Designing Thermally Safe Operation Conditions for Isoperibolic Liquid

Article pubs.acs.org/IECR

Designing Thermally Safe Operation Conditions for Isoperibolic Liquid−Liquid Semibatch Reactors without Kinetic and Solubility Parameters: I. Development of the Procedure for Kinetically Controlled Reactions Zichao Guo,* Liping Chen, and Wanghua Chen Department of Safety Engineering, School of Chemical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China S Supporting Information *

ABSTRACT: In the fine and pharmaceutical chemical industries, designing thermally safe operating conditions for liquid−liquid semibatch reactors (SBRs) is a very important issue to prevent the occurrence of thermal runaway. However, because of the time and money constraints in determining the kinetic and solubility parameters for liquid−liquid reactions in practice, it is essential to develop a kinetic-parameters-free and solubility-parameter-free procedure to determine thermally safe operating conditions for liquid−liquid SBRs. In this work, the primary contribution is to develop a simple procedure, with no requirement of kinetic and solubility parameters, to designing thermally safe operating conditions for liquid−liquid SBRs in which kinetically controlled liquid−liquid reactions occur. For this purpose, a modified definition of QFS operation is proposed first. Then, a series of theoretical tools based on the modified definition of QFS operation are developed to design thermally safe operating conditions. Finally, the practical procedure/with no requirement for kinetic and solubility parameters is introduced. In addition, it is worthwhile to note that the procedure developed in this paper is not valid for completely or even partially diffusion-controlled liquid−liquid reactions and the method proposed in this article can be used only when the reaction rate order is equal to 2.

1. INTRODUCTION In fine and pharmaceutical chemical industrials, so far semibatch reactors (SBRs) are the most frequently used type of reactors to prevent the occurrence of thermal runaway by controlling the heat generation rate through tuning the dosing rate. Unfortunately, thermal runaway incidents are still not completely vanished. Thereby, designing inherently safe operating conditions for SBRs is a highly important issue in the future. One main problem to face in practice is that, because of time and money constraints, generally, the detailed kinetics of the reactions is not likely to be determined. Therefore, it urgently calls for a simple and reliable procedure to design thermally safe operating conditions for SBRs. Along this line, many research works have been executed over the past decades. The pioneers are Hugo and Steinbach,1,2 who observed that an accumulation of the dosed component at reactor temperatures that are too low is the cause of the runaway in homogeneous semibatch reactors. They presented a line in a diagram, separating the region with conditions where the reactions are sufficiently fast and with conditions where runaways may occur. In practice, many highly exothermic reactions occur in liquid−liquid phase systems, such as nitration, oxidation, sulfonation, and so forth. Steensma and Westerterp3,4 were the first to study liquid−liquid reaction systems in SBRs. They defined a target temperature and developed full boundary © 2017 American Chemical Society

diagrams. In these boundary diagrams, a boundary line was plotted to divide the operating regions into three scenarios: QFS (which represents quick onset, fair conversion, and smooth temperature profile), thermal runaway, and no ignition. As stated by Steensma and Westerterp,5 the accumulation of the added reactants in the QFS scenario is harmless. Accordingly, they developed a practical procedure to determine the inherently safe operating conditions for SBRs.6 Maestri and Rota7,8 then demonstrated that, for heterogeneous SBRs, the reaction order of the two reactants can influence the shape and location of the boundary diagrams. As a consequence, the operating conditions determined by boundary diagrams that are based on the assumption of a second reaction order are not always safe. In addition, to prevent the triggering of dangerous decomposition and side reactions, they also developed new diagrams, namely, temperature diagrams.9 Based on these two types of diagrams, simple procedures were proposed for designing safe operating conditions for homogeneous as well as heterogeneous SBRs.10,11 Recently, Bai and co-workers12 constructed a new set of the boundary diagrams for homogeneous semibatch reactions, Received: Revised: Accepted: Published: 10428

June 25, 2017 August 24, 2017 August 28, 2017 August 28, 2017 DOI: 10.1021/acs.iecr.7b02599 Ind. Eng. Chem. Res. 2017, 56, 10428−10437

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Industrial & Engineering Chemistry Research based on their finding that, for QFS and no ignition scenarios, namely, the maximum temperature of synthesis reaction under adiabatic conditions (MTSR) always appeared at the stoichiometric point of dosing period, whereas, in contrast, for the thermal runaway scenario, MTSR always occurred before this time point. It is worthwhile noting that implementation of all the above methods requires knowledge of the kinetic parameters, at least apparent kinetic parameters. However, determination of the kinetic parameters in realistic cases requires professional expertise, especially in the case of heterogeneous reaction systems, in which not only kinetic parameters but also the solubility of reactants are required. This extensively restricts the application of the above method. Therefore, it is desirable to develop a kinetic-parameters-free method to determine thermally safe operating conditions for SBRs. In this sense, Copelli and co-workers13,14 introduced the concept of the topological curve, which is a plot of Tmax/Tc versus the conversion. Using such a curve, QFS operating conditions can be identified without any knowledge of the kinetics. Maestri and Rota15,16 recently developed a general method that was based on a so-called “ψ number criterion”, which allows for ongoing detection of the displacement of the SBR operating regime from safe target conditions. Moreover, the divergence ́ criterion developed by Zaldivar and co-workers17 is also a widely used tool for this purpose. Recently, we have developed a kinetic-parameters-free procedure for isoperibolic homogeneous semibatch reactions.18 It just needs several isothermal RC1 (reaction calorimetry) tests, which are convenient to perform in the laboratory. Nevertheless, as mentioned previously, many highly exothermic reactions belong to liquid−liquid systems, such as nitration, oxidation, sulfonation, and so forth. Together with the statement by Steensma and Westerterp, that runaways in liquid−liquid systems usually occur in kinetically controlled reaction systems, the objective of this work thereby is to develop a simple procedure with no requirement of kinetic and solubility parameters for isoperibolic liquid−liquid semibatch reactions that are only kinetically controlled.

(4) the chemical reaction occurs only in one of the two liquid phases: this situation is very common in industrial processes (such as nitration and oxidations), in which the catalyst (typically a strong acid) is present only in one phase; (5) heat dissipation by the agitator or mixing effect is negligible, relative to the reaction heat, and, consequently the heat effects are only associated with the chemical reaction; and (6) the physicochemical properties of all the components are constant and additive during the entire reaction. 2.1. Mass Balance. The microkinetic rate expression of reaction 1 can be described by a generic power-law expression: ⎧ for reactions occurring in continuous phase ⎪ kcCA,cC B,c r=⎨ ⎪ ⎩ kdCA,dC B,d for reactions occurring in dispersed phase

where k is the reaction rate constant, and CA and CB are the concentration of reactants A and B, respectively. Generally, with respect to kinetically controlled liquid−liquid reactions, the concentration decrease of reactant A/B in the other phase can be neglected.4 Consequently, the equilibrium concentrations at the liquid/liquid interface are equal to the bulk concentrations. Therefore, CA,c and CB,d can be calculated as follows: CA,c = mA CA,dC B,d = mBC B,c

where mA and mB are the distribution coefficients. Obviously, the distribution coefficient is an important parameter in the derivation of an overall reaction rate expression, since it determines the maximum possible concentration in the reaction phase. It is well-known that, for isoperibolic SBRs, the reaction mass temperature is variable. Hence, the dependence of mA and/or mB on temperature should be estimated. ́ 19 reported that the solubility of aromatic comZaldivar pounds in aqueous sulfuric acid could be correlated by ⎛ ΔG ⎞ mArH = exp⎜ − ArH ⎟ ⎝ RT ⎠

2. MATHEMATICAL MODEL Assuming that a single reaction is performed in a liquid−liquid stirred SBR equipped with the cooling jacket: vA A + vB B → C + vDD

(2)

(3)

where ΔGArH is a function of the sulfuric acid strength, the difference in internal pressure of the two immiscible solvents and the molar volume of the aromatic compound. Along this line, eq 3 could be extended to general cases. Accordingly, mA and/or mB can be expressed by

(1)

where A and B are the reactants, C is the desirable product, and D is the side product; vi is the stoichiometric coefficient of component i. The value of vC is assumed to be equal to 1. In addition, reactant B is supposed to be loaded into the reactor initially and reactant A is dosed at a constant rate until the stoichiometric amount of A has been added. The process involves two liquid phases: a continuous phase and a dispersed phase (denoted as the “c-phase” and “d-phase”, respectively, in the following discussion). Generally, the dosed reactant of A exists in the d-phase, while reactant B is presented in the c-phase. In addition, the following assumptions can be stated:

⎛ ΔG ⎞ ⎟ m = exp⎜ − ⎝ RT ⎠

(4)

It is easy to derive the equations relating to the concentration of A in the dispersed phase and the concentration of B in the continuous phase at arbitrary time during the dosing period: CA,d =

C B,c =

(1) the reaction mass is perfectly macromixed; (2) the influence of the chemical reaction on the volume of the single phase is negligible; (3) no phase inversions occur during the reaction duration;

nB,0(θ − XB)vA /vB Vdθ

(5)

nB,0(1 − XB) Vc

(6)

where nB,0 is the initial molar number of reactant B, XB is the conversion of reactant B, Vd and Vc are the volume of dispersed and continuous phases, respectively, and θ is the dimensionless 10429

DOI: 10.1021/acs.iecr.7b02599 Ind. Eng. Chem. Res. 2017, 56, 10428−10437

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Industrial & Engineering Chemistry Research

Table 1. Expressions of the Reactivity Enhancement Factor (RE) and the Function f, for Kinetically Controlled Reactions Occurring in the Dispersed Phase or the Continuous Phase reaction in the dispersed phase, d RE f

mB (θ − XB)(1 − XB)

check for reaction regime

Had =

1 kL,B

(7)

when θ > 1: ⎧ ⎫ dX dτ 1 ⎨Δτad,0 B − ε⎡⎣Wt(1 + ε)(τ − τj)⎤⎦⎬ = ⎭ dθ (1 + εRH) ⎩ dθ (12)

where RH is the ratio between the volumetric heat capacities of component A and component B (RH = (ρcp)d/(ρcp)c); the adiabatic temperature rise is expressed in the dimensionless form (Δτad,0 = (−ΔHr)nB,0/[νB(ρcp)0V0TR]) and Wt represents the Westerterp number23 (Wt = (UA)0td/[ε(ρcp)0V0]), τD = TD/TR, and τj is the dimensionless temperature of dosing component and the jacket coolant (τj = Tj/TR), respectively.

3. THERMALLY SAFE OPERATING CONDITIONS FOR ISOPERIBOLIC LIQUID−LIQUID SBRS The ideal safe behavior of an indirectly cooled SBR can be reasonably identified as that the added component at a constant rate immediately reacts with the component initially present in the reactor. Consequently, the power generated by the exothermic reactions is constant and no accumulation occurs. However, in a realistic case, it is not possible to avoid accumulation. As an alternative solution, designing a QFS operating condition for isoperibolic SBRs, together with the constraint that the maximum temperature reached during the entire process (Tmax) is lower than the maximum allowable temperature (MAT), has been widely accepted in academia.9,21 MAT is usually associated with the decomposition temperature of the reacting mixture and/or the triggering temperature for side reactions. Generally, three operation scenarios in SBRs are present: no ignition (NI), thermal runaway (TR), and QFS. As Westerterp and co-workers stated,3,4 these three scenarios were defined as a result of comparison of the reaction mass temperature profile to a so-called “target temperature” (Tta) profile, as shown in Figure S1 in the Supporting Information: (1) NI: reaction mass temperature does not approach Tta. (2) TR: reaction mass temperature surpasses Tta excessively. (3) QFS: reaction mass temperature approaches Tta rapidly but is lower than Tta. The Tta profile is determined based on the assumption that the added component, at a constant rate, immediately reacts with the component initially present in the reactor; in other words, by no means does accumulation of the added component exist in SBRs. By considering a 5% overestimation, the dimensionless form of Tta (τta) can be expressed as follows:

(8)

(9)

where Da is the Damköhler number of the reactions (Da = kRtDCB,0) and κ is a dimensionless reaction rate constant (κ = exp{[γeff[1 − (1/τ)]}, where γeff = Eeff/(RTR) and τ = T/TR (T is the process temperature, TR is a reference temperature, Eeff is the sum of ΔG and the reaction activation energy (E). Generally, because of the low solubility of the dispersed reactant in the continuous phase in liquid−liquid reaction systems, ΔG ≪ E. In this paper, TR is supposed to be 300 K. The expression of RE and f can be found in Table 1. 2.2. Heat Balance. The heat balance equation for SBRs states that the reaction temperature variation is the result of three enthalpy contributions, related to the dosing stream, the chemical reaction, and the heat removed by the coolant:23 (ρcp)r Vr

< 0.3

dτ 1 = dθ (1 + εRHθ) ⎧ ⎫ dX × ⎨Δτad,0 B − ε[Wt(1 + εθ)(τ − τj) + RH(τ − τD)]⎬ ⎩ ⎭ dθ (11)

The above equation then can be arranged in the dimensionless form as follows: dXB = vADaREκf dθ

(θ − XB)(1 − XB) εθ 1 Hac = k k + L,AC B,c L,A

when θ < 1:

where nB is the molar number of reactant B and Vr represents the effective reaction volume. Vr corresponds to the volume of the continuous phase, in case the chemical reaction occurs in a continuous phase, while Vr corresponds to the dispersed-phase volume, in case the chemical reaction occurs in a dispersed phase. In fact, to determine the location of reaction (in continuous or dispersed phase), qualitative analysis of the reaction mechanism or simple experiment without knowing the kinetics can be made.20−22 For example, nitration reactions by mixing acid and oxidation reactions by aqueous nitric acid usually occur in the aqueous phase; hydrolysis reaction of organic ester by water occurs in the aqueous phase. Therefore, eq 7 can be rewritten as ⎛v t ⎞ dXB = ⎜⎜ B D ⎟⎟rVr dθ ⎝ nB,0 ⎠

mA

k + L,BCA,d < 0.3

form of time t (that is, θ = t/tD; obviously, θ = 1 at the end point of the dosing period). The mass balance equation of reactant B then can be derived as dnB = −vBrVr dt

reaction in the continuous phase, c

dT = ( −ΔHr)rV − (UA)(T − Tj) − (ρcp)D ϕV,A (T − TD) dt (10)

In an ideal case, the overall heat-coefficient multiplied area (UA) is proportional to the liquid volume in the reactor. That is, UA at the final point of dosing period, (UA)f, is equal to (UA)0(1 + ε), in which ε = Vd/Vc. Equation 10 can be conveniently rewritten in the following dimensionless forms: 10430


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Industrial & Engineering Chemistry Research ⎧ Wt(1 + εθ )τ + R τ Δτad,0 c Hd ⎪ + 1.05 θ≤1 ε(Wt(1 + εθ ) + RH) ⎪ Wt(1 + εθ ) + RH τta = ⎨ Δτad,0 ⎪ Wt(1 + ε)τc + RHτd + 1.05 θ>1 ⎪ ε(Wt(1 + ε) + RH) ⎩ Wt(1 + ε) + RH

As shown in Figure S1, the accumulation in the QFS situation remains at a low level over the entire reaction duration and the maximum accumulation is the lowest, compared with that in the TR and NI situations. Based on the above illustration, the so-called boundary diagrams, in which the three situations were separated by a boundary line, were developed to help to select QFS operating conditions for SBRs, as shown in Figure S2 in the Supporting Information. However, it is highly worthwhile to note that, when constructing the heat balance (eq 12) for SBRs, the ideal situation, in which UA is supposed to be proportional to the liquid volume in the reactor, was assumed. Consequently, UA at the end point of the dosing period (i.e., (UA)f) is equal to (UA)0(1 + ε). However, in realistic cases, (UA)f is usually lower than (UA)0(1 + ε), especially in the case of unbaffled reactors.24 Let us assume the extreme situation, at which (UA)f = (UA)0. As an example, a comparison of the ideal situation to the extreme situation in the QFS scenario is shown in Figure 1. It is apparent that the temperature profile in the ideal situation

(13)

is lower than that in the extreme situation in Figure 1a, while the accumulation at θ = 1 presents the opposite trend in Figure 1b. This can be interpreted as follows: the higher the reaction mass temperature, the faster the conversion rate and, consequently, the lower the maximum accumulation. Hence, we suggest a modified definition of QFS operation that should fulfill three constraints: (1) The reaction mass temperatures profile approach Tta profile rapidly; (2) The reaction mass temperature profiles is lower than Tta profile; and (3) Tmax in the assumed extreme situation should be lower than the maximum Tta (denoted as Tta,max) that occurs at the initial time. Accordingly, the procedure to build the traditional boundary diagrams also should be modified. In the following, a simple and general procedure for building the modified boundary diagrams for liquid−liquid SBR is presented. Six dimensionless parameters appear in the mass and heat balances (eqs 9 and 11): vADaRE, ε, γeff, RH, Δτad,0, and Wt. Assign a value to each of these parameters in compliance with its accepted range and set the coolant temperature as a variable parameter. Then, numerical integration of the mathematical model of eqs 9 and 11 with the coolant temperature increasing from a low value to a high value will present the three scenarios of NI, TR, and QFS in turn. Record the two marginal values of coolant temperature between the three scenarios. With respect to the higher one, examine whether the value of Tmax in extreme situation of UA is lower than Tta,max. If yes, it means the operation of SBR at this coolant temperature matches the modified definition of QFS; if no, increase the value of the coolant temperature until the modified QFS scenario is fulfilled. Let us denote the two coolant temperatures to NI-TR and TRQFS points, respectively. Calculate the values of exothermicity (Ex) and the reactivity number (Ry) for NI-TR and TR-QFS points, respectively. Herein, the expression of Ex and Ry are presented as follows: Ry =

Ex =

vADaREκ |τj ε(RH + Wt )

(14)

⎤ Δτad,0 γ eff ⎡ ⎥ 2 ⎢ τj ⎣ ε(RH + Wt ) ⎦

(15)

It is suggested that, for each boundary diagram, the values of Wt, RH, and ε remain constant. Therefore, assign values to the other three parametersvADaRE, γeff, and Δτad,0and determine the corresponding Ex and Ry parameters for NITR and TR-QFS points, respectively. After repeatedly executing such operation, a NI-TR curve and a TR-QFS curve will be obtained in one boundary diagram, as observed in the example shown in Figure S2. Then, with repeating the same procedure for another set of Wt, RH, and ε, a series of boundary diagrams will be obtained, as observed in Figures 2 and 3. It is

Figure 1. Comparison of the ideal and extreme situations for isoperibolic SBR in the QFS scenario: (a) thermal behavior and (b) accumulation behavior. Parameters: ε = 0.4, νADaREκ = 1.8, Wt = 10, Δτad,0 = 0.6, γeff = 38, RH = 1, τj = τD = 1.01. 10431


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illustrate the relation between Rymin and Wt, the plots of Rymin− Wt for liquid−liquid reaction systems occurring in the dispersed and continuous phases are presented in Figures 4

Figure 2. Boundary diagrams for isoperibolic liquid−liquid SBRs, in which kinetically controlled reactions occur in the dispersed phase based on the modified definition of QFS. RH = 1, ε = 0.4, 0.025 < vADaRE < 15, 27 < γeff < 45, 0.2 < Δτad,0 < 0.7.

Figure 4. Inherently safe operating conditions of liquid−liquid SBRs in which single exothermic reactions occur in the dispersed phase. The relationship between Rymin and Wt. 0.025 < vADaRE < 15, 27 < γeff < 45, 0.2 < Δτad,0 < 0.7: (a) influence of RH and (b) influence of ε.

and 5, respectively. We can reasonably expect that if the value of Ry in realistic cases is above the Rymin−Wt curves, the corresponding SBR must be operated in the modified QFS scenario. The influence of RH and ε on the Rymin−Wt curve can also be seen in Figures 4 and 5. From Figures 4 and 5, we can find that regardless of whether the reactions occur in the dispersed phase or in the continuous phase, Rymin−Wt curves with high RH always place underneath that with low RH. In other words, with a constant Wt, Rymin at low RH is higher than that at high RH. This can be interpreted as follows: low RH corresponds to low ratio of the heat capacity of reactant B to the dosing reactant A, which indicates the cooling effect by the dosing flow is discounted. As a consequence, higher Rymin values are required to fulfill the modified QFS operation. Figures 4b and 5b show that the Rymin−Wt curves with low ε always place above those with high ε in the range of relatively low Wt, whereas this trend is inverted in the relatively high Wt range. In addition, it is obvious that, for all the Rymin−Wt curves in Figures 4 and 5, as Wt increases, Rymin will increase at the start. After the maximum Rymin is reached, Rymin starts to decrease with Wt continuously increasing. It is highly worthwhile to note that, in Figure 5, Rymin will reach to zero when Wt is higher than a critical value, which indicates that as long as the value of

Figure 3. Boundary diagrams for isoperibolic liquid−liquid SBRs in which kinetically controlled reactions occur in the continuous phase based on the modified definition of QFS. RH = 1, ε = 0.4, 0.025 < vADaRE < 15, 27 < γeff < 45, 0.2 < Δτad,0 < 0.7.

interesting that the boundary diagrams obtained in the light of the modified definition of QFS operation are the same as the boundary diagrams obtained from the original definition of QFS operation at all. Hence, we do not show the comparison of the boundary diagrams between the two definitions of QFS operation. According to the modified definition of QFS, Tmax must be lower than Tta,max. Therefore, once the QFS operating conditions has been identified, it should examine whether Tta,max < MAT. If yes, then this QFS operating condition can be considered to be thermally safe.

4. THEORETICAL TOOLS FOR DESIGNING THERMALLY SAFE OPERATING CONDITIONS 4.1. Method for Determining the Operating Conditions That Match the Modified Definition of QFS. From Figure S2, it can be reasonably expected that if Ry > Rymin, SBRs are always operated in the QFS region. This conclusion is valid for all the other boundary diagrams in Figures 2 and 3. To 10432


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Figure 6. Relationship between Wtmin and ε for SBRs in which exothermic reactions occur in the continuous phase. RH = 1, 0.025 < vADaRE < 15, 27 < γeff < 45, 0.2 < Δτad,0 < 0.7.

Wt =

(UA)0 t D ε(ρcp)0 V0

(16)

However, tD, which is determined by the Wtmin−ε curve in Figure 6, may be not production-efficient, because the result obtained is conservative. In this case, the modified QFS operation for liquid−liquid reactions that occur in the continuous phase should be designed by the Rymin−Wt tool in Figure 5. From eq 14, the parameters of ε, RH, Wt, and vADaREκ|τj are required to calculate the value of Ry. Wherein, the first three parameters of ε, RH, and Wt can be easily determined as long as the process recipe and the reactor type are provided. To determine the value of vADaREκ|τj, traditionally, it needs the reaction kinetic parameters and the solubility parameter first.10,11 However, determination of the reaction kinetic parameters and the solubility parameter is time-consuming and expensive for liquid−liquid reaction systems. Thereby, it is strongly desirable to develop a kinetic-parameters-free method to obtain the value of vADaREκ|τj. In the following part, we will introduce such a kinetic-parameters-free method. 4.2. Method for Determining the Value of vADaREκ|τj. For an ideal isothermal liquid−liquid SBR, the mass balance equation can be expressed as described by eq 9. From the viewpoint of mathematics, the conversion profiles for isothermal liquid−liquid SBRs with reactions occurring in the continuous phase are only dependent on the values of vADaREκ and ε. If the liquid−liquid reactions occur in the dispersed phase, the conversion profiles are only dependent on the value of vADaREκ. Together with the knowledge that the maximum accumulation, Xac,max, in isothermal SBRs usually occurs at the time point when stoichiometric amount of dosing reactant is added, we can reasonably expect that Xac,max must be a function of νADaREκ. The definition of the accumulation (Xac) of dosed reactant A can be expressed as follows:

Figure 5. Inherently safe operating conditions of liquid−liquid SBRs in which single exothermic reactions occur in the continuous phase. The relationship between Rymin and Wt. 0.025 < vADaRE < 15, 27 < γeff < 45, 0.2 < Δτad,0 < 0.7: (a) influence of RH and (b) influence of ε.

Wt is higher than this critical value, there must be no thermal runaway event happening in the SBRs as long as no failure of the utilities (i.e., the agitator and the temperature control system) occurs. In other words, the temperature of reaction mixture obtained in this case must be lower than the Tta values. For better understanding, one example in Figure S3 is illustrated in the Supporting Information profile. Here, we denote this critical value of Wt to Wtmin. For the sake of example, the relationship between Wtmin and ε at RH = 1 is illustrated in Figure 6. It is obvious that as ε increases, the value of Wtmin decreases. It is strongly significant that the plot in Figure 6 provides a simple way to design the modified QFS operation for kinetically controlled liquid−liquid reactions that occur in the continuous phase with no requirement of the kinetic and solubility parameters of the reactions. For a given process recipe, the information on ε can be easily obtained. Subsequently, the corresponding Wtmin can be determined with the help of plot in Figure 6. From the expression of Wt in eq 16, the parameters, such as (UA)0, (ρcp)0, V0, and tD are required to calculate the value of Wt. Wherein, (ρcp)0 and V0 can be determined from the given process recipes, (UA)0 for industrial reactors can be determined by theoretical calculation based on two-film theory or calibrated by cooling experiments.25 The value of the operation parameter of tD then can be decided based on the above analysis.

⎧ θ − X for θ < 1 Xac = ⎨ ⎩1 − X for θ > 1

(17)

Therefore, plots of νADaREκ vs Xac,max can be constructed by numerically calculating eq 9, as shown in Figure 7. Since the conversion profiles are only dependent on the value of vADaREκ for reactions occurring in the dispersed phase, thereby only one νADaREκ vs Xac,max curve is present in Figure 10433


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5. PRACTICAL PROCEDURE FOR DESIGNING THERMALLY SAFE OPERATING CONDITIONS With the help of the theoretical tools introduced in the above section, thermally safe operating conditions in the laboratory can be easily determined with no requirement of kinetics and solubility parameters for isoperibolic SBRs in which kinetically controlled exothermic liquid−liquid reactions occur. The fourstep procedure for this purpose is represented in the flowchart of Figures 8 and 9 and described in detail as follows.

Figure 7. νADaREκ vs Xac,max plots for isothermal liquid−liquid SBRs in which reactions occur in (a) the dispersed phase and (b) the continuous phase. RH = 1.

7a. Figure 7b shows the plots of νADaREκ vs Xac,max for reactions that occur in the continuous phase, in which three νADaREκ vs Xac,max curves, with respect to ε = 0.2, 0.4, and 0.6, are presented, for the sake of example. It is obvious that, as the value of ε increases, the corresponding νADaREκ at the identical Xac,max increases. It is worthwhile to note that, in realistic cases, the amount of dosed reactant A is usually over the stoichiometric mass. Whereas, Xac,max is equal to Xac that occurs at the time when the stoichiometric amount of the dosed reactant A is dosed. From the above analysis, it is obvious that the value of νADaREκ can be easily estimated through the νADaREκ vs Xac,max curves in Figure 7, as long as Xac,max has been obtained. In fact, Xac,max can be easily determined via sampling analysis in the laboratory. Another widely used method for this purpose is reaction calorimetry, which can provide information on not only Xac,max but also the heat-generated exothermic reactions.26−28 After Xac,max is obtained, the value of νADaREκ for reactions occurring in the dispersed phase, can be directly predicted in Figure 7a. For reactions occur in the continuous phase, ε should be determined based on the process recipe first, then the value of νADaREκ can be directly estimated by the corresponding νADaREκ vs Xac,max curve in Figure 7b. As a consequence, the value of Ry can be easily obtained by substituting the values of νADaREκ, ε, Wt, and RH into eq 14.

Figure 8. Flowchart for designing thermally safe operating conditions at the laboratory scale and scaleup to the industrial plant for liquid− liquid SBRs in which kinetically controlled reaction occur in the dispersed phase.

(1) The thermal stability of reactants, products, and reacting mixture should be investigated first. This can be achieved by carrying out dynamic DSC (differential scanning calorimetry) tests and ARC (adiabatic rate calorimetry) tests.29 The purpose to conduct dynamic DSC tests is usually to detect whether reactants, products, and the reacting mixture are thermally unstable. If so, then a standard heat-wait-search (HWS) test via ARC should be executed. Such a test can provide the maximum allowable temperature (MAT) parameter, which is defined to prevent the triggering of dangerous decompositions or strongly exothermic side reactions.9 (2) Isothermal tests by a reaction calorimeter (RC1) should be conducted to determine the safe and productive operating conditions for isoperibolic SBRs. In fact, RC1 has been widely used in industrial and academia. With the virtues of high cooling capacity and automation, isothermal tests by RC1 is safe and simple to be conducted by researchers. RC1 tests can provide information on the UA of laboratory reactors and cp of the reaction mass through a calibration procedure;21 as a result, RH can be easily obtained. One can then empirically select one set of operating parameters (mainly 10434


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tried first, since prolonging the dosing period generally indicates a reduction in the reaction productivity. If Ry > Rymin, it means that the selected operating parameters belong to the modified QFS scenario. One then must examine whether Tmax reaches MAT or not by comparing Tta,max to MAT. If Tta,max > MAT, then it needs to prolong the dosing period to increase the value of Wt; as a result, Tmax will decrease. Herein, it is worthwhile to state that increasing the coolant temperature will increase both Tmax and Tta,max. In addition, another point that needs to be stated is the fact that prolonging the dosing period does not lower the value of Ry, which can be easily deduced from the mathematical expression of Ry. If Tta,max < MAT, it means the value of Tj for isoperbolic SBRs can be directly set to equal to Tr for isothermal RC1 test and in the same way, tD for isoperbolic SBRs is equal to that for isothermal RC1 test. (3) The operating parameters (Tj and tD) designed by the above procedure must then be experimentally verified through, at least, one isoperibolic RC1 test. In addition, it is worthwhile noting that the above method only addresses normal operating conditions, i.e., the safety aspects linked to failure of the agitator or the temperature control system are not considered. As a consequence, the determined operating parameter may achieve a safe process only when no failure of the utilities occurs, but may be under risk in case of utility failure. This can be addressed by following the belief that, once the operating parameters under normal operation conditions have been determined, the corresponding values of MTSR should be experimentally decided with RC1 tests. If MTSR > MAT, then it should lower the dosing rate until MTSR < MAT. (4) Once a suitable set of operating parameters has been designed and verified in laboratory-scale reactors, scale-up to the pilot scale or industrial plant must be executed. Generally, for single kinetically controlled liquid−liquid reactions, the effect of mass transfer on the reactions can be neglected in the scale-up process. In other words, the macrokinetic regime at the laboratory or pilot-plant scale is the same as that at industrial scale. Accordingly, Tj at the pilot plant or industrial plant should be equal to that at the laboratory plant, while tD for the pilot-plant scale, as well as industrial scale, can be identified using the same Wt number.

Figure 9. Flowchart for designing thermally safe operating conditions at the laboratory scale and scaleup to the industrial plant for liquid− liquid SBRs in which kinetically controlled reaction occur in the continuous phase.

refer to reaction temperature (Tr) and dosing time (tD)) to perform isothermal RC1 tests. After these tests, Xac,max and reaction enthalpy (ΔHr) can be directly obtained through integrating the heat flow curves with RC1 software. Then predict the value of νADaREκ through νADaREκ vs Xac,max plots in Figure 7. In addition, the value of ε can be obtained from the process recipe. Accordingly, Wt and Ry can be calculated with the above obtained parameters. With respect to reactions occurring in the continuous phase, if Wt > Wtmin, it indicates that the selected operating parameters belong to the modified QFS scenario. Then, whether this operation is production-efficient should be examined. If yes, then examine whether the value of Tta,max calculated by eq 18 is lower than MAT. If yes, the selected operating parameters can be identified to be both thermally safe and production-efficient. If Wt < Wtmin, then examine whether Ry < Rymin. It is notable that, with respect to reactions occurring in the dispersed phase, the above examination is not necessary. For liquid−liquid reactions occurring in the both the continuous and dispersed phases, the following procedure should be followed: Tta,max = Tc +

Wtlab =

= Wt ind = lab


ind

(19)

Since the initial heat-transfer surface per unit volume, i.e., (UA)0/V0, is typically lower at industrial scale, the scaled-up dosing time will be higher. As a consequence of the above procedure, the value of Tta,max in scaled-up plants remains equivalent to that at the laboratory plant, while the value of Ry for the industrial scale is higher, which indicates that accumulation in the scaled-up plant should be lower than that in the laboratory plant. In addition, it is notable that the above procedure is developed for kinetically controlled liquid−liquid reactions. One easy way to determine whether the reactions are kinetically controlled is to increase the stirring speed. Since kinetically controlled reactions are not influenced at all by mass-transfer resistances, it should be independent of the interfacial area and, hence, of the degree of agitation. Therefore, once increasing the stirring speed no longer influences the conversion rate of

1.05ΔTad,0 ε[RH + Wt(1 + ε)]


(18)

If Ry < Rymin, it indicates that the selected operating parameters may not belong to the modified QFS scenario. Measures then should be taken to increase the value of Ry until Ry > Rymin. Generally, two measures can be adopted to achieve this purpose: increasing reaction temperature and prolonging dosing period. Increasing the reaction temperature should be 10435



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ACKNOWLEDGMENTS This work has been supported by the Fundamental Research Funds for the Central Universities.

liquid−liquid reactions, the reactions can be reasonably expected to be kinetically controlled. In fact, for more evidence, the Ha criterion can be applied. As shown in Table 1, if Ha < 0.3, it indicates that the liquid−liquid reactions are kineticscontrolled. However, it is worthwhile to note that estimating the value of Ha requires information on the reaction rate constant.

6. CONCLUSION The primary contribution of this work is to develop a simple procedure, with no requirement of kinetic and solubility parameters, to designing thermally safe operating conditions and scaling-up from laboratory or pilot plants to industrial plants for isoperibolic SBRs in which kinetically controlled liquid−liquid reactions occur. For the above purpose, we suggest a modified definition of QFS operation first. Accordingly, a series of theoretical tools based on the above modified definition of QFS operation are developed for designing thermally safe operating conditions, as shown in Figures 4−7. However, the theoretical tools shown in this work do not contain all the cases of interest in practice. Thereby, in a realistic case, the corresponding theoretical tools should be built following the procedure presented in this work. Once built, the practical procedure introduced in section 5 can be followed to design thermally safe operating conditions for liquid−liquid SBRs in which kinetically controlled reactions occur. In addition, it is worthwhile noting that the procedure developed in this paper cannot be applied for completely or even partially diffusion-controlled liquid−liquid reactions. Diffusion-controlled liquid−liquid reactions will be studied in the future. In the end, we would like to underline that the method proposed in this article can be used only when the reaction rate order is equal to 2. Some knowledge (even if only qualitative) on the kinetics of the process is required to assess the reliability of the second-order kinetics hypothesis. In particular, reactions with autocatalytic behavior should be excluded.

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NOMENCLATURE A = heat exchange surface area, m2 C = instantaneous concentration, mol m−3 cp = specific heat capacity, J kg−1 K−1 Da = Damköhler number at the reference temperature (TR); Da = kRtdosCB,0 DL = diffusion coefficient E = activation energy, J mol−1 Ex = exothermicity number f = function of the dimensionless time and conversion of component B; f = (1 − ξB)(θ − ξB)/(1 + εθ) (for θ < 1) or f = (1 − ξB)2/(1 + ε) (for θ > 1) Ha = Hatta number k = kinetic rate constant, m3 mol−1 s−1 kL = mass transfer coefficient m = distribution coefficient MAT = maximum allowable temperature, K nB = number of moles of component B r = instantaneous reaction rate, mol m−3 s−1 RH = ratio of the volumetric heat capacities of the dosed component A and B; RH = (ρcp)A/(ρcp)B RE = reactivity enhancement factor Ry = reactivity number t = time, s T = temperature, K Tta = the target temperature, K U = overall heat transfer coefficient, W m−2 K−1 V = actual volume of the reactor content, m3 Wt = Westerterp number; Wt = (UA)0tdos/(ε(ρcp)0V0) X = conversion

Greek Symbols

γ = dimensionless activation energy; γ = E/(RTR) ε = relative volume increase at the end point of the feed period θMTSR = dimensionless instant corresponding to MTSR θ = dimensionless time; θ = t/tdos κ = dimensionless reaction rate constant; κ = exp{γ[1 − (1/ τ)]} ν = stoichiometric coefficient ρ = density of the reaction mixture, kg m−3 τ = dimensionless temperature; τ = T/TR ϕV,A = volumetric dosing rate of component A, m3 s−1 ΔHr = enthalpy of reaction, J mol−1 ΔTad,0 = adiabatic temperature rise under initial conditions; ΔTad,0 = (−ΔHr)nB,0/(νB(ρcp)0V0), K Δτad,0 = dimensionless form of ΔTad,0; Δτad,0 = ΔTad,0/TR

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b02599. Characteristic profiles for isoperibolic SBRs (Figure S1); one typical boundary diagram for isoperibolic SBRs based on the modified definition of QFS (Figure S2); an example to illustrate that there must be no thermal runaway event happening in the SBRs as long as no failure of the utilities in the case that Wt > Wtmin (Figure S3) (PDF)

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Subscripts and Superscripts

ac = accumulation A, B, C, and D = components c = continuous phase d = dispersed phase + = dose eff = effective end point = end point of the dosing period 0 = initial f = final j = jacket temperature max = maximum

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 25 84315526. Fax: +86 25 84315526. E-mail: [email protected]. ORCID

Zichao Guo: 0000-0002-2288-1174 Notes

The authors declare no competing financial interest. 10436


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min = minimum n = value of n in eq 15 p = process r = reaction ta = target temperature

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Designing Thermally Safe Operation Conditions for Isoperibolic Liquid

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