Facile Approach To Design Thermally Safe Operating Conditions for

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Facile Approach To Design Thermally Safe Operating Conditions for Isoperibolic Homogeneous Semibatch Reactors Involving Exothermic Reactions Zichao Guo,*,† Wei Feng,‡ Shiran Li,† Peng Zhou,† Liping Chen,† and Wanghua Chen† †

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Department of Safety Engineering, School of Chemical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China ‡ Xi’an Modern Chemistry Research Institute, Xi’an 710065, China S Supporting Information *

ABSTRACT: To prevent thermal runaway incidents in semibatch reactors (SBRs), designing thermally safe operating conditions should be considered first. Regretfully, determination of the reaction kinetics for organic reactions is a timeconsuming and money-consuming process, especially in the fine chemical and pharmaceutical industries. Therefore, it is desirable to develop kinetic-parameter-free methods that can design the right operating conditions. Along this line, a new facile approach to design thermally safe operating conditions for isoperibolic homogeneous SBRs has been developed in this article. This approach is developed on the basis of deep insights into the phenomenon that the MTSR vs reaction temperature profiles for isothermal exothermic reactions present an “S” shape. The approach has been experimentally validated using the hydrolysis reaction of acetic anhydride. However, more work should be carried out to extend this method to more complex reaction systems, like consecutive reactions.

1. INTRODUCTION

In the last decades, many works have been published to design thermally safe operating conditions for isoperibolic SBRs.3−12 Most of these works require detailed reaction kinetics, at least the apparent kinetics. However, determination of the kinetic models and parameters is time-consuming and money-consuming in realistic cases. Therefore, it is desirable to develop simple and reliable approaches without the requirement of kinetics to design thermally safe operating conditions for SBRs. For this purpose, in our opinion, safety criteria that can distinguish between the safe and runaway situations with no requirement of the detailed reaction kinetics should be developed first. Then feasible procedures to design the right operating conditions should be developed considering the safety criteria.

Thermal runaway accidents are undesirable in the chemical industrials. Unfortunately, they are still not completely gone. Blassubramanian and Louvar1 have revealed that 26% of the major petrochemical plant accidents are due to thermal runaway. In the fine chemical and pharmaceutical industries, semibatch reactors (SBRs) are the most frequently used type of reactors to prevent thermal runaway events by controlling the heat generation rate through tuning the dosing rate. To prevent thermal runaway incidents in SBRs, designing the right operating conditions should be considered first.2 It is well-known that the temperature control strategies in SBRs mainly include isothermal, isoperibolic, adiabatic, and other nonisothermal ones. Since the isoperibolic temperature control strategy is simple to be operated and popular in practice, the isoperibolic SBRs are considered in this article. The isoperibolic mode means the coolant temperature keeps constant while the reaction mixture temperature changes. © XXXX American Chemical Society

Received: Revised: Accepted: Published: A

April 20, 2018 July 16, 2018 July 20, 2018 July 20, 2018 DOI: 10.1021/acs.iecr.8b01720 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research The first attempt to define such a safety criterion was carried out by Hub and Jones.13 They state that when the first and second derivatives of reactor temperature are simultaneously ́ et positive, the reactions are in a runaway situation. Zaldivar al.14−16 proposed a more advanced criterion, that is, divergence criterion, which is derived from the dynamical systems theory and characterization of chaotic attractors in dynamical systems. This divergence criterion states that when the divergence of the dissipative reactive systems exceeds 0 at a segment of the reaction path, the reaction is under a potential thermal loss risk. The main advantage of this divergence criterion is that it only requires temperature measurements. Westerterp proposed that on the basis of the comparison between the reaction mixture temperature profile and a so-called target temperature profile, isoperibolic SBRs could be classified into three situations: no ignition, thermal runaway, and QFS (quick onset, fast conversion, and smooth temperature profiles).3,4 Herein, QFS is the desirable situation. On the basis of the concept of QFS, Copelli and co-workers17−19 introduced the theoretical tool of the topological curve, which was a plot of Tmax/Tc vs the conversion. With the topological tool, QFS operating conditions can be identified without any knowledge of the kinetics. Maestri and Rota recently developed a new kinetic free SBR monitoring method to allow for online detection of the runaway accidents in the SBRs and designing safe operating conditions for SBRs.20−22 Their method is developed on the basis of the X number. In our recent papers,23,24 a feasible kinetic-parameter-free procedure for designing thermally safe operating conditions for isoperibolic SBRs is developed. It just needs several isothermal RC1 (reaction calorimetry) tests, which are convenient to conduct in a laboratory. Regretfully, this procedure can be used only when the reaction order is equal to 2. In this article, a new feasible procedure to design thermally safe operating conditions for isoperibolic homogeneous SBRs will be developed. The procedure is suitable to the reactions of arbitrary order and is developed on the basis of the phenomenon that the MTSR vs reaction temperature profiles for isothermal exothermic semibatch reactions present an “S” shape.25 This article will be constructed as follows: first, the dimensionless mathematical model for isoperibolic homogeneous SBRs will be briefly introduced; second, the theory background of the approach will be introduced; third, development of the facile approach to design thermally safe operating conditions for isoperibolic SBRs will be discussed on the basis of the previous sections; then the practical procedure to design thermally safe operating conditions will be introduced in detail; finally, a homogeneous hydrolysis reaction will be taken as a case study to validate the procedure.

A is dosed at a constant rate until the stoichiometric amount of A is added. The heat generated by the reaction is removed by a flow of coolant through a coil or a jacket. In addition, some more assumptions are made as follows: • The reaction is irreversible and nonautocatalytic. • The reaction mixtures are perfectly macromixed. • The reaction volume variation during the reaction is negligible, and the increase in reaction volume is only because of the dosing of component A. • The heat generated from physical phenomenon like mixing is negligible. The reaction enthalpy is the only source of heat generation. • The physicochemical properties of all the components are constant during the whole reaction. • The overall heat transfer coefficient multiplied by heat transfer area (UA) is proportional to the liquid volume in the reactor. The mass and heat balance equations for these reaction systems can be expressed as follows23 l dXB o o o = vADaRE·κ ·f o o o dθ o o o ÄÅ o o ÅÅ o dτ 1 ÅÅΔτad,0 dXB − ε(Wt(1 + εθ )(τ − τc) m = o ÅÅ o o θ ε θ + d (1 R ) ÅÇ É dθ o H o ÑÑ o o ÑÑ o o τ τ + − R ( )) ÑÑ o H dos o ÑÑÖ o n

(1)

where f is expressed as (1 − XB) (θ − XB) /(1 + εθ) if θ < 1 or (1 − XB)(n+m)/(1 + ε) (n+m−1) if θ ≥ 1. It can be easily derived from the above model that the value of NB/vB must be higher than that of NA/vA over the whole dosing period. Herein, NB and NA refer to the mole numbers of components B and A in the SBRs. Therefore, the accumulation of component A at dimensionless time θ < 1 can be expressed by n

Xac = θ − XA

m

(n+m−1)

(2)

The maximum value Xac can be expressed as Xac,max, which generally appears at the endpoing of dosing period, namely θ = 1, for isothermal semibatch reactions.26

2. DIMENSIONLESS MATHEMATICAL MODEL Assuming that a single bimolecular reaction is performed in a semibatch homogeneous stirred reactor equipped with the cooling jacket: vA A + vB B → C + vDD

3. THEORY TOOLS 3.1. Relationship between νADaRE·κ and Xac,max in Isothermal Homogeneous SBRs. For isothermal homogeneous SBRs, the mathematical model can be represented by the first mass balance equation in eq 1. It can be reasonably derived that, for isothermal SBRs with constant feed rate, Xac,max is a function of νADaRE·κ and ε. Hugo found that, for a reaction of second order in isothermal SBRs, there is26 2 1 νADaRE·κ = π Xac,max 2 (3)

where components A and B are pure or diluted with inert solvents, C and D are the products of the reaction, and vi is the stoichiometric coefficient of components. Let us assume that the product C is the desired product; then the value of vC can be reasonably set to 1. Moreover, we assume that the reactant B is loaded into the reactor vessel initially and then component

as long as νADaRE·κ > 6. One can find that no information on ε is included in eq 3. In contrast, we numerically calculated the mathematical model for isothermal SBRs and obtained the quantitative relationships between νADaRE·κ and Xac,max at different values of ε for second-order reactions, as shown in Figure 1. It is obvious that B

DOI: 10.1021/acs.iecr.8b01720 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research at different values of ε, the values of Xac,max are different at the same value of νADaRE·κ.

Figure 2. Linearly quantitative relationships between (νADaRE·κ)−1/2 and Xac,max for reactions of second order at different ε. νADaRE·κ ranges from 6 to 100. Dots are calculated data, and solid lines are fitting lines.

Figure 1. Quantitative relationships obtained by numerically calculation between νADaRE·κ and Xac,max at different values of ε for isothermal homogeneous SBRs with second-order exothermic reactions involved.

Table 1. Fitting Results of the Relationship between (νADaRE·κ)−1/2 and Xac,max for Isothermal Homogeneous SBRs

Rearangement of eq 3 gives the following form Xac,max =

2 (νADaRE·κ )−1/2 π

(4)

Then one can easily find from eq 4 that the relationship between νADaRE·κ−1/2 and Xac,max is linear. Inspired by this, we speculate that the data in Figure 1 comply with the following expression Xac,max = a(νADaRE·κ )−1/2 + b

(5)

(6)

From the mathematic model for isothermal homogeneous SBRs, one can find that only two constants are involved, namely, νADaRE·κ and ε. Hence, we can reasonably conclude that the constant a in eq 5 is only a function of ε. To validate this, the curves of (νADaRE·κ)−1/2 vs Xac,max are provided in Figure 2 and the fitting results are listed in Table 1. In Figure 2, it is obvious that Xac,max linearly increases with increasing (νADaRE·κ)−1/2. The values of adjusted R2 in Table 1 for all the cases in Figure 2 are higher than 0.999, verifying the validity of eq 6. Inspired by the above results, we further speculate that, for reactions of arbitrary order, the quantitative relationship between νADaRE·κ and Xac,max in isothermal homogeneous SBRs obeys Xac,max = a(νADaRE·κ )−1/(n + m)

a

adjusted R2

0.1 0.5 0.9

0.83176 0.95224 1.05482

0.99999 0.99986 0.99964

please refer to the Supporting Information. It is apparent that the values of Xac,max present a linear relationship with (νADaRE·κ)−1/(n+m) at different ε in all the cases in Figure 3. The values of the adjusted R2 for all the cases are higher than 0.999, which evidence the linear relationship between (νADaRE·κ)−1/(n+m) and Xac,max. The values of a for the four cases in Figure 3 are listed in Table S1. 3.2. Criterion To Check the Minimal Value of MTSR. MTSR (defined as the maximum temperature of synthesis reaction under adiabatic conditions) is an essential safety indicator for assessment of thermal runaway, which can be expressed by the following formula

where a and b are the constants. It is clear that when the value of νADaRE·κ goes to infinity, Xac,max must go to 0. Hence, we can reasonably expect that b in eq 5 must be equal to 0. Thus, eq 5 can be simplified as Xac,max = a(νADaRE·κ )−1/2

ε

MTSR = T + Xac,max ·ΔTad,f

(8)

where T is the reaction temperature and ΔTad,f = (−ΔHr)nB,0/ (ρcp)fVf) is the adiabatic temperature rise. As is well-known, for isothermal SBRs, the value of Xac,max will decrease with increasing reaction temperature. Accordingly, it can be reasonably expected that at least for high exothermic reactions there exists an optimum reaction temperature at which the MTSR reaches its minimal value. Hugo26 and our previous work25 have numerically and experimentally demonstrated this trend. For better understanding, a typical example is shown in Figure 4. It is obvious that MTSR reaches the minimal value at point B. From the math viewpoint, at point B the following expression is fulfilled

(7)

where n and m represent the reaction orders of components A and B, respectively. To validate eq 7, four calculated examples with different kinetic orders are shown in Figure 3. For more examples,

dXac,max dMTSR = 1 + ΔTad, f =0 dT dT

(9)

From eq 7, we easily get C


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Industrial & Engineering Chemistry Research

Figure 3. Four examples to show the linearly quantitative relationships between (νADaRE·κ)−1/(n+m) and Xac,max at different ε. The values of νADaRE·κ range from 6 to 100. Dots represent calculated data, and solid lines represent fitting lines. Key: (a) n = 1, m = 0.5; (b) n = 1, m = 0; (c) n = 1, m = 1.5; (d) n = 1.5, m = 0.5.

d(νADaRE·κ ) E = (νADaRE·κ ) dT RT 2

(11)

Then let us substitute eq 11 into eq 10 and obtain dXac,max dT

=

−a(νADaRE·κ )−1/(n + m) − 1 E × n+m RT 2 × (νADaRE·κ )

= −a(νADaRE·κ )−1/(n + m) ×

E RT (n + m) 2

(12)

Substituting eq 7 into eq 12 gives the following expression: dXac,max dT Figure 4. MTSR vs T profile for isothermal operation: νADaRE = 4, γ = 40, ε = 0.3, Δτad,f = 0.5, n = 1, m = 1.

dXac,max dT

−a(νADaRE·κ )−1/(n + m) − 1 d(νADaRE·κ ) = n+m dT

=−

E Xac,max RT (n + m) 2

(13)

Substituting eq 13 into eq 9 gives ΔTad,f E dMTSR Xac,max = 0 =1− dT RT 2(n + m)

(10)

(14)

Arrange eq 14 and give

From the definitions of Da = kn,mtdCB,0n+m−1 and κ = exp(γ(1−

ΔTad,f ·E RT 2

1/τ)), we can also get D

−

n+m =0 Xac,max

(15) DOI: 10.1021/acs.iecr.8b01720 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 5. (a) Values of adiabatic criterion profile for isothermal SBRs with reaction temperature τr = 1.055. Temperature profiles for isoperibolic SBRs are shown in (b)−(d): (b) Wt = 10; (c) Wt = 20; (d) Wt = 40. vADaRE = 1.8, ε = 0.4, γ = 38, RH = 1, Δτad,0 = 0.6, τc = τdos = 1.055, n = 1, m = 1.

ΔTad,f ·E

On the basis of the above analysis, we can reasonably expect that, for the points in the range from B to C in Figure 4, the following expression is fulfilled ΔTad,f ·E RT

2

−

n+m T1) with the same feed rate should be conducted in the RC1 reactor. If eq 21 is not fulfilled, two options can be employed: increasing T or prolonging tdos until eq 21 is fulfilled. In general, the first one is preferable, because prolonging tdos leads to longer addition times, which negatively affect the time cycle and productivity. In contrast, increasing the reaction temperature shows the following merits: the reaction is faster resulting in a shorter time-cycle and cooling capacity is increased due to the increase in temperature difference between reaction mass and coolants. If eq 21 is fulfilled, it means an isoperibolic SBR with an initial temperature of T2 is in the QFS situation. Then it needs to be checked whether Tta,max calculated by eq 18 is lower than MAT. If Tta,max < MAT, the isoperibolic homogeneous SBRs can be considered as thermally safe. If not, it needs to increase tdos until Tta,max < MAT.

(19)

where Xstoi is the conversion at stoichiometric point, Qstoi is the heat production before the stoichiometric point, the subscript “stoic” is the stoichiometric point where stoichiometric amounts of component A are added into the SBRs. Since the value of ΔTad,f for a given exothermic reaction is constant, MTSR2 > MTSR1 is equivalent to ij y i y jjT − Q stoi,2 zzz > jjjT − Q stoi,1 zzz jj 2 zz jj 1 z j (mcp)stoi z j (mcp)stoi zz k { k {

(20)

Arranging eq 20 gives (mcp)stoi (T2 − T1) ≥ (Q stoi,2 − Q stoi,1) (mcp)stoi ΔT ≥ ΔQ stoi

or (21)

Once eq 21 is fulfilled, the corresponding isoperibolic homogeneous SBRs with the initial temperature of T2 must be in the QFS situation.

5. PRACTICAL PROCEDURE FOR DESIGNING THERMALLY SAFE OPERATING CONDITIONS In this part, a feasible procedure to design thermally safe operating conditions for isoperibolic homogeneous SBRs will be developed on the basis of the previous discussion and one can also refer to the flowchart in Figure 6: (1) In general, to avoid triggering the second decompostion reaction, the thermal stability of reactants, products, and reacting mixture should be experimentally determined first. This can be achieved by carrying out dynamic differential scanning calorimetry (DSC) tests and/or adiabatic rate calorimetry (ARC) tests. The dynamic DSC tests are conducted usually to scan whether reactants, products, and the reacting mixture are thermally stable. If not, then at least a standard heat-wait-search (HWS) ARC test should be F


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Figure 7. Experimental heat flow profiles in the isothermal mode: (a)70 °C; (b)75 °C; (c) 85 °C; (d) 90 °C.

Tta,max =

ΔTad,0 WtTc + RHTdos + 1.05 ε(Wt + RH) Wt + RH

6. CASE STUDY 6.1. Reaction. In this part, the hydrolysis reaction of acetic anhydride (AAh) in acetic acid (AAc)/water mixture will be used to prove the feasibility of the previously proposed procedure. This reaction obeys the typical N-order kinetics.5 AAh and acetic acid (AAc) were purchased from Aladdin reagent co., Ltd. 6.2. Apparatus and Calorimetric Tests. All the experiments were carried out in a RC1Mettler Toledo reaction calorimeter of a nominal volume of 1 L. The reactor is equipped with a stainless temperature sensor, a stainless calibration heater, and an anchor agitator. The calibration heater allows measuring the heat transfer coefficient and the heat capacity of the reaction mixture by carrying out a calibration procedure. A dosing pump is used to introduce the water into the reactor. The heating−cooling system using a single heat transfer fluid (silicone oil) works within a temperature range of −50 to +200 °C. First, four isothermal tests at 70, 75, 85, and 90 °C are carried out. A mixture of 287.9 g AAh and 56.21g AAc was loaded into the RC1 reactor. Herein, AAc was added to increase the solubility of water in the organic phase and keep the reacting mixture always homogeneous. Once the temperature has been stable at the set temperature, 50.76 g of water was dosed within 3800 s at a constant rate. The mole ratio of AAh to water is 1:1. After the exothermic signals are insignificant, two droplets of sulfuric acid were introduced into the reactive mixture to expedite the hydrolysis reaction. It

(22)

(3) Then the operating parameters obtained by the above procedure must be experimentally verified by at least one isoperibolic RC1 test. Note that the operating parameters obtained at this stage only take into account normal operating conditions; i.e., the safety aspects linked to failure of the agitator or the temperature control system are not considered. From the inherent safety point of view, triggering of the secondary decomposition or side reactions should be prevented even in the case of utility failure. This inherent safety can be achieved if MTSR < MAT. If MTSR > MAT, then the dosing rate should be lowered until MTSR > MAT. It should be kept in mind that the initial temperature of the isoperibolic SBRs should not be increased at this stage, because increasing the initial temperature will increase the value of MTSR. It is obvious that designing thermally safe operating conditions for isoperibolic homogeneous SBRs via the above procedure does not require any information on the kinetic parameters of the exothermic reactions at all. Compared to the procedures we developed in our previous works,23 the main virtue in this work is that the procedure in this article is suitable to arbitrary order reactions; in other words, there is no need to determine the kinetic orders completely. In contrast, our previous method is only rigorously valid to second-order reactions. G


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Figure 8. Experimental temperature profiles in the isoperibolic mode: (a) 90 °C; (b) 75 °C.

should be stated that the heat of dilution of sulfuric acid can be neglected because the mass of the added two sulfuric acid droplets is negligible relative to the mixture mass. Then, two isoperibolic tests at 75 and 90 °C are conducted. The mass ratio and the dosing rate of water are identical with the above isothermal tests. No sulfuric acid is dosed in the isoperibolic operation mode. 6.3. Results. The experimental heat flow profiles at isothermal mode are shown in Figure 7. Integration of the heat flow profiles gives the values of Qstoi of 107.4 kJ, 114.5 kJ, 126.3 kJ and 130.0 kJ at 70 °C, 75 °C, 85 and 90 °C, respectively. The average heat capacity of the final mixture (cp) from the RC1 calibration tests is about 2.36J J/(K × g). In addition, the average reaction heat (ΔHr) is determined to be −53.17 kJ/mol. As a result, the value of ΔTad,0 can be calculated as follows ΔTad,0 =

Moreover, from Figure 8b, we can observe that the isoperibolic operation at 75 °C is in thermal runaway situation, which is in good agreement with the eq 24.

7. CONCLUSION To deal with the problem that determination of the reaction kinetics is time-consuming and money-consuming for organic reactions, a new feasible procedure to design thermally safe operating conditions for isoperibolic homogeneous SBRs has been developed in this article. The main merit of this new method is no requirement of kinetic parameters of the reactions. This method is developed on the basis of an insight into the phenomenon that the MTSR vs reaction temperature profiles for isothermal exothermic reactions present an “S” shape and the adiabatic criterion developed in our previous papers. This method states that as long as the values of MTSR increase with the reaction temperature under isothermal operation, the corresponding isoperibolic homogeneous SBRs must be in QFS situation, regardless of the values of Wt. To follow this procedure, several isothermal RC1 tests should be conducted first to determine the operating parameters for isoperibolic homogeneous SBRs and then at least one isperibolic RC1 test is required to verify the QFS operation. In addition, the hydrolysis reaction of acetic anhydride in acetic acid/water mixture was conducted on a RC1 calorimeter and the results obviously evidenced the validation of the procedure. In the end, it should keep in mind that the method in this article is developed for single exothermic homogeneous reactions with no autocatalytic behavior. More work should be carried out to extend this method to reactions with autocatalytic behavior and even more complex reactions, like consecutive reactions.

−ΔHr × Nwater 53.17 kJ/mol × 3.12 mol = m0 × cp ,0 344.1 g × 2.17 J/(°C × g)

= 222.17 °C

(23)

where m0 and cp,0 are the mass and heat capacity of the initial mixture of AAh and AAc, respectively. Substituting these Qstoi into eq 21 gives the following equations (mcp)stoi (T75° C − T70° C) = 4.99 kJ < (Q stoi,75° C − Q stoi,70° C) = 7.1 kJ

(24)

(mcp)stoi (T90° C − T85° C) = 4.99 kJ > (Q stoi,90° C − Q stoi,80° C) = 3.7 kJ

(25)

■

From eq 25, we can expect that the isoperibolic operation at 90 °C must be in QFS situation. This expectation has been verified by the RC1 temperature profile, as shown in Figure 8a. The Tta temperature lines in Figure 8 are determined following the equation

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.8b01720.

l ΔTad,0 o Wt(1 + εθ)Tc + RHTdos o o θ≤1 + 1.05 o o Wt(1 + εθ) + R o ε(Wt(1 + εθ) + RH) o H o Tta = m o o ΔTad,0 o Wt(1 + ε)Tc + RHTdos o o θ>1 + 1.05 o o o Wt ε R ε Wt (1 ) ( (1 + + + ε) + RH) o H n

Fitting results between (νADaRE·κ)−1/(n+m) and Xac,max for Figure 3 (Table S1); three examples to show the l i n e a r ly q u a n t i t at i v e r e l a t i o n s h i p s b e t w e e n (νADaRE·κ)−1/(n+m) and Xac,max at different ε(Figures S1, S2 and S3) (PDF)

(26) H


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ΔHr = enthalpy of the reaction, J·mol−1 ΔTad,0 = adiabatic temperature rise at initial conditions ΔTad,0 = (−ΔHr)nB,0/(νB(ρCp)0V0), K Δτad,0 = dimensionless form of ΔTad,0 Δτad,0 = ΔTad,0/TR ϕV,A = volumetric flow rate

AUTHOR INFORMATION

Corresponding Author

*Z. Guo. E-mail: [email protected], Tel: +86 025 84315526; Fax: +86 025 84315526. ORCID

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Zichao Guo: 0000-0002-2288-1174 Notes

SUBSCRIPTS AND SUPERSCRIPTS ac = accumulation ad = adiabatic condition A, B, C, D = components c = cooling medium cf = cooling failure dos = dosing 0 = initial f = final m = kinetic order for reactant that are initially loaded in the SBRs max = maximum min = minimum n = kinetic order for reactant that are dosed into the SBRs r = reaction R = reference ta = the target temperature stoi = stoichiometric

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been financially supported by National Key R&D Program of China (2017YFC0804701-4) and the Fundamental Research Funds for the Central Universities (30917011312).

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NOMENCLATURE A = heat exchange surface area, m2 C = instantaneous concentration, mol·m−3 cp = specific heat capacity, J·g−1·K−1 Da = Damköhler number at the reference temperature (TR) Da = kRtdosCB,0 E = activation energy, J·mol−1 f = function of the dimensionless time and conversion of component B k = kinetic rate constant, m3·mol−1·s−1 k0 = pre-exponential factor MAT = maximum allowable temperature, K MTSR = maximum temperature of synthesis reactions under adiabatic condition Q = heat generated by the reactions m = mass N = mole number RH = ratio of the volumetric heat capacities of the dosed component A and B RH = (ρcp)A/(ρcp)B RE = the reactivity enhancement factor RE1 = (νB/νA)1−n t = time, s T = temperature, K Tcf = temperature obtained under adiabatic conditions Tta = the target temperature, K TR = reference tmperature, 300 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

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REFERENCES

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GREEK SYMBOLS γ = dimensionless activation energy γ = E/RTR ε = relative volume increase at the end point of the feed period ε = VA/VB 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 I


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Facile Approach To Design Thermally Safe Operating Conditions for

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