Model-Based Design For Inhibition Of Thermal Runaway In Free

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Kinetics, Catalysis, and Reaction Engineering

Model-Based Design For Inhibition Of Thermal Runaway In Free Radical Polymerization Guanyang Liu, and Benjamin A Wilhite Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b02007 • Publication Date (Web): 12 Aug 2019 Downloaded from pubs.acs.org on August 12, 2019

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Industrial & Engineering Chemistry Research

MODEL-BASED DESIGN FOR INHIBITION OF THERMAL RUNAWAY IN FREE RADICAL POLYMERIZATION Guanyang Liu,1,2 Benjamin A. Wilhite1,2* 1Artie

McFerrin Department of Chemical Engineering, Texas A&M University, 3122 TAMU, College Station, Texas 77843, United States

2Mary

Kay O’Connor Process Safety Center, Texas A&M University, 3122 TAMU, College Station, Texas 77843, United States

*Author to whom correspondence should be addressed: [email protected]

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract This work demonstrates the extension of runaway criteria originally developed by Van Welsenaure and Froment and the more recent divergence criterion to the “short-stopping” of batchwise free-radical polymerization processes for mitigating thermal runaway via injection of free-radical scavengers in response to early detection of imminent runaway. A transient, nonisothermal well-mixed batch reactor model is developed for free-radical solution polymerization of methyl methacrylate (MMA) as an illustrative example, with instantaneous addition of inhibitor occurring at predetermined injection or response time. Three parameters of the “short-stopping” mitigation procedure are investigated, specifically the injection time, inhibitor quantity, and inhibitor activity. Model results are analyzed in terms of Semenov (ψ) and critical Semenov number (ψc) trajectories in time. Results indicate the general criterion for sufficient “short-stopping” recipe is to ensure a negative time-derivative of local Semenov number upon inhibitor injection (i.e., dψ/dt|t = tinj < 0), as opposed to the far more conservative criteria of reducing local Seminov number to below the corresponding local critical Seminov number, i.e. ψ(t) < ψc(t), at the time of injection. The proposed criterion is also compared with a similarly modified first-derivative divergence criterion, and exhibits better performance than the latter.

Keywords Free-radical polymerization, Thermal runaway, Inhibition, System analysis

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1. Introduction Thermal runaway of an exothermic reaction occurs when heat generation via reaction exceeds the heat removal capacity of the cooling system, resulting in temperature autoacceleration of the reacting mass. Without effective mitigation measures, consequences are violent boiling of reaction fluids and/or vapor generation of secondary reactions, both of which can result in overpressurization and subsequent thermal explosion [1]. Of particular interest is the mitigation of thermal runaway in batch polymerization reactors; a recent study of thermal runaway incidents in batchwise processes within the United Kingdom (UK) indicated that the majority of such incidents occurred in polymerization units, which accounted for 41 of 189 incidents from 1962 – 1987, and 10 of 30 from 1988 – 2013 [2]. As a key component of emergency response against thermal runaway, theoretical runaway criteria are capable of reliably predicting onset of thermal runaway while allowing sufficient response times for implementing mitigation strategies [3 - 9]. The first such criteria was provided by Semenov et al. [3] for the case of a potentially explosive mixture under storage. By assuming negligible reaction prior to runaway, geometrical criteria in terms of a critical Semenov Number (𝜓𝐶) was developed based upon the heat generation and removal trajectories corresponding to a minimum heat removal capacity necessary for preventing thermal runaway. Subsequent geometrybased criteria accounting for non-zero reactant uptake prior to runaway were developed by Thomas and Bowes [4] and Adler and Enig [5]. Van Welsenaere and Froment [6, 7] developed criteria for the case of a batch reactor with power-law reaction kinetics, in which thermal runaway is to be avoided despite near or complete conversion of reactants. The resulting algebraic criteria enables rapid estimation of conditions necessary for runaway to occur. However, aforementioned criteria based on geometry properties of temperature and/or conversion profile are not able to indicate the

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intensity of runaway. Sensitivity-based criteria have been developed to overcome this limitation [8 - 9]. Varma et al. [8] characterized runaway via parametric sensitivity of the system such that intensity of runaway can be related to the magnitude of the sensitivity. Zaldivar et al. [9] introduced Lyapunov exponent to define sensitivity, and proposed that runaway can be represented using the system divergence. This divergence (div) criterion states that if system divergence exhibits a positive value on a segment of the reaction trajectory, the system is operating in runaway region. By reconstruction of state space and time series analysis, div criterion can be further implemented for online runaway detection using only real-time temperature measurements [10 - 11]. Sensitivitybased criteria have been validated for multiple chemical process [12 - 15], including polymerization [13]. Common mitigation measures include either venting to prevent over-pressurization or evacuation of the reacting mass to a larger quenching tank; however these measures require additional equipment for post-treatment of the discharge which can significantly increase cost [16]. A promising alternative mitigation technique for batchwise free-radical polymerization is to “shortstop” thermal runaway via injection of free-radical scavengers (e.g., Quinone or nitrocompounds, originally developed for stabilizing monomer solutions during transportation and storage) in sufficient quantities as to halt polymerization reaction [17 - 20]. A thorough experimental investigation of runaway inhibition in batchwise polymerization reactors via freeradical injection was reported by Rowe [21], who investigated multiple inhibitors for quenching the thermal runaway of styrene polymerization in a lab-scale adiabatic calorimeter. More recently, Russo et al. [22] and Ampelli et al. [23] employed calorimetry to investigate the inhibition of thermal runaway in the batchwise polymerization of methyl methacrylate (MMA). While these reports demonstrate the feasibility of free-radical injection as a means to quench thermal runaway

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in free-radical polymerization batchwise processes, there remains a need for quantitatively understanding dynamic competition between runaway and inhibition kinetics for developing robust design criteria for effective “short-stopping” of free-radical polymerization reactors. In the present work, a system-level reactor model with polymerization and inhibition kinetics is developed to apply classical reaction engineering principles to investigate dynamic competition between free-radical polymerization and inhibition. The uniqueness of this work lies in: (1) development of a process model that enables quantitative analysis of how inhibition recipes can affect the quenching performance in a dynamic runaway scenario, and its use for developing (2) explicit criteria for successful inhibition design. The free-radical solution polymerization of methyl methacrylate (MMA) carried out in a batch reactor is taken as a demonstrative example employed by industry [24]. By treating the inhibitor as a free radical scavenger, the rates of polymerization and inhibition are derived based on classical free-radical mechanism [25], employing the method of moment approach originally proposed by Ray et al. [26]. Model results are analyzed in terms of temperature trajectories and both local (ψ) and critical (ψc) Semenov number to identify an explicit criterion for effective inhibition.

2. Theoretical 2.1 Methyl methacrylate (MMA) solution polymerization kinetics The current work employs MMA solution polymerization as an illustrative industrial case to investigate inhibition of thermal runaway. With presence of solvent, the polymerization recipe can be adjusted to the scenario where exothermic polymerization is the driving force of thermal runaway, instead of gel effect. Following a free radical mechanism [25], the kinetics of MMA

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solution polymerization can be described by ( Ⅰ) – ( Ⅴ), where n and m are positive integers indicating chain length. Ethyl acetate (EAc) and Azobis-isobutyronitrile (AIBN) are used as solvent and initiator (I), respectively. The mechanism consists of free radical initiation, propagation of live polymer radicals (Pn•), and termination by combination or disproportionation of two radicals to form dead polymer products (Dn). Since the model is developed to investigate thermal runaway behavior and not product distribution, side reactions (e.g., chain transfer) can be neglected, following previous analysis by Ray [26]. (a) Initiation: I

𝑘𝑑

[Ⅰ]

2𝑅 ∙

R ∙ +M

𝑘𝑖

[Ⅱ]

𝑃1 ∙

(b) Propagation: 𝑘𝑝

[Ⅲ]

𝑃𝑛 ∙ +𝑀 𝑃𝑛 + 1 ∙ (c) Termination 𝑃𝑚 ∙ + 𝑃𝑛 ∙

𝑃𝑚 ∙ + 𝑃𝑛 ∙

𝑘𝑡𝑐

[Ⅳ]

𝐷𝑚 + 𝑛

𝑘𝑡𝑑

[Ⅴ]

𝐷𝑚 + 𝐷𝑛

The rate expression of monomer uptake is obtained by applying quasi-steady state assumption (QSSA) to the total live polymer radical concentration (or zeroth moment of free radical concentration, λ0 which represents the level of reactivity inside the reactor), following the analysis of Ray et al. [26]. The kinetic parameters of MMA polymerization in the EAc solvent are taken

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from Brandrup and Immergut [27] and shown in Table 1. Note that higher order of moment is not calculated, since the main interest of the current work is the global temperature behavior under thermal runaway, and not polymer chain and/or molecular weight distribution. 𝑟𝑀 = ― 𝑘𝑝[𝑀]𝜆0

(1)

𝑟𝐼 = ― 𝑘𝑑[𝐼]

(2)

𝜆0 =

2𝑓[𝐼]𝑘𝑑

(3)

𝑘𝑡

2.2 Inhibition kinetics As an emergency response of thermal runaway, chemical inhibitors (X) are injected into the reactor vessel. Given that the inhibitor acts as a free radical scavenger, it is assumed to react with live polymer radicals to form dead products as shown in [VI]. Therefore, the inhibited polymerization mechanism comprises [I] – [VI]. 𝑘𝑥

[Ⅵ]

𝑃𝑛 ∙ +𝑋 𝐷𝑛

By neglecting volume contraction of the reaction mass, species balance of free radicals present in the reactor can be expressed based on the inhibition mechanism as follows. For species balance of free radicals generated by the initiator: 𝑑[𝑅 ∙ ] 𝑑𝑡

(4)

= 2𝑓𝑘𝑑[𝐼] ― 𝑘𝑖[𝑅 ∙ ][𝑀]

To manage a large number of live polymer species inside the reactor, kth moment of live polymer radicals is introduced as:

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∞

𝜆𝑘 = ∑𝑛 = 1𝑛𝑘[𝑃𝑛 ∙ ]

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(5)

Species balance of growing polymer radicals are written with zeroth moment, λ0inh: 𝑑[𝑃1 ∙ ] 𝑑𝑡

= 𝑘𝑖[𝑅 ∙ ][𝑀] ― 𝑘𝑝[𝑃1 ∙ ][𝑀] + (𝑘𝑡𝑐 + 𝑘𝑡𝑑)𝜆𝑖𝑛ℎ 0 [𝑃1 ∙ ]

(6) 𝑑[𝑃𝑛 ∙ ] 𝑑𝑡

= 𝑘𝑝([𝑃𝑛 ― 1 ∙ ] ― [𝑃𝑛 ∙ ])[𝑀] ― (𝑘𝑡𝑐 + 𝑘𝑡𝑑)𝜆𝑖𝑛ℎ 0 [𝑃𝑛 ∙ ] ― 𝑘𝑥[𝑋][𝑃𝑛 ∙ ], n ≥ 2

(7) Thus, the species balance of zeroth moment can be derived as: 𝑑𝜆𝑖𝑛ℎ 𝑑𝑡

2

𝑖𝑛ℎ = 𝑘𝑖[𝑅 ∙ ][𝑀] ― 𝑘𝑝[𝑃𝑛 ∙ ][𝑀] ― (𝑘𝑡𝑐 + 𝑘𝑡𝑑)(𝜆𝑖𝑛ℎ 0 ) ― 𝑘𝑥[𝑋]𝜆0

(8)

To simplify the model and characterize the effect of inhibitor, QSSA is applied to all the free radical species. Further algebraic manipulation yields a quadratic equation of λ0inh:

(𝑘𝑡𝑐 + 𝑘𝑡𝑑)𝜆𝑖𝑛ℎ2 + 𝑘𝑥[𝑋]𝜆𝑖𝑛ℎ 0 ―2𝑓𝑘𝑑[𝐼] = 0

(9)

By solving Eq. (9) analytically, the expression of λ0inh can be obtained as Eq. (10). Note that kt represents the sum of ktc and ktd: 𝜆𝑖𝑛ℎ 0 =

― 𝑘𝑥[𝑋] + (𝑘𝑥[𝑋])2 + 8𝑓[𝐼]𝑘𝑑𝑘𝑡

(10)

2𝑘𝑡

Thus, the uptake rate expressions of MMA monomer, initiator, and inhibitor consumption in the inhibited polymerization can be expressed as the following: 𝑖𝑛ℎ 𝑟𝑖𝑛ℎ 𝑀 = ― 𝑘𝑝[𝑀]𝜆0

(11)

𝑟𝑖𝑛ℎ 𝐼 = ― 𝑘𝑑[𝐼]

(12)

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𝑖𝑛ℎ 𝑟𝑖𝑛ℎ 𝑋 = ― 𝑘𝑥[𝑋]𝜆0

(13)

2.3 Stirred tank batch reactor model A common cause of thermal runaway in batch reactors is process control loop failure, either due to human error, poor design and/or mechanical failures, leading to system cooling being exceeded by heat generation via reaction, which has caused multiple incidents [2]. As such, the case of full cooling with complete loss of control loop (i.e., isoperibolic) can be taken as a worst-case scenario during thermal runaway following Westerterp et al. [28 - 30] and as an appropriate case for investigating thermal runaway with and without inhibitor injection in the present work. While isoperibolic operation is not realistic for nominal batchwise polymer production, it becomes quite useful when studying safety aspects of batch operations [13, 28 - 30] as it maintains constant external cooling thus allowing heat release behavior to be solely explored. Resulting species and energy balances for a well-mixed batchwise stirred reactor carrying out the liquid phase, homogeneous isoperibolic solution polymerization of MMA are shown as Eqs. (14) - (17). Note that the current model is written with inhibition kinetics, and the reactor model for uninhibited polymerization can be accommodated by replacing zeroth moment of inhibition kinetics, λ0inh with that of polymerization kinetics, λ0 and neglect Eq. (17). During polymerization, increasing viscosity of the reaction media may lead to a decrease of termination rate and heat transfer coefficient (known as the “gel” or “Tromsdorff” effect) [24]. This may introduce an additional ‘lag time,’ coinciding with spatial variations in temperature and reaction rates prior to achieving homogeneous conditions inside the reactor [31, 32]. However, the present study focuses on investigating inhibition behavior in absence of complicating transport

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effects, thus providing a foundation for subsequent work accounting for the effect of mixing behavior of inhibitors and significant viscosity variation during polymerization. As “gel” effect is neglected in the present model, the solvent EAc is only taken as a diluent without any influence on kinetics. The values of physical properties of the reactive media is estimated by considering an ideal mixture of MMA and EAc. The reaction volume (i.e., the volume of reacting mass) is assumed to be 1 L, with heat transfer coefficients and available heat removal area via cooling jacket based upon a typical 1 L reaction vessel. Values selected for fluid properties and reactor parameters are summarized in Table 2.

𝑑𝑇

𝑈𝑎

( ) 𝑑𝑡 = 𝜌𝑐𝑝𝑉 𝑇𝑐 ― 𝑇 +

𝑑[𝑀] 𝑑𝑡

𝑑[𝐼] 𝑑𝑡

―Δ𝐻𝑟 𝜌𝑐𝑝

𝑘𝑝[𝑀]𝜆𝑖𝑛ℎ 0

(14)

= ― 𝑘𝑝[𝑀]𝜆𝑖𝑛ℎ 0

(15)

= ― 𝑘𝑑[𝐼]

(16)

= ― 𝑘𝑥[𝑋]𝜆𝑖𝑛ℎ 0

(17)

𝑑[𝑋] 𝑑𝑡

2.4 Systems analysis of reactor model To further generalize the application of the developed model, the energy balance is recast in dimensionless form by introducing the dimensionless temperature deviation, η, monomer conversion x, initiator conversion xi, and dimensionless time τ (a characteristic time scale of kp,iλ0,i is selected based on the overall uninhibited rate expression): 𝜂=

𝑇 ― 𝑇0

[𝑀0] ― [𝑀]

𝑇0

[𝑀0]

𝛾, 𝑥 =

, 𝑥𝑖 =

[𝐼0] ― [𝐼] [𝐼0]

, 𝜏 = 𝑡 ∙ 𝑘𝑝,𝑖𝜆0,𝑖

8


(18)

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where γ is the lumped dimensionless activation energy. The dimensionless energy balance shown in Eq. (19) accommodates both thermal runaway and inhibition scenarios since the only difference of the two lies in the expression of zeroth moment (λ0 or λ0inh). 1 𝑑𝜂 𝐵 𝑑𝜏

1

= 𝜓(𝜂𝑐 ― 𝜂) + 𝑅 (1 ― 𝑥)

(19)

where R is dimensionless rate of uninhibited or inhibited polymerization:

𝑅=

𝑘𝑝𝜆0 𝑘𝑝,𝑖𝜆0,𝑖

or 𝑅 =

𝑘𝑝𝜆𝑖𝑛ℎ 0

(20)

𝑘𝑝,𝑖𝜆0,𝑖

The resulting adiabatic temperature rise, B, is defined as the ratio of heat generation to the heat capacity of the reaction mass scaled by dimensionless activation energy, and represents the magnitude of thermal runaway. The Semenov number (ψ), defined as the ratio of heat generation to cooling rates scaled by dimensionless activation energy, delineates the potential for thermal runaway. The initial state is traditionally selected as the appropriate reference for defining all dimensionless groups; as such, the dimensionless activation energy is defined according to the rate of uninhibited polymerization. In the present analysis in which each point in time is a candidate for inhibitor injection, these dimensionless parameters can be cast in terms of an instantaneous reference state, as shown in the following: 𝐵(𝑡) =

𝜓(𝑡) =

( ―∆𝐻)[𝑀] 𝜌𝑐𝑝𝑇

(21)

𝛾

( ―∆𝐻)𝑉 ∙ 𝑘𝑝𝜆𝑖𝑛ℎ[𝑀] 𝑈𝑎𝑇

(22)

𝛾

1

𝛾=

𝐸𝑝 + 2(𝐸𝑑 ― 𝐸𝑡)

(23)

𝑅𝑇0

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Van Welsenaere Froment [6, 7] rigorously derived geometry-based criteria for thermal runaway in a batch reactor with overall first order kinetics in terms of critical Semenov number (ψc) defined as Eq. (24). Results from numerical simualtions are used to calculate the local, or time-dependent, values of Semenov and critical Semenov number (ψ, ψc) at each time step.

(

1

1

𝜓𝑐(𝑡) = 1 + 𝑄 +

𝑄

)(𝜂 ― 𝜂 ) ∗ 𝑒𝑥𝑝 ( ―

2

𝑐

𝑎

𝜂𝑐

1 + 𝜂𝑐/𝛾)

(24)

where Q and ηc are defined as the following: 𝛾

𝜂𝑐 = 2[(𝛾 ― 2) ― 𝛾(𝛾 ― 4) ― 4𝜂𝑎]

[

1+4

𝑄=

𝐵(𝑡) 𝜂𝑐 ― 𝜂𝑎

(25)

]

―1 ―1

(26)

2

For all simulations, time-trajectories of local and critical Seminov number are obtained via solution of Eq. (22) and (24) at each time-step to further analyze the impact of inhibition. All simulations are performed based on an illustrative case [33] with initial monomer concentration [M0] = 4.5 mol/L (Φm = 0.478), initial initiator concentration [I0] = 0.055 mol/L, initial temperature T0 = 330 K, and cooling temperature Tj = 300 K, sufficient for runaway in the absence of mitigation procedures.

2.5 Numerical methods The reactor model with polymerization kinetics composed of Eqs. (14) – (16) is solved numerically using a pre-packaged solver of ordinary differential equations (ode45) in MATLAB programming environment to study the temperature and conversion profile of thermal runaway.

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Analysis of trajectories of the corresponding Semenov (ψ) and the critical Semenov number (ψc) during thermal runaway is performed by solving Eq. (22) and (24) with the simulation results. Simulations follow a two-stage procesure to capture the impact of inhibitor injection at a predetermined injection time, tinj > 0. Since chemical inhibitors are injected during thermal runaway, the reactor model with polymerization kinetics (Eqs. (14) - (16)) is initially solved to generate the temperature trajectory of uninhibited thermal runaway. At selected injection timing, the model equations with inhibition kinetics, composed of Eqs. (14) - (17), are then solved numerically with initial conditions as the solution of polymerization reactor model at tinj.

2.6 Divergence criterion applied to inhibition model A generalized thermal runaway criterion based on the system model divergence has been developed [9 - 11] and validated for multiple chemical process [12 - 15]. This divergence theory states that the system is in runaway region when divergence exhibits a positive value at a point along the reaction path. The system divergence is defined as a scalar quantity equal to the trace of the Jacobian matrix of the system of ordinary differential equations governing system dynamics. For complex kinetics such as polymerization, divergence calculation can be simplified by only considering temperature and the reactions that contribute to heat release [9, 13]. Thus, the divergence of the present free-radical polymerization with inhibition model can be expressed as Eq. (27):

(𝑑𝑇𝑑𝑡)

∂

𝑑𝑖𝑣 =

∂𝑇

(𝑑[𝑀] 𝑑𝑡 )

∂

+

(27)

∂[𝑀]

The temperature component of Eq. (27) is calculated as the following:

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(𝑑𝑇𝑑𝑡)

∂

𝑈𝑎

= ― 𝜌𝑐𝑝𝑉 +

∂𝑇

(

―Δ𝐻𝑟 𝜌𝑐𝑝

𝑑𝜆𝑖𝑛ℎ 0

𝑑𝑘𝑝

𝜆𝑖𝑛ℎ 0 𝑑𝑇 + 𝑘𝑝

𝑑𝑇

)

Page 14 of 44

(28)

where the derivatives in Eq. (28) are calculated:

𝑑𝜆𝑖𝑛ℎ 0 𝑑𝑇 𝑑𝑘𝑝 𝑑𝑇

(

― 𝐸𝑝𝑘𝑥 +

=

=

)

𝐸𝑝(𝑘𝑥[𝑋])2 + 4𝑓[𝐼]𝑘𝑑𝑘𝑡(𝐸𝑑 + 𝐸𝑡)

(𝑘𝑥[𝑋])2 + 8𝑓[𝐼]𝑘𝑑𝑘𝑡

― 𝐸𝑡( ― 𝑘𝑥[𝑋] + (𝑘𝑥[𝑋])2 + 8𝑓[𝐼]𝑘𝑑𝑘𝑡)

(29)

2𝑅𝑇2𝑘𝑡 𝐸𝑝

(30)

𝑘 𝑅𝑇2 𝑝

The monomer concentration component of Eq. (27) is expressed as:

(𝑑[𝑀] 𝑑𝑡 )

∂

∂[𝑀]

= ― 𝑘𝑝𝜆𝑖𝑛ℎ 0

(31)

Note that divergence of inhibition model can be converted to that of polymerization model by setting inhibition kinetics kx and inhibitor concentration [X] to “null”. The divergence criterion will be evaluated for predicting the inhibition performance in Section 3.4.

3. Results and discussions 3.1 Baseline case of thermal runaway in isoperibolic batch reactor From a typical experimental work of polymerization by Maschio et al. [33], the polymerization recipe, i.e., initial monomer concentration [M0] = 4.5 mol/L (correspondingly, initial monomer volume fraction Φm = 0.478 assuming MMA molecular weight of 100.1 g/mol and density of 940 g/L), initial initiator concentration [I0] = 0.055 mol/L, initial temperature T0 = 330 K, and cooling temperature Tj = 300 K, is particularly selected such that free radical chemistry is the driving force of thermal runaway while gel effect exhibits negligible impact on temperature auto-acceleration.

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For the baseline case, the resulting temperature trajectory and conversion profile during thermal runaway in the absence of inhibitor injection are shown in Figure 1(a). Under isoperibolic operation, reaction heat immediately begins exceeding system cooling, leading to a rapid rise in temperature to a maximum value coinciding with complete depletion of monomer and initiator. Once reactants are depleted, temperature decreases monotonically due to constant external cooling. As shown in Figure 1(b), real-time Semenov and critical Semenov numbers, ψ(t) and ψc(t), both increase with temperature to a maxima coinciding with near complete depletion of monomer. Prior to the time of maximum reactor temperature, ψ(t) is always greater than ψc(t), indicating that system cooling alone is insufficient to prevent runaway. A comparison of thermal runaway temperature behavior for the baseline case using the present model neglecting gel effect with a modified version of said model including gel effect [34] (detailed in supplemental material) is presented in Figure 2, which confirms that gel effect is negligible for this recipe which corresponds to the case where free radical polymerization time scale is significantly less than that of gel effect.

3.2 Analysis of inhibition via injection of free radical scavengers Under the assumption of homogeneous and immediate mixing of inhibitors upon injection, the design of inhibitor injection, or “short-stopping,” recipes for mitigating thermal runaway in batchwise free-radical polymerization processes can be reduced to three primary parameters, specifically injection time (tinj), inhibitor quantity ([X0]), and inhibitor activity. Assuming identical activation energy for inhibition and propagation, inhibitor activity can be described as the ratio of inhibition to free-radical propagation rate constants, referred to as inhibitor relative activity, θ, shown in Eq. (32):

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𝑘𝑥

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(32)

𝜃 = 𝑘𝑝

Due to the dynamic nature of thermal runaway, the quantity ([X0]) and activity (θ) of inhibitor required to prevent thermal runaway depends on injection timing (tinj), which is dictated by the early warning detection system. The impact of design parameters on thermal runaway mitigation for the base case shown in Figure 1 are presented below, alongside analysis in terms of timedependent Semenov and critical Seminov numbers (ψ, ψc) prior to and after inhibitor injection.

3.2.1. Influence of injection timing Predicted temperature trajectories of inhibited free-radical polymerization over a range of injection times are shown in Figure 3(a) for the case of [X0] = [I0] = 0.055 mol/L, and θ = 0.01. When inhibitor is injected at tinj ≥ 10 min, thermal runaway is delayed for a period of time before occurring at a reduced magnitude. The earlier the injection is performed, the longer time thermal runaway is delayed and the lower the magnitude of the maximum temperature. The effect of injection timing is in a qualitative agreement with experimental studies of inhibition of thermal runaway of styrene polymerization by Rowe [21]. Time trajectories of Semenov and the critical Semenov number (ψ(t), ψc(t)) prior to and after inhibitor injection are presented in Figure 3(b) – 3(e). A sudden decrease of ψ at tinj is observed in all cases due to the sudden change of reaction rate by inhibitor injection, otherwise the trend of ψ trajectories is similar to that of temperature profiles. Under runaway conditions, a spike is observed in trajectories of ψc immediately before the temperature maximum as observed in Figure 1. For the case of sufficient quenching or effective inhibition, ψ keeps decreasing after injection until intersecting with ψc. 14


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3.2.2. Influence of inhibitor quantity Values of inhibitor quantity are selected from 50% to 120% of [I0], the initial molar concentration of initiator, as the amount of inhibitor is practically supposed to be small, and comparable to that of initiator of polymerization [21 - 23]. The temperature behavior of inhibition with different inhibitor quantity is shown in Figure 4(a) with tinj = 10 min and θ = 0.01. A larger amount of inhibitor leads to a more effective quenching of thermal runaway, which is in agreement with previous experimental studies by Russo et al. [22] and Ampelli et al. [23] of inhibition of MMA polymerization. Sufficient quenching of thermal runaway is observed as inhibitor quantity reaches 120% of initial initiator quantity. Figure 4(b) – 4(e) present trajectories of local Semenov ψ(t) and critical Semenov number ψc(t) during inhibition with different inhibitor quantity. In the cases of insufficient quenching as shown in Figure 4(b), 4(c) and 4(d), ψ(t) remains consistently above ψc(t), despite an instantaneous reduction of ψ at tinj. As before, in the case of sufficient quenching, ψ decreases after tinj until eventually falling to a value below ψc.

3.2.3. Influence of inhibitor strength As mentioned above, inhibitor strength is described by the inhibitor relative activity θ. The values of θ are selected from 0.01 to 0.04 for demonstrating loss of criticality at tinj = 12 min and [X0] = 100% [I0]. Temperature trajectories presented in Figure 5(a) indicate that a value of θ = 0.4 is sufficient to shortstop runaway.

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Corresponding behavior of Semenov ψ(t) and critical Semenov number ψc(t) during inhibition with different inhibitor strength is shown in Figure 5(b) – 4(e). As before, sufficient quenching corresponds to a decreasing ψ(t) upon inhibitor injection.

3.3 Development of inhibition criterion Based on the discussion above, a descending Semenov number ψ after injection resulting in an eventual intersection with ψc was consistently observed for cases of sufficient quenching. Thus, a mathematical criterion for effective inhibition, or sufficient quenching as to prevent runaway, is proposed in Eq. (33). This represents a less conservative criterion than the traditional criterion of ψ < ψc at tinj following Van Welsenaere Froment [7]. 𝑑𝜓

𝑑𝑡 |𝑡𝑖𝑛𝑗

(33)

Model-Based Design For Inhibition Of Thermal Runaway In Free

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