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Experimental Design of Topological Curves to Safely Optimize Highly Exothermic Complex Reacting Systems Sabrina Copelli, Marco Derudi, and Renato Rota* Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica “G. Natta” via Mancinelli 7, 20131 Milano, Italy
Angelo Lunghi and Christian Pasturenzi Stazione Sperimentale per i Combustibili viale A. De Gasperi 3, 20097 S. Donato M.se, Italy ABSTRACT: Strongly exothermic solution homopolymerizations are a class of chain reactions particularly difficult to be optimized from both a safety and a productivity viewpoint. Particularly, lots of side undesired reactions (e.g., backbiting, propagation of tertiary radicals, chain transfer to monomer or solvent, etc.), which affect the selectivity with respect to the desired product, and relevant mass and heat transfer problems, due to the increasing system viscosity, take place during such syntheses. Under these unavoidable operating conditions, it is difficult to employ theoretical procedures that are able to safely optimize the analyzed process, because the development of a reliable mathematical model is often not affordable or too time-consuming. In this work, it is shown that the topological criterion theory and its related optimization procedure can be used to optimize experimentally (through a dedicated set of isoperibolic reaction calorimetry tests) a complex reacting system even if its reaction scheme and all information about the kinetics are not available.
1. INTRODUCTION Polymerizations are known to be one of the most frequent causes of thermal runaway (that is, loss of the reactor temperature control occurring whenever the rate of heat removal through the cooling system becomes lower than the rate at which the heat is evolved by the synthesis reactions) in fine chemical and pharmaceutical industries. According to Barton and Nolan,1 about 30% of thermal runaway incidents can be ascribed to polymerizations (homopolymerizations and copolymerizations) and cross-link reactions so that such processes may be considered the most frequent causes of industrial registered runaways. Moreover, if bulk and solution polymerizations are considered, the increasing viscosity of the reacting mixture within the reaction progress and the high system exothermicity (up to 100 J/mol) constitute two relevant additional problems to be faced in order to safely optimize and scale-up a process involving these reaction typologies. In industrial plants, fast and strongly exothermic bulk or solution polymerizations are often carried out in semibatch reactors (SBRs), where the heat released is controlled by a reactant (usually called coreactant) feeding rate providing a low accumulation level. Obtaining such conditions is fundamental for the safe operation of a polymerization reactor. As the matter of fact, if a loss of temperature control would occur, dangerous boiling phenomena with consequent stable foam formation and undesired side reactions that affect both process safety and productivity may be triggered. Moreover, if cross-linking phenomena are triggered because of the high temperature values reached during the runaway, a carbonization of the whole reacting mixture may occur. Such a phenomenon is undesired because it implies irreparable loss of the entire reactor body. Therefore, the amount of coreactant accumulation and the maximum temperature above which undesirable phenomena r 2011 American Chemical Society
may be triggered (maximum allowable temperature, MAT) are two key pieces of information needed to perform a safe optimization of such semibatch (SB) processes. Unfortunately, all methods for the detection of the so-called “runaway boundary” (that is the system parameters in correspondence of which runaway phenomena are triggered) and the optimum set of process operating conditions need a detailed mathematical model of the analyzed synthesis to be predictive.29 However, when dealing with polymerizations, which involve a great number of reactions whose temperature dependent kinetics is very difficult to be estimated, the development of a reliable mathematical model able to correctly simulate the system thermal behavior in a wide range of operating conditions often is not affordable or is too time-consuming. In this work, the topological criterion and its related optimization procedure1012 have been extended to a process that is difficult to be mathematically modeled. In other words, it has been shown that, through a dedicated set of reaction calorimetry tests, the topological curve can be easily drawn (therefore clearly identifying both runaway and safe and productive operating regions) without any model simulation and, consequently, without knowing the kinetic of the system. The solution homopolymerization of butyl acrylate in ethyl acetate thermally initiated by 2,20 -azobis(2-methylpropionitrile), carried out in an indirectly cooled semibatch reactor using the isoperibolic control mode, has been used as a case-study.
2. TOPOLOGICAL CRITERION The topological criterion theory states that, for any semibatch process carried out in the isoperibolic control mode, the Received: January 4, 2011 Accepted: July 27, 2011 Revised: June 29, 2011 Published: July 28, 2011 9910
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Industrial & Engineering Chemistry Research boundary between runaway and “quick onset, fair conversion, smooth temperature profile” (QFS13) conditions with respect to a desired product (which are considered the most safe and productive for an isoperibolic SB process, if compatible with the system MAT constraint) is identified by an inversion of the topological curve showing a concavity toward right.11 This curve can be constructed by solving the mathematical model which describes the analyzed system varying the most relevant system operating parameter (in the polymerization case, the dosing time) and, then, reporting onto a reduced phase portrait all the obtained maximum reactor temperatures divided by the initial reactor temperature, τmax = Tmax/T0, and the conversion with respect to the desired product in correspondence of such a maximum, ζmax. If the complete reaction scheme and all information about the kinetics are not available, it is not possible to model the reacting system and, consequently, the topological curve cannot be predicted. However, in the presence of only one exothermic reaction, the topological curve may be generated also by measuring, through a set of isoperibolic tests carried out in an RC1 laboratory calorimeter, the maximum reactor temperature and the calorimetric conversion in correspondence of such a maximum, Z tmax _ rxn ðtÞ dt Q ζmax, cal ¼ 0 ^ rxn 3 mÞ ðΔH
3. CASE-STUDY As a case study, the free radical solution homopolymerization of butyl acrylate (BA, g99%, Sigma-Aldrich) in ethyl acetate (EtOAc, g99%, Sigma-Aldrich) has been investigated. This polymerization is thermally initiated by 2,20 -azobis (2-methylpropionitrile) (AIBN, g98%, Fluka) and it is performed in an isoperibolic semibatch reactor refrigerated by an external jacket. The following kinetic scheme shows the main reactions that may be involved in the synthesis.14 kd
1 initiation
I sf 2Rs1
2 propagation
Rsn + M sf Rsn+1
3 backbiting
Rsn sf Rtn
ksp
kbb
ktp
4 propagation from tertiary radicals Rtn + M sf Rsn+1 5 chain transfer to monomer
kstr, M
Rsn + M sf Rs1 +Pn kttr, M
Rtn + M sf Rs1 +Pn 6 termination by combination
kssr
Rsn + Rsm sf Pn+m kttt
Rtn + Rtm sf Pn+m kstt
Rsn + Rtm sf Pn+m Here, I represents the initiator, Rs is a secondary radical, Rt is a tertiary radical, M is the monomer and P is the polymer (with n, m = (1, ..., ∞)). As it can be seen, a great number of kinetic constants has to be estimated if a reliable simulation of the system thermal behavior is required. However, since the desired goal is to optimize a process that will be then scaled-up from laboratory to full plant scale, high purity degree reactants, essential to determine reliable kinetic constants values, cannot be employed during laboratory
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tests because such reactants will not normally be used for the industrial synthesis. This further complicates the theoretical study of the process at the various scales since impurities have a great influence onto the reactions paths so that kinetic parameters determined through laboratory-scale tests using high pure reactants cannot be safely assumed to simulate reliably the thermal behavior of an industrial reactor. Moreover, when high monomer/polymer concentrations are employed (and this will be the case), a correct modeling of the Trommsdorff effect (or gel effect)15 cannot be disregarded. Such a phenomenon, which often takes place during a free radical polymerization at intermediate or high degrees of conversion, consists of an autoacceleration of the polymerization rate, and it is due to diffusion limitations that slow down the termination reactions leaving the propagation and the initiation reactions unaffected. This effect is highly undesired in industrial application because it causes a fast and dramatic increase of the temperature of the reacting medium, often leading to instabilities, hot spots, and erratic reactor behavior. Actually, modeling the gel effect is difficult because only few and, often, unreliable correlations among conversion, viscosity increase, and global mass and heat transfer coefficients are available.16 Therefore, an experimental approach for optimizing the process is often preferred with respect to a theoretical one.
4. INSTRUMENTS AND TECHNIQUES Before starting whatever synthesis at the RC1 scale, it is useful to perform calorimetric screening tests (e.g., differential scanning and accelerating rate calorimetry tests) in order to characterize completely all thermal behaviors of reactants and products that will be involved into the process. After that, a set of isoperibolic semibatch RC1 tests must be planned to design the desired topological curve (which is parametrized with respect to the dosing time, as required for chain reactions11) able to detect the QFS conditions for the process here analyzed. 4.1. Differential Scanning Calorimetry (DSC). This calorimetric technique allows for substances thermo-chemical characterization by comparing the thermal behavior of a sample with that one of a reference. Particularly, the instrument is able to record the rate at which the sample develops or absorbs heat (dQ/dt) during a transformation and to generate characteristic diagrams that report heat power exchanged between sample and reference versus temperature (dQ/dt vs T) or time (dQ/dt vs t). Moreover, these diagrams show the number and the characteristics of all the thermal effects, the temperatures (or times) at which these effects take place and, finally, their importance. In this work two different types of DSC tests have been carried out: dynamic (sample holders, stainless steel; medium pressure; Viton/120 μL/closed/nitrogen atmosphere; sample heating rate, 5 °C/min; investigated temperature range, 30280 °C) and isothermal (sample holders, stainless steel; medium pressure; Viton/120 μL/closed/nitrogen atmosphere; sample investigated temperature, 70 °C) tests. 4.2. Accelerating Rate Calorimetry (ARC). ARC is an adiabatic calorimeter particularly suitable to study the thermal stability of homogeneous reacting systems. In this work it has been used in the “HEAT”-“WAIT”-“SEARCH” (HWS) mode by carrying out the following dynamic standard test: the sample is warmed up (HEAT) by a radiant heater at a desired temperature, then the instrument waits (WAIT) until all temperatures are stabilized, and, finally, it starts to search for exothermic effects (SEARCH), namely, a self-heating rate of reaction mass into the 9911
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Industrial & Engineering Chemistry Research sample larger than 0.02 °C/min. This research terminates when either a predetermined time is passed (15 min) or a sample selfheating rate that exceeds 0.02 °C/min is detected. If an exothermic effect is revealed, the instrument automatically collects temperature and pressure data as functions of time, shifting to adiabatic mode until the reaction ends (self-heating rate lower than the selected limit). If an exothermic effect is not revealed, a new sequence of HWS is started at a higher temperature. From a single HWS ARC test, it is possible to obtain several pieces of information, including initial and final temperature of any exothermic effect, sample self-heating rate at any temperature, adiabatic temperature increase, pressure at any temperature, and pressure increase rate. Results obtained are strictly dependent on sample holder thermal inertia, Φ. Consequently, experimental data have to be corrected to take into account this effect. Temperature and pressure operating ranges between the different test typologies may be conducted varying from 25 to 500 °C and from 1 to 170 bar, respectively. 4.3. Reaction Calorimetry (RC1). This calorimeter is a laboratory reactor of 1 L capacity, equipped with an external jacket to heat or cool the reaction mass, a thermocouple and a calibration probe, which is necessary in order to determine massspecific heat capacities (^cp) and global heat transfer coefficients (U). Particularly, in the RC1 evaluation software it is possible to take into accounts the variation of ^cp and U parameters by superimposing different constraints and ranging modes. For the case-study here considered, as the overall heat transfer coefficient varies from U0 to Uf (because of the reacting mixture viscosity increase), it has been superimposed in the RC1 software solution of the heat balance that the overall heat transfer coefficient varies linearly with the stirrer torque (To) collected by the RC1. Particularly, this last parameter is measured because it is used to calculate the appropriate power to be supplied to the stirrer so as to keep the stirring rate at a desired and constant value: the higher the stirrer torque is, the higher the power to be supplied is and, consequently, the higher the mixture viscosity is. For each test (UA)0 and (UA)f have been determined supplying a constant and known heat power for 10 min and, then, measuring the temperature difference between the reactor and the jacket that allows the removal of all the heat supplied to the system.
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Figure 1. ARC thermal characterization of the final reaction mixture: sample amount, 3.02 g; initial temperature, 30 °C. (A) Sample temperature and can pressure as a function of the time (HWS standard cycles); (B) sample self-heating rate (°C/min) and can pressure rate (bar/min) as functions of temperature (°C).
Table 1. Results Obtained by the Standard HWS ARC Test on the Analyzed Reacting Mixture
5. EXPERIMENTAL OPTIMIZATION 5.1. Reactants and Product Thermal Stability through DSC Tests. Dynamic DSC tests have been performed onto 2,20 -
azobis(2-methylpropionitrile) and polybutylacrylate, while both dynamic and isothermal DSC tests have been carried out onto butyl acrylate with and without AIBN. As it can be noticed, DSC tests in static air atmosphere have not been considered because oxygen often works as an inhibitor with respect to these polymerization reactions. For what concerns 2,20 -azobis(2-methylpropionitrile) (AIBN), the typical behavior of a thermally instable crystalline solid, which melts (effect starting at about 94 °C) and immediately after that decomposes (exothermic effect starting at about 104 °C), has been observed. Moreover, at 70 °C a small endothermic effect has been detected: such a phenomenon can be ascribed to a transition between two different AIBN crystalline phases. On the contrary, neither endothermic nor exothermic effects have been observed for polybutylacrylate (PBA) in the investigated temperature range.
sample amount (BA, EtOAc, AIBN), g
3.12 (1.14, 1.95, 0.03)
sample holder mass, g
13.91
sample holder volume, ml
8
filling degree, g/mL
0.39
Φ factor,
2.15
exothermic effect initial temperature, °C
65
initial pressure, bar
1.4
maximum self-heating rate (Φ corrected), °C/min adiabatic temperature increase (Φ corrected), °C
4.65 101
exothermic effect final temperature, °C
112
final pressure, bar
3.4
reaction enthalpy, J/g
187
Finally, for the dynamic DSC test on BA without AIBN, it can be observed that the substance is stable until 150 °C. Then an exothermic phenomenon ascribed to the self-initiated butyl acrylate polymerization starts. On the contrary, if maintained at 70 °C (isothermal test), BA is stable for over 300 min, so that no self-initiated polymerization is predicted to occur at this temperature. With the addition of AIBN to pure BA and the repeat 9912
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Figure 2. Scheme of the Experimental Optimization Procedure.
performance of the isothermal test at 70 °C, it can be observed that the homopolymerization starts immediately after all instrument thermocouples have reached equilibrium at the initial temperature. From this DSC test, the reaction heat is equal to 507 J/g; such a value is very close to the one reported in the literature.15 5.2. Reacting Mixture Thermal Stability Through an ARC Test. Reacting mixture stability has been analyzed through a standard HWS test. Average reaction mass-specific heat capacity (=1.85 kJ/(kg K)) has been preliminarily determined by calibrations in an RC1 equipment. Figure 1A shows temperature and pressure profiles measured in the ARC experiment performed by loading all reactants into the sample holder at the initial temperature of 30 °C: in correspondence of 65 °C an exothermic effect followed by a pressure increase (that, starting from about 77 °C, can not be ascribed to the temperature increase only) can be recognized (see Figure 1A and Figure 1B). This phenomenon extinguishes at 112 °C (as it can be observed from the selfheating rate and pressure rate reported in Figure 1B as a function
of temperature) and it has been ascribed to the thermally initiated polymerization of BA combined with the reacting mixture boiling (the occurrence of this phenomenon is evidenced by the lower reaction enthalpy detected from this test that is 187 J/g vs 504 J/g). Moreover, after this exothermic effect no decomposition event takes place until 300 °C, where the experiment has been terminated. According to these issues, the MAT parameter has been set equal to 77 °C (the temperature at which the pressure increase has been significantly detected). All the details concerning this test are reported in Table 1. 5.3. Reaction Optimization Using the Topological Criterion and an RC1 Equipment. To optimize experimentally the analyzed process, the following procedure summarized in Figure 2 has been used. (1) First of all, a standard recipe (that must be the same for all experimental tests) has to be decided. In this case study, ethyl acetate and AIBN are loaded into the reactor at ambient temperature and, then, the instrument is heated up to 70 °C in 20 min using an isothermal control mode. After this time, the 9913
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Table 2. Process Recipe, Reactor Cooling System, and Thermodynamic Parameters for the Free Radical Solution Homopolymerization of Butyl Acrylate in Ethyl Acetate Thermally Initiated by AIBN Recipe and Reactor Cooling System initial load and dosed stream cooling system 205 g
ethyl acetate
external jacket
2g
AIBN
silicon oil
120 g
butyl acrylate
(UA)0 = 2.234 (W/K) (UA)f = 1.266 (W/K)
Thermodynamic and Thermochemical Parameters ΔHrxn,p
64.6 kJ/mol
MAT
77 °C
control mode is shifted to isoperibolic (Tj = 70 °C) providing a waiting time of 15 min to allow for the reactor and jacket temperatures to equilibrate. Finally, butyl acrylate is dosed by means of a pump. Reactant amounts and reactor features are reported in Table 2. Then, starting from a minimum dosing time of 15 min, a sequence of isoperibolic RC1 tests at different dosing times must be carried out. Particularly, the topological curve can be drawn by collecting, for each experiment, the maximum reactor temperature and the calorimetric conversion in correspondence of such a maximum and reporting these couples of data onto a suitable reduced phase portrait (whose axes are the maximum reactor temperature with respect to the initial reactor temperature, τmax = Tmax/T0, and the reaction conversion in correspondence of such a maximum, ζmax). Once three experimental points (τmax,ζmax,cal) are available, QFS inversion can be sought. When a concavity of the experimental topological curve toward right is detected, this means that the QFS boundary has been found10 and, in order to obtain the desired QFS operating conditions and stop the iterative procedure, the following constraints must be checked: τmax , n 3 T0 < MAT
ð1Þ
ζdos, n g ζdos, min
ð2Þ
where n represents the last test carried out, MAT is the maximum allowable temperature for the considered system as measured from the ARC test (which has been found to be equal to 77 °C), and ζdos,MIN is the minimum calorimetric conversion required at the end of the dosing time (in this case it has been imposed as 0.78. High conversion degrees are difficult to obtain in solution homopolymerizations because of the difficulties in material and heat transfer efficiencies occurring when viscosity increases). If check 1 and 2 are fulfilled, the desired optimum dosing time has been found, if not another experiment must be carried out increasing the dosing time as required by the procedure summarized in Figure 2. (3) Once the optimum dosing time has been established, the time at which the maximum conversion is reached must be sought. Practically, such a time is considered to be that one corresponding to a calorimetric conversion variation of about 0.0001 min1. The real stop time will be lower than this value, and it will be determined by a cost/benefit analysis.
Experimental temperature and calorimetric conversion vs time profiles obtained from RC1 tests at increasing dosing times are shown in Figure 3. For each experiment, maximum reactor temperature and calorimetric conversion in correspondence of such a maximum and at the end of the dosing period have been collected in Table 3. Moreover, each run has been classified into one of the three different thermal behavior classes available for an isoperibolic SB reactor whose operating variable to be optimized is the dosing time: runaway (RW, that is before the conversion inversion shows a concavity toward right, sharp temperature profile characterized by high maximum temperature increase with respect to the initial reactor temperature and violent conversion variations either after or before the end of the dosing period), QFS (that is after the conversion inversion shows a concavity toward right, smooth temperature profile characterized by medium maximum temperature increase with respect to the initial reactor temperature and almost linear conversion variations during the dosing period) and STARVING (STV, that is before the conversion inversion shows a concavity toward left, squared temperature profile characterized by low maximum temperature increase with respect to the initial reactor temperature and a forced linear conversion during the dosing period). Such a classification is reported in Table 3. Particularly, in this specific case-study no starving runs have been carried out because experimental tests have been stopped after the optimum detection (that was obviously before the STV inversion). Finally, the experimental topological curve can be easily generated using the data reported in Table 3, as shown in Figure 4. It can be noticed that the QFS boundary is detected for a dosing time equal to 55 min (run 7). In correspondence of this point, a local minimum (since it is referred to the investigated dosing time range) of the calorimetric conversion has been observed. On the contrary, the first run able to satisfy all constraints provided by the experimental optimization procedure is that one with a dosing time equal to 65 min (run 8). This experiment can be considered optimized from both the safety (Tmax = 75.4 °C is lower than the MAT = 77 °C) and productivity points of view (ζdos = 0.78 is equal to the minimum desired conversion at the end of the dosing period, ζdos,MIN). (4) Once the optimum dosing time has been detected at the laboratory scale, a simple and reliable correlation must be provided in order to scale this time to the industrial plant. Such a correlation can arise, for instance, from an order-of-magnitude procedure which requires that the cooling numbers, Co, at RC1 and industrial scales being equal.8 These factors represent the ratios between the characteristic time at which the heat evolved by the exothermic reactions is accumulated into the system and the characteristic time at which this heat is removed by the cooling system. Referring to the laboratory and industrial scale, they are roughly related to each other as follows: URC1 3 tdos, RC1 ¼ Coind CoRC1 ¼ const 3 DRC1 Uind 3 tdos, ind ¼ const 3 ð3Þ Dind tdos, ind Dind URC1 ¼ tdos, RC1 DRC1 3 Uind
ð4Þ
if CoRC1 = Coind, the ratio of industrial to laboratory dosing time is roughly proportional to the product of two factors: The first 9914
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Figure 3. Experimental temperature (A) and calorimetric conversion (B) vs time profiles for the free radical solution homopolymerization of butyl acrylate in ethyl acetate thermally initiated by AIBN (see Table 2 and Table 3). The initial reactor temperature decrease is due to the dosing of cold monomer (at about 15 °C).
Table 3. Experimental Results Obtained for the Free Radical Solution Homopolymerization of Butyl Acrylate in Ethyl Acetate Thermally Initiated by AIBN run ()
tdos (min)
Tmax,exp (°C)
ζmax,exp ()
ζdos,exp ()
tstop (min)
ζstop ()
1 2
15 20
85.07 82.82
0.89 0.85
0.48 0.88
54 37
0.98 0.97
RW RW
3
25
82.55
0.73
0.85
92
0.97
RW
4
30
81.95
0.64
0.76
141
0.98
RW
5
35
81.80
0.57
0.76
161
0.98
RW
6
45
81.34
0.50
0.77
188
0.98
RW
7
55
76.70
0.42
0.77
221
0.98
RW/QFS
8
65
75.39
0.80
0.78
140
0.97
QFS
one (which is the ratio of industrial to laboratory reaction vessel characteristic dimensions) is always much higher than 1 and, mostly, implies a dosing time increase when moving from laboratory to industrial scale. The second one (which is the ratio of laboratory to industrial effective overall heat transfer coefficients) decreases as industrial reactor cooling efficiency increases: such a
experimental classification
behavior is logical, since a high reactor cooling efficiency allows for a lower dosing time. This factor can be obtained through either simple measurements, in both RC1 and real-size reactors, or estimated using a standard correlation17 such as (for propeller stirrers) URC1/Uind = (LRC1/Lind)1/3(NRC1/Nind), with L and N being the blade width and the stirring speed, respectively. 9915
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t = time, s or min U = overall heat transfer coefficient, W/(m2 K) Subscripts and Superscripts
Figure 4. Experimental design of the topological curve for the free radical solution homopolymerization of butyl acrylate in ethyl acetate thermally initiated by AIBN. T0 = 70 °C, tdos ∈ (1565) min and recipe as in Table 2: (O) runaway, (b) QFS, (—) runaway region, (---) QFS region.
Therefore, once the characteristic dimensions and the cooling system of the industrial reactor are known the optimum scaled dosing time can be obtained from the laboratory one by the means of eq 4.
6. CONCLUSIONS In this work it has been shown that the topological criterion theory is useful to optimize experimentally a complex reacting system with only one exothermic reaction even if the complete kinetic scheme is not available. In this case the topological curve can be drawn by measuring the maximum reactor temperature and the calorimetric conversion in correspondence of such a maximum. These pieces of information can be easily obtained by a set of isoperibolic RC1 tests carried out at different dosing times. Moreover, it has been confirmed that the QFS inversion represents a thermal behavior boundary between runaway and QFS conditions which can be detected experimentally. On the basis of such a result a suitable experimental optimization procedure has been proposed using the free radical solution homopolymerization of butyl acrylate in ethyl acetate thermally initiated by AIBN as a case-study. ’ AUTHOR INFORMATION Corresponding Author
*Fax. +39 0223993180. E-mail:
[email protected].
’ NOMENCLATURE A = heat transfer surface, m2 Co = cooling number, const(U 3 tdos/D), ^cp = mass-specific heat capacity, kJ/(kg K) D = reactor diameter, m k = kinetic constant, s1 or m3 s/kmol L = blade length, m m = number of monomer units in a radical chain, mass, g or kg MAT = maximum allowable temperature, °C N = stirring speed, min1 n = run index number of monomer units in a radical chain, Q_ rxn = heat power released by the exothermic reaction, W QFS = quick onset, fair conversion, smooth temperature profile operating conditions RW = runaway operating conditions STV = starving conditions T = reactor temperature, °C or K
cal = calorimetric conversion dos = dosing period holder = sample holder in an ARC test ind = industrial reactor j = jacket max = maximum value of a quantity min = minimum value of a quantity mix = reacting mixture n = run index RC1 = laboratory reactor 0 = start of the dosing period Greek Symbols
^ rxn = reaction enthalpy, cal/g or J/g ΔH ζ = conversion with respect to the desired product τ̅ = T/Tcool, dimensionless temperature with respect to coolant temperature, Φ = 1 + (mholder 3 ^cp,holder)/(mmix 3 ^cp,mix), inertia factor,
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