Thermal decomposition of hexanitrostilbene at low temperatures

Thermal decomposition of hexanitrostilbene at low temperatures

Journal of Analytical and Applied Pyrolysis 58–59 (2001) 569– 587 www.elsevier.com/locate/jaap Thermal decomposition of hexanitrostilbene at low temp...

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Journal of Analytical and Applied Pyrolysis 58–59 (2001) 569– 587 www.elsevier.com/locate/jaap

Thermal decomposition of hexanitrostilbene at low temperatures Th. Rieckmann a,*, S. Vo¨lker b, L. Lichtblau a, R. Schirra c a

Uni6ersity of Applied Sciences Cologne, Department of Chemical Engineering and Plant Design, Betzdorfer Str. 2, D-50679 Cologne, Germany b 42 Engineering, 6on-Behring-Str. 9, D-34260 Kaufungen, Germany c Dynamit Nobel GmbH, Kaiserstraße 1, D-53839 Troisdorf, Germany Accepted 11 October 2000

Abstract The thermal decomposition of hexanitrostilbene (HNS), a well known heat resistant explosive, has been investigated by simultaneous TGA/DTA at heating rates between 0.05 and 40°C min − 1. Depending on the temperature/time history, the reaction takes place either in the solid phase or in the liquid phase after melting of the sample. In order to observe the solid phase reaction, experiments with constant heating rates well below 2.5°C min − 1 have to be performed. Therefore, it is impossible to judge the thermal stability of solid HNS using kinetic models derived from DSC experiments at heating rates of 10 – 20°C min − 1, as is the standard procedure. In this work, a formal kinetic model has been developed for the thermal decomposition of high bulk density HNS in its solid phase. The model consists of three consecutive reaction steps (1) a three dimensional phase boundary reaction, dominantly a sublimation, (2) an autocatalytic decomposition reaction, and (3) a slow reaction of fractal order, supposedly a high temperature pyrolysis of primary solid products. The model was used to simulate the stability of HNS under isothermal conditions at temperatures below 300°C. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Explosives; Hexanitrostilbene (HNS); Pyrolysis; Thermal stability; Formal reaction kinetics; Modelling; TGA/DTA; Multivariate regression

* Corresponding author. Tel.: + 49-221-82752212; fax: + 49-221-82752202. E-mail address: [email protected] (T. Rieckmann). 0165-2370/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 3 7 0 ( 0 0 ) 0 0 1 7 7 - 7

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Nomenclature a, b, c, d, e, p concentration of reactants, intermediates and products cp heat capacity, kJ kg − 1 K − 1 D diameter, mm Ea activation energy, kJ mol − 1 ZsubH heat of sublimation, kJ mol − 1 −1 coefficient of Arrhenius equation, log10 log(k0/s ) m sample mass, mg n reaction order r reaction rate, kg m − 3 s R ideal gas constant, 8.314 J mol − 1 K − 1 t time, s T temperature, °C h conversion heating rate, °C min − 1 i u heat conductivity, W m − 1 K − 1 z density, kg m − 3

Fig. 1. Hexanitrostilbene, HNS.

1. Introduction During crude oil production, certain explosives are used for perforation of source rocks. As the average depths of the wells and, as a consequence the ambient

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temperatures in the wells are increasing, heat resistant explosives are of growing interest for the petroleum industry. Modern explosives have to stand temperatures up to 260°C for several hours without noticeable thermal decomposition. Hexanitrostilbene (HNS, Fig. 1) is a common example of this group of heat resistant explosives [1,2]. HNS is a nitro-aromatic compound with very low vapour pressure [3], high vacuum stability and a melting temperature documented between 316 and 321°C [2,4]. The thermal stability of HNS in the temperature range between its melting temperature and approximately 70°C below is of primary interest for technical applications. Usually, the thermal stability of heat resistant explosives is investigated by differential scanning calorimetry (DSC) under non-isothermal conditions. Standard experiments are performed at constant heating rates of 10, 15, or 20°C min − 1. Under these conditions, HNS is showing a strong exothermic peak at temperatures between 300 and 350°C, depending upon the initial mass, the particle size, the composition and the modification of the sample, as well as the heating rate. The decomposition is described as a single reaction with first order kinetics and a linear or non-linear regression technique is applied to calculate the kinetic parameters with separate evaluations for every experimental heating rate [5]. The obtained kinetic parameters are then used to predict the rate of thermal decomposition at lower temperatures, e.g. between 250 and 300°C. Our own initial tests on the stability of HNS with a combined TGA/DTA (STA) system indicated three prominent findings (1) at standard heating rates of 10, 15 and 20°C min − 1, HNS is melting before its decomposition starts, (2) melting and decomposition are separated and separable events, and (3) it is impossible to model the thermal decomposition by a single reaction of first order, which was shown by a Friedman analysis [10] and a formal kinetic analysis by multivariate regression [8]. The obvious problem associated with describing the decomposition of a solid at temperatures below its melting point using kinetic data obtained from molten samples stimulated us to look for conditions under which the decomposition of crystalline samples could be observed.

2. Methods Experiments have been performed with a STA (simultaneous thermal analysis) and a TGA/MS system. The multivariate non-linear regression technique has been applied for calculating kinetic parameters. The HNS type HNS HBD (high bulk density) used in this investigation was prepared by Dynamit Nobel GmbH, Troisdorf, Germany.

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2.1. Thermogra6imetric analysis The STA apparatus is a STA 503 from Ba¨hr GmbH, Germany. The maximum load of the STA is 1000 mg, and mass variations of 9200 mg can be detected. The maximum resolution is 1 mg. The STA apparatus has a horizontal weighing beam that provides an unusually low drag in the weighing direction, and the effect of drag was further reduced by using helium as a low viscosity purge gas, thus avoiding significant corrections of the baseline. A temperature calibration has been performed for all heating rates by analyzing the DTA signal from the melting peak of the pure substances indium (Tm =156.6°C), KClO4 (Tm = 299.4°C) and Ag2SO4 (Tm =426.4°C). The experiments have been performed with HNS HBD (high bulk density), which consists of spherical particles with an average diameter of 0.5 –1.5 mm (Fig. 8). The experiments entailed variation of the heating rate i and the initial sample mass mo. Initial sample masses of 1, 2, 5, 10 and 20 mg have been investigated in open crucibles made from Al2O3. High purity helium (Linde, Germany, purity 99.999%) was used as purge gas with a gas flow rate of 1.7 l h − 1. The absence of mass transfer limitations on the gas side of the liquid –gas (solid –gas) interphase was demonstrated by different gas flow rates. Two sets of constant heating rates have been applied (1) standard high heating rates of 2.5, 5, 10, 15, 20 and 40°C min − 1, as well as (2) very low heating rates of 0.05; 0.2; and 0.4°C min − 1. The linearity of the T(t) curves was excellent for all heating rates. In addition, gaseous reaction products have been analyzed for one HNS HBD sample (m0 =7 mg, i =3°C min − 1) with a TGA/MS system (Netzsch TGA with Balzers Thermostar MS) at the Max-Planck-Institut fu¨r Kohlenforschung in Mu¨hlheim/Ruhr, Germany.

2.2. Modelling of the sample temperature distribution Derivation of kinetic models from TGA analyses is based on the presumption of a homogeneous temperature throughout the sample. In order to check this assumption, the spatial temperature distribution in the sample has been modelled. The temperature difference between the thermocouple of the TGA and the different spatial layers of the sample mainly depend on the heating rate and the initial sample mass as well as on the heat of reaction, the thermal conductivity and the heat capacity of the sample. To approximate this difference, a simplified heat transport reaction model was applied. The crucible contents of the TGA were modelled as a dynamic system with distributed parameters assuming plate geometry (as ‘worst case’). The overall rate of reaction is approximated by an irreversible reaction of 1st order: HNS “ solids+ volatiles, assuming the validity of the Arrhenius expression for the temperature dependency of the rate constant.



r(z, T) = k0 exp −



EA z RT

(1)

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Table 1 Model parameters for calculating the temperature gradients into the sample during thermogravimetric analysis of HNS HBD with constant heating rate log(k0/s−1) Ea ZsubH cp u z0 D

20.2 267 kJ mol−1 180 kJ mol−1 2.5 kJ kg−1 K−1 0.24 W m−1 K−1 1740 kg m−3 1 mm

The kinetic parameters were estimated in this work and are summarized in Table 1. The material balance and the enthalpy balance result in a set of two meshed partial differential equations (PDE). Material balance (z = −r(z, T) (t

(2)

Enthalpy balance z0cp

(T ( 2T = u 2 =( −DRH)r(z, T) (t (t

(3)

Initial conditions T(t= 0) =T0

(4)

z(t = 0)= z0

(5)

Boundary conditions (T (x

)

=0

(left boundary)

(6)

x=0

T x = 0.5D =T(t) =it + T0

(right boundary)

(7)

The PDE system was transposed into a set of 2n ordinary differential equations (ODE) by the finite differences technique (calculated with n=128 finite differences). ( 2T (Tj − 1 −2Tj +Tj + 1) : (x 2 Dx 2

(8)

for j = 2, n − 1 finite element. At the left boundary ( j= 1) ( 2T ( −Tj +Tj + 1) : (x 2 Dx 2 and at the right boundary ( j= n)

(9)

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( 2T (Tj − 1 −2Tj +T x = 0.5D ) : (10) (x 2 Dx 2 The simultaneous numerical solution of the ODE system was performed using Matlab (The MathWorks, Inc.).

2.3. E6aluation of model parameters and numerical computation procedure Formal kinetic models have been calculated to describe the total mass loss of the STA experiments at low heating rates. The material balance of the crucible contents is described as a dynamic system with concentrated parameters, which results in a single respectively in a set of ordinary differential equations (ODE). The simultaneous numerical solution of the ODE system and the estimation of the kinetic parameters by the least squares (LSQ) method were performed using the software package THERMOKINETICS (Netzsch GmbH, Germany). NETZSCH THERMOKINETICS is a software module for the evaluation of thermokinetic experiments. The differential equations are calculated using a 5th-degree Runge-Kutta embedding method with automatic optimization of the interpolation nodes for numerical solution of the reaction equations (Prince –Dormand method). To minimize the deviance LSQ, an improved version of the hybrid procedure described in [6] is used. This is a combination of a derivation-free, regularized Gauss –Newton procedure (Marquardt procedure) with a step-length optimization [7]. The kinetic parameters have been fitted by simultaneously running the optimization routine and the ODE solver according to the technique of multivariate regression [8]. The task of non-linear regression is the iterative calculation of the minimum sum of least squares. In case the multivariate regression technique is applied, the total LSQ yields into a sum of LSQ calculated with data from the different experimental runs m

n

LSQ = % % (yexp k, i −ymod k, i )2.

(11)

k=1 i=1

The experimental values yexp k, i are the normalized mass, y= m(T)/m0(t=0) from STA. Data for all low heating rates (i=0.05, 0.2 and 0.4°C min − 1) have been used simultaneously. The model values ymod k, i result from the numerical solution of the respective ODEs. The multivariate analysis bases on the assumption that the kinetic parameters have to be identical for all experimental conditions in the modelled parameter range. This constraint facilitates the choice of an appropriate model considerably and improves the model validity. The kinetic analyses of this work is based on formal kinetic models. Those models include one, two, or multi-step processes, in which the individual steps are linked as parallel, competitive or consecutive reactions. The intermediate and final products are to be viewed as pseudo components and may comprise a whole variety of chemical species. Arrhenius equations were used for the temperature dependent rate constants. The following irreversible reaction types and corresponding reaction equations [9] turned out to be applicable for the kinetic description of the decomposition of HNS HBD (Table 2).

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Table 2 Code and reaction types used for the kinetic description Code

f(e, p)

Reaction type

F0 F1 Fn R3 Cn−x

1 e en 3e 2/3 e n (1+kcatx)

Reaction of zero order Reaction of first order Reaction of nth order 3D phase boundary reaction Autocatalytic reaction of nth order with catalysis by x

de/dt = −k0 exp( − Ea/RT) f(e, p)

(12)

3. Results

3.1. Sample temperature distribution The modelling results of the temperature distribution in HNS HBD samples during TGA runs according to our heat transport reaction model are displayed in Fig. 2. At a constant heating rate of 40°C min − 1, the temperature of a HNS HBD

Fig. 2. Calculated temperature distribution for the different spatial layers in an HNS HBD sample of m0 =5 mg at i = 40°C min − 1 according to the heat transport reaction model.

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Fig. 3. Influence of initial sample mass on the liquid phase decomposition of HNS, i= 10°C min − 1, m0 =1, 2, 5, 10 and 20 mg

sample of 5 mg closely follows the temperature of the carrier gas with a maximum temperature difference between the spatial layers of the HNS HBD sample and the carrier gas of less than 1.5°C. At a heating rate of 10°C min − 1 the maximum temperature difference is less than 0.4°C. The results indicate that the assumption of a homogenous temperature distribution can be applied without significant error.

3.2. Influence of initial sample mass The influence of the initial sample mass on decomposition has been investigated for HNS HBD by STA. The solid phase reaction did not depend upon initial sample mass as long as the sample consisted of uniformly dimensioned spheres, which were spread in the crucible without covering each other. Conversely, the results of the melt phase reaction were altered with varying initial sample mass as displayed in Fig. 3 for a heating rate of 10°C min − 1. Initial sample masses of 1, 2, 5, 10 and 20 mg have been investigated and the presence of heat and mass transfer could be detected. With increasing sample mass, the volatilization of reaction products is delayed due to mass transfer limitations and the final mass is possibly increased by secondary reactions. With a sample mass of 20 mg, the heat release of the exothermic reaction exceeds the maximum heat flow rate out of the reaction zone and the sample deflagrates.

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3.3. Thermogra6imetry-mass spectrometry The TGA-MS experiment has been conducted with a HNS HBD sample at a heating rate of 3°C min − 1. CO and NO were the main reaction products with by-products being CO2 and H2O. NO2 and H2 could not be detected. The NO production coincided with the exothermic decomposition, while the CO2, CO and especially the H2O trace exhibited long tails to higher temperatures. This supports the interpretation of the last reaction step being a carbonization of the primary solid product.

3.4. Simultaneous thermal analysis STA experiments have been performed with HNS HBD. The results of thermogravimetric experiments with heating rates of 0.05, 0.2, 0.4, 2.5, 5, 10, 15, 20 and 40°C min − 1 for a constant sample mass of 5 mg are shown in Fig. 4. The figure displays a distinctive separation between experiments with low (0.05; 0.2; 0.4°C min − 1) and high (2.5, 5, 10, 15, 20, 40°C min − 1) heating rates regarding the curve form as well as the attained final mass. This behaviour becomes clear by looking at the STA results as they are given in Figs. 5–7 for heating rates of 40, 2.5 and 0.4°C min − 1. At a heating rate of 40°C min − 1, the melting peak is well

Fig. 4. Thermogravimetric analysis of HNS HBD, i= 0.05, 0.2, 0.4, 2.5, 5, 10, 15, 20 and 40°C min − 1.

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Fig. 5. STA results for HNS HBD, i= 40°C min − 1, ( from TGA, from DTA).

separated from the reaction peak and the sample melts completely before significant mass loss occurs. By lowering the heating rate, the reaction peak approaches the melting peak. At a heating rate of 2.5°C min − 1, the melting peak is still observable and the vast majority of the sample reacts in the molten state. At a heating rate of 0.4°C min − 1 (or less), the melting peak disappears and the sample reacts completely in the solid state before the melting temperature is reached. Inspection of the crucible contents after the experiments confirmed the fact that the reactions took place in solid respectively liquid phase. The residues from samples exposed to low heating rates retained the initial spherical shape and diameter of the virgin material, whereas the residues from samples decomposed at high heating rates formed an even layer at the bottom of the crucible. In Figs. 8 and 9, SEM pictures of virgin HNS HBD and of HNS HBD after decomposition at a linear heating rate of 0.4°C min − 1 are shown. Throughout the solid phase reaction, the spherical HNS HBD particles keep their shape and diameter. The porosity increases whilst the bulk density decreases. All experiments seem to entail multi reaction processes. The curves for low heating rates are characterized by a slow mass loss at low temperatures that is gradually accelerated with rising temperature as a fast exothermic reaction overtakes control. This is followed by a slow reaction over a broad temperature range, most probably a secondary pyrolysis of solid reaction products. The curves for high heating rates show almost no mass loss at temperatures below the melting point.

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The fast decomposition of the molten samples takes place via at least two reactions with different heat release and is also followed by a slow reaction, comparable to the last reaction of the solid state process.

3.5. Kinetic models For the prediction of thermal stability of HNS at low temperatures, a formal kinetic model has been derived for the decomposition in solid phase. For this model, experiments with i =0.05, 0.2 and 0.4°C min − 1 have been considered. The model describes the total mass loss of the samples and was calculated using the thermogravimetric results from STA. The advantages of using the mass signal for kinetic modelling compared to the heat flow rate signal lie in its higher validity and resolution and its accessibility over a broad range of experimental conditions. Furthermore, the mass signal is directly proportional to the extent of reaction, even if the process contains endothermic and exothermic steps simultaneously. To minimize the effect of heat and mass transfer on the developed kinetic models, results from experiments with an initial sample mass of 5 mg have been used for the calculations, which was the best compromise regarding base line stability during long time experiments and resolution of the STA. The model free Friedman analysis [10] was used to gain first insights into the reaction process. The Friedman analysis bases upon a supposed first order reaction

Fig. 6. STA results for HNS HBD, i =2.5°C min − 1, ( from TGA, from DTA).

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Fig. 7. STA results for HNS HBD, i =0.4°C min − 1, ( from TGA, from DTA).

Fig. 8. SEM pictures of virgin HNS HBD sample.

and yields the apparent kinetic parameters for successive conversion stages. A Friedman analysis for the solid phase reaction yielded in strong variations of the apparent activation energy with the extent of reaction (Fig. 10) and confirmed that the data sets can not be fully described by a single first order reaction. Additionally, references to the presence of an autocatalytic reaction step were obtained [11].

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To check, whether the simple one-step models with first order kinetics (F1) da = −k01 exp( − EA 1/RT)a dt

(13)

respectively zero order kinetics (F0)

Fig. 9. SEM pictures of HNS HBD sample after pyrolysis at i = 0.4°C min − 1 in helium, maximum temperature: 450°C.

Fig. 10. Friedman analysis — energy plot, i=0.05, 0.2 and 0.4°C min − 1.

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Table 3 Kinetic data for the decomposition of solid phase HNS HBD 1

Log(k0.1/s−1) Ea,1 (kJ mol−1) Log(k0.2/s−1) Ea,2 (kJ mol−1) n2 Log(kcat/s−1) Log(k0.3/s−1) Ea, 3 (kJ mol−1) n3 Mass loss by step 1 Mass loss by step 2 Mean of residues

2

3

1

A“B“C“D

A “ B, F0

15.5 217 17.8 234 0.914 0.0283 64.0 712 10.1 0.119 0.325 0.356

18.6 251

1.089

da = −k01 exp( − EA 1/RT) dt

1

A “ B, F1 20.2 267

1.038

(14)

for solid phase decomposition reactions are nevertheless sufficiently reliable for technical applications, best fits have been evaluated. The resulting kinetic parameters are listed in Table 3. The inapplicability of F1 and F0 models are demonstrated in Figs. 11 and 12, which display the respective best fit for low heating rates. The F1 model can not predict correctly the technically important onset of the reaction with varying heating rate. The F0 yields better results regarding the onset of reaction. Nevertheless, the experiments show an accelerated decomposition rate with conversion that cannot be described by a F0 mechanism. Both models completely fail in the description of the high temperature stages of the experiments. Consequently, a multi step reaction model has been developed under consideration of the qualitative facts described above and using a minimized number of kinetic parameters. The solid phase reaction could best be modelled by three consecutive reactions A “ B “ C “ D of the type R3, Cn and Fn (Table 1). The results are shown in Fig. 13, for kinetic parameters see Table 3. da = −k01 exp( − EA 1/RT)3a 2/3 dt db = k01 exp( − EA 1/RT)3a 2/3 −k02 exp(− EA 2/RT)b n(1+kcatc) dt dc = k02 exp( − EA2/RT)b n(1 + kcatc)− k03 exp(− EA 3/RT)c n dt

(15)

The first reaction step R3, a three dimensional phase boundary reaction, has an activation energy of 217 kJ mol − 1. It can be interpreted as being dominated by a mass transfer limited sublimation of the virgin solid. The calculated activation energy of this reaction step is in the order of the sublimation enthalpy of the

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sublimation of HNS, which was measured to be 180 kJ mol − 1 between 160 and 205°C [3]. The second reaction step Cn was modelled as an autocatalytic reaction with autocatalysis by the intermediate pseudo component C. The calculated activation energy is 234 kJ mol − 1 and the reaction order is close to one. Autocatalysis during decomposition of nitro compounds is known to be caused by NOx [12,13]. Whether the active species are NO, a main product of this reaction step as detected by mass spectrometry, or highly activated NO2, which dissociates to NO and O, can not be determined by this investigation. The third reaction step Fn, supposedly a secondary pyrolysis of the complex primary product, has a very high formal activation energy of 712 kJ mol − 1 and a reaction order of ten. This reaction type is a possibility to include a great number of individual reactions with distributed activation energies into one formal reaction [14]. It should be noticed that an excellent correspondence exists between the activation energies of the consecutive reactions of the derived formal kinetic model and the conversion dependent activation energies from the Friedman analysis. The kinetics of the thermal decomposition of HNS have also been described by Kony et al. [5]. The authors performed isothermal DSC runs and determined the kinetic parameters from a linear regression of the classical Arrhenius plot ln k=

Fig. 11. Best fit model of F1 type, i= 0.05, 0.2 and 0.4°C min − 1 symbols: experimental data, lines: model prediction.

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Fig. 12. Best-fit model of F0 type, i= 0.05, 0.2 and 0.4°C min − 1, symbols: experimental data, lines: model prediction.

f(1/T). In the liquid state for a conversion hB 0.25, a first order reaction was assumed with the kinetic parameters log(k0/s − 1)= 13.9 and Ea = 170 kJ mol − 1. The conversion range 0.25Bh B 0.75 was modelled by a Prout –Tompkins reaction with the kinetic parameters log(k0/s − 1) =14.0 and Ea = 164 kJ mol − 1. The evaluation of the reactions in solid state yielded no reasonable results with kinetic parameters being extremely high. Furthermore, the kinetic analysis of this reaction using DSC measurements may be additionally complicated by the concurrent presence of endothermic and exothermic phenomena. Zeman [15] cited the kinetic parameters Ea = 183.8 kJ mol − 1, log(k0/s − 1)= 12.0 for the solid phase decomposition of HNS. These first order kinetics did not match our experimental data. The onset of the decomposition was predicted for much higher temperatures and the variation of the decomposition curves with different heating rates were not estimated correctly. This emphasizes the value of the multivariate analysis for the evaluation of kinetic data and the choice of an appropriate reaction model.

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3.6. Thermal stability of HNS HBD at low temperatures The global kinetic model for the solid phase decomposition from this work thus should allow a much more reliable prediction of the thermal stability of HNS HBD at low temperatures. The reaction of HNS HBD under isothermal conditions at a temperature of 260°C has been simulated with our model. In Fig. 14, the resulting concentrations of intermediate and final pseudo components for a total reaction time of 200 h are shown. The conversion is a very slow process under these conditions. It takes around 15 h before the sublimation is accompanied by a noticeable decomposition of HNS HBD, and more than 30 h until half of the initial sample mass has reacted. For the prediction of time dependent stability of HNS HBD at temperatures relevant to crude oil production, isothermal simulations of the conversion at temperatures between 260 and 295°C have been performed. The results are shown in Fig. 15 for a total exposition time of 12 h; 280°C are needed to observe a

Fig. 13. Multi step reaction model for the solid phase decomposition of HNS HBD, i= 0.05, 0.2 and 0.4°C min − 1.

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Fig. 14. Concentration of pseudo components for the isothermal reaction of HNS HBD at 260°C for the 1 2 3 multi step reaction model A “ B “ C “ D (R3– Cn– Fn).

significant conversion, starting after around 10 h. With increasing temperature, the onset of conversion is shifted to 2 h for 295°C.

4. Conclusions To predict the thermal decomposition of HNS HBD successfully, care must be taken if the reaction will take place in the crystalline or molten state. In this work, a formal kinetic model for the solid phase reaction of HNS HBD has been developed that describes the behaviour of HNS HBD in the investigated parameter range adequately. The model comprises multiple steps to simulate physical and chemical processes. STA measurements are less restricted compared to DSC measurements and turned out to provide a much better basis for this kind of investigation. The non-linear multivariate regression is an essential method to derive kinetic models. This technique is the only way to decide between different reaction models and obtain a global model giving reliable results for the whole parameter range.

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.

1

2

3

Fig. 15. Conversion under isothermal conditions for the multi step reaction model A “ B “ C “ D (R3– Cn – Fn).

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