 can be interpreted as the re-ignition of reaction 1, because of the heat of reaction 2, and it will be indicated as the marginal ignition of reaction 2 (namely, the marginal ignition of the entire system), MI2. A further increase in Tcool leads to a fourth inversion in the C-space, which corresponds to the coolant temperature at which the first inversion in the D-space occurs. Both of these turning points are labeled as “D” in Figures 4 and 5. This confirms that the same physical phenomenon is represented in two different spaces. Analogously, another (but less evident) inversion in the C-space occurs at the coolant temperature, which is the same as that of the second inversion in the D-space (point E in Figures 4 and 5), which individuates the QFS2 boundary. However, numerical problems can arise for the detection of QFS2 on the C-space, because of the very low values of ξC(τjmax) in the QFS2 region, while it can be easily identified in the D-space, as previously discussed.

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Table 2. Experimental Classification of Various Experimental Conditions17 and Predictions of the Proposed Criteriona run

Co

Tcool,exp [K]

experimental classification

proposed criterion

1 2 3 4 5

21 21 21 21 21

272 281 290 304 333

QFS1 MI2 RW2 QFS2 QFS2

QFS1 MI2 RW2 QFS2 QFS2

a

Operating and model parameters are as described in Table 1.

Figure 12. Comparison between experimental data (denoted by solid squares (9) and taken from ref 17) and criterion predictions. Dots represent the boundaries between different system behaviors. For each experimental point, run numbers (experimental classification) are also reported, as summarized in Table 2.

4. Comparison with Experimental Results The proposed criterion can easily identify QFS conditions both for the intermediate C and for the final product D, through the combined use of C-space and D-space. When these spaces show the presence of a runaway loop, there exists a region where runaway (or excessive accumulation) occurs and, consequently, a QFS boundary can be identified. In particular, the second inversion in the C-space identifies the QFS1 boundary, while the second inversion in the D-space identifies the QFS2 boundary. However, when no loops can be evidenced on these spaces, the system is invariably safe and no QFS2 conditions can be identified, even if excessive accumulation regions exist, while QFS1 conditions can be always identified. The experimental results previously reported in the literature16,17 for the reactive system here analyzed are summarized in Table 2, and they are compared with the proposed criterion in Figure 12 for Co ) 21. We can see that there is good agreement between the predictions of the proposed criterion and the experimental findings, in particular, for what concerns QFS conditions. Moreover, for Co ) 65, the proposed criterion predicts invariably safe conditions, in agreement with the experimental findings.

5. Optimization Procedure From the proposed criterion, which allows one to easily identify QFS conditions, the optimization procedure follows straightforward. In the following, C will be considered the desired product; however, the extension to the case where D is the desired product is straightforward. As used for SB processes, given the process recipe, the two operating parameters to be optimized are the coolant temperature (Tcool) and the dosing time (tdos). In particular, tdos should be minimum to obtain the maximum productivity of species C, accounting for the safety constraints. The optimization procedure can be summarized in the following steps: (1) Definition of a range of coolant temperatures [Tcool,MIN; Tcool,MAX] and a range of dosing times [tdos,MIN; tdos,MAX]. Such ranges depend on the process constraints, such as crystallization phenomena, maximum allowable temperature, maximum dosing stream supplied by pumps, etc.; (2) Identification of the QFS1 coolant temperature (Tcool,2°INV,C) in the C-space, using the current value of tdos (which is equal to tdos,MIN at the first iteration) and varying Tcool over its entire range of variation; (3) Under the QFS1 operating conditions, the maximum temperature (TMAX) must be checked, with respect to the experimentally determined MAT. If TMAX > MAT, the dosing time should be increased and step 2 should be repeated; (4) To obtain a suitable level of optimization, the conversion to species C at the end of the dosing time is required to be larger than a given threshold value ξC,dos, e.g., 60%. If this is not true, the dosing time should be increased and step 2 should be repeated; (5) After Tcool and tdos values such that QFS1 conditions are attained and TMAX < MAT, as well as ξC(tdos) > 60% (or any other threshold value), the time at which the maximum conversion to species C occurs must be identified: such a time will be the stop time of the process. The optimization procedure can be suitably complemented with a different method for identifying runaway boundaries, like the sensitivity one, as well as with dedicated experiments. While the first would ensure that the operating conditions identified through the aforementioned optimization procedure are not beyond the runaway boundary of the entire system, the latter would validate, on a laboratory scale, the optimization procedure itself for the specific case of interest. Therefore, the following two steps can usefully complement the proposed procedure: (1) Using parametric sensitivity coefficient criterion, the Tcool value of the entire system runaway boundary for the identified operating conditions is computed; such a value (that should correspond, apart from numerical approximation, to the MI2 boundary: the third inversion in the C-space) must be higher than the Tcool value identified in the optimization procedure. This step is a crossverification through two independent criteria that the identified operating conditions lie outside the thermal runaway region; and (2) The optimization procedure is performed initially for a laboratory-scale reactor (e.g., an RC1 one), and an experimental run is performed under the identified laboratory-scale optimal conditions to verify the reliability of the predictions of the optimization procedure. After these predictions are validated on the laboratory scale, the optimization procedure (which can be regarded, in this case, as a safe scale-up procedure) is performed for the real-size SBR.


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value) and those on the pre-exponential factor values are lower than approximately (10%, the optimum coolant temperature values determined by the optimization procedure are practically not affected by such errors (i.e., (1 K). Conclusion In this work, a new topological criterion that is able to identify runaway and QFS conditions for multireactions systems have been developed and validated by comparison with literature experimental data. Such a criterion makes use of suitable phase portraits that are generated by solving the system of ordinary differential equations (ODEs) that describes the process analyzed. Using such phase portraits, with reference for the sake of example to a system characterized by two consecutive reactions, it is possible to identify not only QFS1 and QFS2 conditions, but also all the marginal ignition conditionssthat is, MI1 and MI2sin agreement with the results of the classical boundary dynamics (BD) method, suitably adapted to multireactions systems.18 Moreover, a safe optimization procedure, which allows one to maximize reactor productivity, with respect to the intermediate species (C), has been developed and validated by comparison with experimental findings. Appendix The Damko¨hler numbers for reactions 1 and 2 (Da1 and Da2, respectively) are given by eqs A1 and A2, respectively:

Figure 13. Scheme of the optimization procedure. Table 3. Optimal Operating Conditions for the Reacting System Analyzeda Optimal Operating Conditions

Tcool tdos TMAX tstop

predicted by the optimization procedure

found experimentallyb

280 K 1.3 h 291 K 3.3 h

272 K < Tcool < 281 K 1h >285 K not determined

Dimensionless model parameters values: (UA)0 ) 4.3 W/K; RH ) 0.6; n ) m ) p ) q ) 1; γ1 ) 37.7; γ2 ) 40; ∆τad,1 ) 0.373; ∆τad,2 ) 1.21; ε ) 0.4; ξB,0 ) 0.035; ξN,0 ) 4.74; mH0,1 ) 6.6; mH0,2 ) 2.2; and Tdos ) 293 K; see Table 1. b Data taken from ref 17. a

The entire procedure is summarized in the scheme shown in Figure 13. Using this procedure, implemented in a suitable software tool, the optimal operating conditions for the reaction system described by reactions 1 and 2 have been computed, and they are compared with the experimentally determined optimum conditions17 in Table 3. The good agreement shown in Table 3 strongly supports the reliability of the proposed optimization procedure. To investigate the sensitivity of the optimization procedure, with respect to the experimentally determined parameters involved in the model, a sensitivity analysis should be always performed by varying all the parameters considered, one by one. For the case considered here, as long as the uncertainties on the activation energy values are lower than approximately (2 kJ/(mol K) (which means approximately (2% of the actual

Da1 ) k1,rifCA,dosn+m-1tdos

(A1)

Da2 ) k2,rifCA,dosp+q-1tdos

(A2)

The dimensionless concentration functions of reactions 1 and 2 (f1 and f2) are given as eqs A3 and A4, respectively: f1 )

{

m (ϑ - ξC - ξD)n(ξB0 + ξC)

ε1-mϑn m (1 - ξC - ξD)n(ξB0 + ξC) ε1-m

f2 )

{

ψ1(C) ⇒ (for 0 e ϑ < 1)

ψ1(C) ⇒ (for ϑ g 1)

(A3) ξpC(ξB0 + ξC)q ε1-qϑp p ξC(ξB0 + ξC)q

ψ2(C) ⇒ (for 0 e ϑ < 1)

(A4) ψ2(C) ⇒ (for ϑ g 1)

ε1-q The full expressions for the increase in dimensionless adiabatic temperature of reactions 1 and 2 (∆τad,1 and ∆τad,2, respectively) are given as follows: ∆τad,1 )

∆τad,2 )

˜rxn1)nA,dos (-∆H F ˜c˜cp,cVr0Trif ˜rxn2)nA,dos (-∆H F ˜c˜cp,cVr0Trif

(A5)

(A6)

Nomenclature A ) heat-transfer area of the reactor (associated to the jacket and/ or the coil) [m2] A ) species A A ) pre-exponential factor [m3/(kmol s)]

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B ) species B BD ) boundary diagram C ) molar concentration [kmol/m3] C ) species C Co ) cooling number; Co ) (UA)0tdos/F˜ cc˜p,cVd) c˜p ) molar heat capacity [kJ/(kmol K)] D ) species D Da1 ) Damko¨hler number for reaction 1 (see Appendix) Da2 ) Damko¨hler number for reaction 2 (see Appendix) Eatt ) activation energy [kJ/kmol] Ex ) exothermicity number f1 ) dimensionless concentration function of reaction 1 (see Appendix) f2 ) dimensionless concentration function of reaction 2 (see Appendix) H(C) ) Hammett’s acidity function (from ref 17) ˜ rxn ) reaction enthalpy [kJ/kmol] ∆H k ) reaction rate constant; k ) A exp[-Eatt/(RT)] [m3/(kmol s)] m ) in reaction 1, reaction order with respect to species B m ) equilibrium distribution coefficient mH ) Hammett’s coefficient MAT ) maximum allowable temperature [°C or K] n ) in reaction 1, reaction order with respect to species A n ) number of moles of a species [mol] p ) in reaction 2, reaction order with respect to species C q ) in reaction 2, reaction order with respect to species D r ) reaction rate relative to the liquid volume of the phase in which reaction occurs [kmol/(m3 s)] R ) gas constant; R ) 8.314 kJ/(kmol K) RE1 ) reactivity enhancement factor of reaction 1; RE1 ) (mA)n RE2 ) reactivity enhancement factor of reaction 2; RE2 ) (mC)p RH ) heat-capacity ratio; RH ) F˜ dosc˜p,dos/F˜ 0c˜p,0 Ry ) reactivity number sR,β ) normalized sensitivity coefficient of R, with respect to the β system constitutive parameter; sR,β ) β/R dR/dβ t ) time [s] T ) temperature [K] TD ) temperature diagram U ) overall heat-transfer coefficient [kW/(m2 K)] V ) volume [m3] Superscripts/Subscripts A ) refers to species A acq ) refers to the aqueous phase (continuous phase) ad ) adiabatic A,dos ) refers to the total number of moles of co-reactant A dosed att ) refers to the activation energy B ) refers to species B B,0 ) refers to the initial catalyst amount C ) refers to species C c ) continuous phase cool ) coolant D ) refers to species D d ) dispersed phase dos ) dosing stream or dosing time H ) in the heat capacity ratio RH HNO3 ) nitric acid i ) index of the species j ) index of model constitutive parameters MAT ) at the maximum allowable temperature max (or MAX) ) maximum value of a quantity or at the maximum value of a quantity N,0 ) refers to the initial nitric acid amount QFS ) “quick onset, fair conversion, smooth temperature profile”

r ) refers to the reactor rif ) refers to the reference conditions stop ) refers to the stop time x ) in the exothermicity number Ex y ) in the reactivity number Ry 0 ) start of the dosing period 1 ) refers to the first reaction (eq 1) 2 ) refers to the second reaction (eq 2) 2°INV ) refers to the second inversion in the C- or D-space Greek Symbols ∆τad,1 ) increase in dimensionless adiabatic temperature of reaction 1 (see Appendix) ∆τad,2 ) increase in dimensionless adiabatic temperature of reaction 2 (see Appendix) γ ) dimensionless activation energy; γ ) Eatt/(RTrif) ξ ) dimensionless concentration of species i; ξ ) ni/nA,dos ε ) relative volume increase at the end of the semibatch period; ε ) Vdos/Vr0 κ ) dimensionless kinetic constant; κ ) exp{γ[1 - (1/τ)]} F˜ ) molar density [kmol/m3] ϑ ) dimensionless time; ϑ ) t/tdos τ ) dimensionless temperature; τ ) T/Trif jτ ) dimensionless temperature with respect to coolant temperature; jτ ) T/Tcool jτmax ) maximum dimensionless temperature, with respect to coolant temperature; jτmax ) TMAX/Tcool eff τeff cool ) effective dimensionless coolant temperature; τcool ) RHετdos + Coε(1 + εϑ)τcool/[Coε(1 + εϑ) + RHε] ψ(C) ) dimensionless concentration function; ψ(C) ) exp(-mHH(C)) φ ) constitutive model parameter (K or dimensionless)

Literature Cited (1) Varma, A.; Morbidelli, M.; Wu, H. Parametric SensitiVity in Chemical Systems; Cambridge University Press: Cambridge, U.K., 1999. (2) Strozzi, F.; Zaldı´var, J. M.; Kronberg, A. E.; Westerterp, K. R. Online runaway detection in batch reactors using chaos theory techniques. AIChE J. 1999, 45, 2429. (3) Zaldı´var, J. M.; Cano, J.; Alo´s, M. A.; Sempere, J. M.; Nomen, R.; Lister, D.; Maschio, G.; Obertopp, T.; Gilles, E. D.; Bosch, J.; Strozzi, F. A general criterion to define runaway limits in chemical reactors. J. Loss PreV. Process Ind. 2003, 16, 187. (4) Bosch, J.; Strozzi, F.; Zbilut, J. P.; Zaldı´var, J. M. On-line runaway detection in isoperibolic batch and semibatch reactors using the divergence criterion. Comput. Chem. Eng. 2004, 28, 527. (5) Zaldı´var, J. M.; Bosch, J.; Strozzi, F.; Zbilut, J. P. Early Warning detection of runaway initiation using non-linear approaches. Int. J. Nonlinear Sci. Numer. Simul. 2005, 10, 299. (6) Hugo, P.; Steinbach, J. Praxisorientierte Darstellung der thermischen Sicherheitsgrenzen fu¨r den indirekt geku¨hlten Semibatch-Reaktor. Chem. Ing. Tech. 1985, 57, 780. (7) Steensma, M.; Westerterp, K. R. Thermally safe operation of a cooled semi-batch reactor. Slow liquid-liquid reactions. Chem. Eng. Sci. 1988, 43, 2125. (8) Steensma, M.; Westerterp, K. R. Thermally safe operation of a semibatch reactor for liquid-liquid reactions. Slow reactions. Ind. Eng. Chem. Res. 1990, 29, 1259. (9) Steensma, M.; Westerterp, K. R. Thermally safe operation of a semibatch reactor for liquid-liquid reactions. Fast reactions. Chem. Eng. Technol. 1991, 14, 367. (10) Alo´s, M. A.; Nomen, R.; Sempere, J. M.; Strozzi, F.; Zaldı´var, J. M. Generalized criteria for boundary safe conditions in semi-batch processes: Simulated analysis and experimental results. Chem. Eng. Process. 1998, 37, 405. (11) Maestri, F.; Rota, R. Temperature diagrams for preventing decomposition or side reactions in liquid-liquid semibatch reactors. Chem. Eng. Sci. 2006, 61, 3068.

Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010 (12) Maestri, F.; Copelli, S.; Rota, R.; Gigante, L.; Lunghi, A.; Cardillo, P. Simple procedure for optimally scaling-up fine chemical processes. I. Practical tools. Ind. Eng. Chem. Res. 2009, 48, 1307. (13) Maestri, F.; Copelli, S.; Rota, R.; Gigante, L.; Lunghi, A.; Cardillo, P. Simple procedure for optimal scale-up of fine chemical processes. II. Nitration of 4-chlorobenzotrifluoride. Ind. Eng. Chem. Res. 2009, 48, 1316. (14) Maestri, F.; Rota, R. Thermally safe operation of liquid-liquid semibatch reactors. Part I: Single kinetically controlled reactions with arbitrary reaction order. Chem. Eng. Sci. 2005, 60, 3309. (15) Maestri, F.; Rota, R. Thermally safe operation of liquid-liquid semibatch reactors. Part II: Single diffusion controlled reactions with arbitrary reaction order. Chem. Eng. Sci. 2005, 60, 5590. (16) van Woezik, B. A. A.; Westerterp, K. R. The nitric acid oxidation of 2-octanol. A model reaction for multiple heterogeneous liquid-liquid reactions. Chem. Eng. Process. 2000, 39, 521.

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(17) van Woezik, B. A. A.; Westerterp, K. R. Runaway behaviour and thermally safe operation of multiple liquid-liquid reactions in the semibatch reactor. The nitric acid oxidation of 2-octanol. Chem. Eng. Process. 2001, 41, 59. (18) Copelli, S.; Derudi M.; Maestri, F.; Rota, R. Safe operating conditions for semibatch processes involving consecutive reactions with autocatalytic behaviour. Submitted to Chem. Eng. Sci., 2009. (19) Steinbach, J. Safety Assessment for Chemical Processes; WileyVCH Verlag GmbH: Weinheim, Germany, 1999.

ReceiVed for reView October 28, 2009 ReVised manuscript receiVed February 23, 2010 Accepted March 25, 2010 IE901679Q

Topological Criteria to Safely Optimize Hazardous Chemical

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