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AKTS Thermal Safety Software

AKTS-Thermal Safety Software enables the calculation of the Time to Maximum Rate under adiabatic conditions (TMRad). Finite Element Analysis (FEA) extends the

application of calculating methods to analyze the thermal behavior under non-adiabatic conditions. FEA enables the determination of the impact that substance properties and

container properties can have on the reaction progress. This analysis can then be used to determine critical design parameters such as the critical radius for a container, the

necessary thickness of insulation, and the influence of the surrounding temperature on storage and transport safety. The method enables the prediction of the heat accumulation

process and the reaction progress for any surrounding temperature profile (isothermal, stepwise, periodic temperature variations, temperature shock and real

atmospheric temperature profiles). Key applications for AKTS method are found in the chemical, pharmaceutical and food industries, for self-reactive chemicals, explosives and thermal

hazards for dangerous goods. Analysis and specific safety concepts produced for customers by ??AKTS Thermal Safety Software?? are optimized forcost-effectiveness and apply state-of-the-art technology.

Introduction
 
In the early nineties of the last century several rocket motors of a ground to air missile started burning after a fire in a rock cavern. Due to this heavy accident the Swiss General Staff were interested to understand the time of ignition for stored rocket motors after a fire in a storage facility. One example of a rocket motor is shown in figure 1. The motor contains two propellants: Boost propellant and sustain propellant (glycerol nitrate, cellulose nitrate, stabilizers and other components).


Fig. 1 Left: Rocket motor for a ground to air missile. Right: Rocket propellant. Boost propellant (dark), sustain propellant (hell).
 
Generally, all energetic materials liberate heat during decomposition. Processing, design, quality control, and operational applications all require an evaluation of thermal hazards and an ability to predict the safety limits and the course of the decomposition process in extended temperature ranges [1-5]. The present paper is designed to answer the following issues:
 
? What is the reaction progress of the propellants under any temperature profile?
? Does a thermal hazard exist?
? At what temperature does the thermal hazard begin?
? What is the Time to Maximum Rate under adiabatic conditions (TMRad) at any temperature?
? What is the temperature of maximum self-heating?
? How much influence does isolation/heat transfer have on the heat accumulation conditions?
? What are the critical storage container sizes or transport temperatures?
? What influence do the surrounding temperature profiles have on the reaction progress and on the heat accumulation conditions?
 
Experimental and analysis process
 
Collection of experimental data and baseline determination
 
The analysis process requires determination of the kinetic characteristics of the reaction. The kinetic parameters can be extracted in principle from experimental data gathered by any of the following dynamic thermo-analytical methods: DSC (Differential Scanning Calorimetry), DTA (Differential Thermal Analysis), TG (Thermogravimetry) or EGA (Evolved Gas Analysis TG-MS or TG-FTIR). The application of the thermoanalytical techniques is widely recognized for the characterization of the degradation of solids [6].


Fig. 2 DSC heat flow curve and the baseline subtracted heat flow curve of the sustain propellant at 0.25 ?C/min under isochoric conditions (closed crucibles).
 
The correct kinetic analysis of a decomposition reaction has at least three major stages: (1) collection of experimental data; (2) computation of kinetic parameters, and (3) prediction of the reaction progress for required temperature profiles applying determined kinetic parameters. Kinetic evaluation of the reactions should be carried out with thermoanalytical data obtained at several heating rates (non-isothermal) or temperatures (isothermal) to ensure reliable results. Kinetic methods that use single heating-rate experimental results should be avoided because they tend to produce highly ambiguous kinetic descriptions [7,8]. At least three heating rates or temperatures are recommended. Applying the results obtained by thermoanalytical techniques (DSC), the kinetic analysis presented in this paper enables accurate prediction of the reaction progress of materials in a broad temperature range that may be difficult to explore for time, sensitivity or safety reasons. The presented study focuses on the evaluation of the experimental results obtained by DSC under isobaric (open crucibles) and isochoric (closed crucibles) conditions. The thermal behaviour of both propellants was examined at different heating rates ranging from 0.25 to 10 ?C/min. Figure 2 shows the DSC heat flow curve and the baseline subtracted heat flow curve of the sustain propellant at 0.25?C/min under isochoric conditions (closed crucibles). Generally, the application of straight-line form for the baseline is incorrect. The tangential area-proportional baseline is the most widely applied because of its correction possibilities [9]. The recorded signal results not only from the heat of the reaction but is additionally affected by the change of the specific heat of the mixture reactant-products during the progress of the reaction. The baseline determination can significantly influence the determination of the kinetic parameters of the reaction. Moreover, the correct baseline determination should be intimately combined with the computation of the kinetic parameters for the investigated reaction. Advanced mathematical procedures are therefore necessary for an objective calculation of the most appropriate baseline for each DSC signal [1].
 
Determination of the kinetic characteristics
 
The noticeable weakness of the ?single curve? methods (determination of the kinetic parameters from single run recorded with one heating rate only) has led to the introduction of ?multi-curve? methods over the past few years, as discussed in the International ICTAC Kinetics project [7, 10-12]. Only series of non-isothermal measurements carried out at different heating rates can give a data set, which generally contains the necessary amount of information required for full identification of the complexity of a process [7-8, 10-13]. This data set usually contains:
- the relationship between specific conversion xi and temperatures for different heating rates (non-isothermal mode).
- the relationship between specific conversion xi and time for different temperatures (isothermal mode).


Fig. 3 (A) Friedman analysis of decomposition of the sustain propellant under isobaric conditions (open crucibles).
(B) Activation energy as a function of the reaction progress for decomposition of the boost and sustain propellant under isochoric (DSC closed crucibles) and isobaric conditions (DSC open crucibles).
 
Commonly applied are the following isoconversional methods known as: Friedman [14] and Ozawa-Flynn-Wall [15-16]. A detailed analysis of the various isoconversional methods (i.e. the isoconversional differential and integral methods) for the determination of the activation energy has been reported in the literature by Budrugeac [17]. The convergence of the activation energy values obtained by means of a differential method (Friedman) with those resulted from using integral methods with integration over small ranges of reaction progress x comes from the fundamentals of the differential and integral calculus. Friedman analysis, based on the Arrhenius equation, applies the logarithm of the conversion rate dx/dt as a function of the reciprocal temperature at different degrees of the conversion. As f(x) is constant at each conversion degree xi, the method is so-called ?isoconversional? and the dependence of the logarithm of the reaction rate over 1/T is linear with the slope of m = E/R as presented in figure 3 A. Decomposition reactions are often too complex to be described in terms of a single pair of Arrhenius parameters and the commonly applied set of reaction models. As a general rule, these reactions demonstrate profoundly multi-step characteristics. They can involve several processes with different activation energies and mechanisms. In such situation the reaction rate can be described only by complex equations, where the activation energy term is no more constant but is dependent on the reaction progress x (Econst but E=E(x)) (Fig. 3B) [14-16]. Figure 4 presents the normalized non-isothermal DSC-signals as a function of the temperature or time for the decomposition of the double base rocket motor (boost- and sustain- propellants). Experimental data are represented as symbols. Solid lines represent the calculated signals. The comparison between experimental and calculated reaction extents indicates that the applied numerical technique enables accurate prediction of the reaction course under any experimentally chosen temperature mode.


Fig. 4 Advanced kinetic description of normalized non-isothermal DSC-signals as a function of the temperature for the decomposition of double base rocket motor propellants. Experimental data are represented as symbols, solid lines represent the calculated signals. The values of the heating rate in ?C/min are marked on the curves.
Closed crucibles (isochoric conditions): (A1): Boost propellant, (B1): Sustain propellant.
Open crucibles (isobaric conditions): (A2): Boost propellant, (B2): Sustain propellant.
 
Kinetics and milligram scale - thermal stability predictions
 
Verification of numerical computations and confidence interval for the prediction of the reaction progress
 
Verification of numerical computations can be readily achieved by graphical comparison of the calculated reaction progress or rate with subsequently obtained experimental data. Figure 5A presents the comparison of experimental and calculated DSC signals for the step-wise temperature program and isothermal conditions. In this program, heating with a rate of 3 ?C/min was followed by an isothermal run at 160?C. Accurate prediction of the thermal stability is achieved. In this case, the predicted reaction rate has been calculated for a temperature of 160?C, laying inside the observed temperature window used for the determination of the kinetic parameters under non-isothermal conditions (from 140?C for the beginning to 250?C at the end of the reaction, see Fig. 4 A1). In fact such a prediction of the propellant behaviour refers to an interpolation of the detected reaction rate from other experimental temperature profiles (non-isothermal conditions) but still in a temperature range where the thermal event occurs. However, for many applications it is relevant to examine how accurate is the prediction of the reaction rate for isothermal temperatures lying below the onset temperature observed under non-isothermal conditions. Therefore let us determine if the model used for the revelation of potentially hazardous behaviour of substances at elevated temperatures is still valid for the prediction of the reaction rate at prolonged times exposures at lower temperatures such as 110?C and 100?C for the double base propellant (Fig. 5B). At lower temperature the thermal event will occur over a much longer period of time with some confidence interval which has to be determined for the predictions (lower limit, mean value, upper limit).


Fig. 5 (A) Experimental data for the boost propellant decomposition (step-wise temperature program: heating rate of 3?C/min followed by an isothermal run at 160?C, closed crucibles i.e. isochoric conditions) are presented together with the reaction rate predictions. Experimental data are represented as symbols, solid lines represent the calculated relationships of da/dt over t. Accurate prediction of the thermal stability is achieved. (B) Predicted (solid line) and experimental reaction rates (points) of the rocket propellant under isothermal conditions (110?C and 100?C). The reaction is strongly autocatalytic because the initiation rate of the reaction is low leading to a long induction time under isothermal conditions. The figure at the top illustrates the reaction extent (DSC, normalized signals) of the rocket propellant with the confidence interval (95% probability) as a function of time under isothermal conditions (T = 110?C, closed crucibles/isochoric conditions).

During the determination of correct course of the baseline for all differential signal types, like DTA and/or DSC measurements and the calculation of the kinetic parameters for a decomposition reaction, the predictions give the nullcentral tendencynull, for which the chance of the good reproducibility on subsequent measurements is maximal. A Gaussian distribution is expected around this nullbest valuenull. The illustration of these remarks for the investigation of the rocket propellant (isochoric conditions) is presented in figure 5B. The predicted and experimental rates of the self-accelerating reactions are shown when the rocket propellant is held isothermally at 110?C and 100?C. This system is strongly autocatalytic, the rate of the initiation is low, leading to a long induction time under isothermal conditions. The reaction remains undetected for a relatively long period of time because the product catalysing the self-acceleration is formed slowly and/or the reaction begins when the stabilizers have been used up. When the reaction accelerates the rate increases so rapidly that it may lead to a runaway. For the rocket propellant, the mean value of the prediction for reaching the reaction progress of 65% (which corresponds to the maximum rate under isothermal conditions at T=110?C) is 47 hours. The lower and upper limits of the confidence intervals are 38 and 58 hours, respectively. These values indicate that there is a 95% probability that the mean time required for reaching 65% reaction progress is greater than 38 hours and lower than 58 hours. Similar measurements were done for 100?C showing again the long induction time under isothermal conditions. Extrapolation at lower temperature than the beginning of the reaction under non-isothermal conditions is therefore possible for in-depth awareness of the propellant thermal behaviour under varied eventualities. As a safety measure the confidence interval of the reaction progress is required to determine the precise the range of validity of the prediction. In that way very significant time/expertise savings can be achieved compared to real time analysis which can extend over prolonged periods, even years.
 
Cyclic temperature changes and prediction of the reaction progress under temperature mode corresponding to real atmospheric temperature changes
 
The kinetic parameters calculated from the non-isothermal experiments enable prediction of the reaction progress for any temperature mode: isothermal, non-isothermal, stepwise and therefore intermediate intervals in the heating rate, expressed, e.g. in oscillatory temperature modes. The prediction of the reaction progress in oscillatory temperature mode (widely applied in temperature-modulated calorimetry) is given below.


Fig. 6 (A) Reaction extents a of the decomposition of boost propellant in isothermal (50?C) and oscillatory (50?C 40?C, 24 h period) temperature modes (isochoric conditions, closed crucibles). (B) Predictions of the extent of boost propellant decomposition for the New Dehli and Moscow temperature profiles under isochoric conditions (closed crucibles). The substance exposed to daily temperature changes decomposes only slightly in New Dehli (about 0.25% after 60 years). In Moscow the reaction will progress even much less in this period of time.
 
Figure 6A shows the reaction extents a of the decomposition of boost propellant in isothermal (50?C) and oscillatory (50?C 40?C, 24 h period) temperature modes (isochoric conditions, closed crucibles). The arithmetic mean temperature (50?C) of the oscillatory temperature mode is the same as in the isothermal experiment. However, the presence of the temperature amplitudes greatly influences the reaction progress. The prediction of the decomposition of the boost propellant under isochoric conditions at 50?C with 40?C amplitude and 24h period indicates that after 3.5 months the sample is fully decomposed. For the same mean temperature (50?C) under isothermal conditions, the boost propellant will decompose only slightly after 12 months (<1%).
 
One of the main reasons for investigating the kinetics of thermal decompositions of solids is the need to determine the thermal stability of substances, i.e. the temperature range over which a substance does not decompose at an appreciable rate. The correct prediction of the reaction progress of unstable materials such as some pyrotechnics, propellants, food, drugs, polymers, etc. under ambient conditions requires accurate knowledge of both:
- the kinetic parameters
- the exact temperature profile for a given climate
 
As an example one may predict the extent of the decomposition of the boost propellant in New Dehli and Moscow. The temperature profile used for the calculations is the average of all daily minimal and maximal temperatures recorded for each day of the year between the years 1961 and 1990. These temperature fluctuations will be applied in the calculations of the thermal properties with cyclic temperature changes over the years. The calculation of the kinetic parameter E (activation energy) as a function of the reaction progress (Fig. 3B) enables calculation of the thermal stability of the propellant expressed in figure 6B as a function of time. The boost propellant under isochoric conditions exposed to daily temperature changes in these two places decomposes only slightly over this period. These results show that within 60 years, the reaction reaches about 0.25% in New Dehli. In Moscow, the reaction will progress even much less in this period of time.
 
Kinetics and scale up - thermal stability and safety analysis
 
Adiabatic conditions: Calculation of time of maximum rate under adiabatic conditions (TMRad) from non-isothermal DSC measurements
 
The precise prediction of the reaction progress in adiabatic conditions is necessary for the safety analysis of many technological processes. Calculations of an adiabatic temperature-time curve for the reaction progress can also be used to determine the decrease of the thermal stability of materials during storage at temperatures near the threshold temperature for triggering the reaction. Due to limited thermal conductivity, a progressive temperature increase in the material can easily take place, resulting in an explosion.
 
Several methods have been presented for predicting the reaction progress of exothermic reactions under heat accumulation conditions [18-22]. However, because decomposition reactions usually have a multi-step nature, the accurate determination of the kinetic characteristics strongly influences the ability to correctly describe the progress of the reaction. The use of simplified kinetic models for the assessment of runaway reactions can, on the one hand, lead to economic drawbacks, since they result in exaggerated safety margins. On the other hand, it can cause dangerous situations when the heat accumulation is underestimated. For adiabatic self-heating reactions, incorrect kinetic description of the process is usually the main source of prediction errors.
 
To be able to assess the probability of occurrence of a decomposition reaction, it is necessary to replace simple rules by sound methods based on reaction kinetics. The concept of Time to explosion or TMRad (Time to Maximum Rate under adiabatic conditions) is of great utility for that purpose [23]. A commonly used approach for the determination of the TMRad applies the following formula with the arbitrarily chosen zero-order reaction [24]:
 
TMRad = c_p R To²/(qo Ea) where:
 
c_p [J/kg/K]: specific heat,
To [K]: initial temperature of the runaway,
qo [W/kg]: maximum specific heat flux measured during an isothermal exposure at the temperature To,
Ea [J/mol]: activation energy of the reaction,
R [= 8.31431 J/mol/K]: ideal gas constant.

However, when applying the above approach to predict the TMRad the only one, simplified zero-order kinetic equation is used by fitting the reaction/decomposition exotherms by the Arrhenius relationship. This method gives unfortunately a very rough approximation of the TMRad due to the severe assumptions made concerning both, the kinetics and the constancy of the value of the activation energy. As presented in figure 3B, the activation energy is strongly dependent on the reaction progress for the considered compounds. In addition, it can be observed that the different experimental conditions (isochoric/isobaric) strongly influence the dependence of the activation energy on the reaction extent. The solution of the problem should therefore be achieved numerically. The computations have to consider the dependence of the activation energy on the reaction progress. For predictions with a certain level of accuracy, advanced kinetic analysis is therefore required because most decomposition reactions are complex combinations of several steps.
 
The decomposition of the boost propellant is a highly exothermal process. Using the reaction heat (ΔHr) and the heat capacity (cp), one can calculate the reaction progress due to self-heating for different values of ΔTad (with ΔTad = ΔHr/cp). In figure 7A, the simulated T-time relationships with a starting temperature of 100?C are presented for ΔTad = 2651?233?C (boost propellant, isochoric conditions). Figure 7B presents the starting temperature and corresponding adiabatic induction time TMRad relationship. The confidence interval was determined for 95% probability. The inset (Fig. 7C) presents the heat rate curves of the boost propellant under isochoric conditions for the different starting temperatures.


Fig. 7 (A) Adiabatic runaway curves for the boost propellant (isochoric conditions) showing the confidence interval for the prediction (Tbegin=100?C and ΔTad=ΔHr/cp=2651?233?C). The confidence interval was determined for 95% probability. (B) Starting temperature and corresponding adiabatic induction time TMRad relationship of the boost propellant under isochoric conditions. The choice of the starting temperatures strongly influences the adiabatic induction time (confidence interval: 95% probability). (C) Heat rate curves versus temperature for the boost propellant under isochoric conditions.

Table 1 Starting temperatures for TMRad of 8 h and 10 days for the boost and sustain propellants under isochoric and isobaric heat accumulation conditions.

The adiabatic induction time is defined as the time which is needed for self-heating from the start temperature to the time of maximum rate (TMRad) under adiabatic conditions. Depending on the decomposition kinetics and ΔTad, the choice of the starting temperatures strongly influences the adiabatic induction time and, therefore, the boundary conditions valid for achieving necessary safety (e.g. storage or transport of self-reactive substances). It can be observed that the reaction is more exothermal (higher ΔTad) and less stable (lower starting temperatures for the same TMRad) for the boost propellant than for the sustain propellant. The same observation is also valid for the isochoric conditions (closed crucibles) compared to the isobaric conditions (open crucibles).


Non-adiabatic conditions: application of Finite Element Analysis (FEA) for the determination of heat balances

The second field of application for numerical simulation techniques in process safety is the solution of partial differential equations as they are encountered in the heat conduction problems. Applications of Finite Element Analysis (FEA) and appropriate decomposition kinetics enable the determination of the effect of scale and geometry of the container as well as the heat transfer, thermal conductivity and ambient temperature on the heat accumulation conditions. This analysis enables the optimal choice of critical design parameters of the containers such as critical radius, insulation and safe storage or transportation conditions (i.e. determination of the best storage container size, insulation and/or optimal transport temperatures). Using the generalized heat balance over one layer element in the confinement wall, we can relate the heat transfer in each layer. Thermal energy can be transferred into a bounded region by conduction, convection, or radiation. For some systems, the mathematical problem can be reduced to the conduction of heat, to which the discussion will be largely confined, but the other modes of heat transfer may occur at the boundaries. The scheme of the grid-point distribution applied for calculating the temperature profile in each layer is presented in Fig. 8. The function of the heat balance can be singular at the interface of the different layers and at the beginning of the heat transfer process (times around 0). Therefore the grid-point distribution must be chosen with variable step lengths. The generation of adaptive meshes allows achievement of a desired resolution in localized regions and decreases by orders of magnitude the calculation time. Grid points are added in regions of high gradients to generate a denser mesh in that region, and subtracted from regions where the solution is decaying or flattening out. FEA and kinetics enables calculations of the temperature gradients using finite element methods and considering the heat transfer progress in the multi-layers [1]. It can be assumed that the heat transfer obeys Fourrier?s law (rate of heat transfer is proportional to the temperature gradient). The equations were developed using coordinates (x, y and t) where the whole surface of the volume will be derived from the different pre-defined geometries.


Fig. 8 Scheme of the multilayer confinement. The simulation of the whole multilayer confinement reduces to the analysis of a single layer and can then be extended from layer to layer. The grid-point distribution is chosen with variable step lengths in the heat transfer direction as well as in the time direction.


Fig. 9 Slow cook-off experiments of the rocket motor (A) and simulation (B). The predicted temperature of ignition was 124?C. It is in good agreement with the slow cook-off experiments (126?C).

The heat balance can be expressed by the following equations:



or for cylindrical coordinates



where J is a geometry factor which is dependent on the type of recipient: J=0 for the infinite plate, J=1 for the infinite cylinder and J=2 for the sphere. The above equation has been extended by the consideration of heat produced by the decomposition reaction Qr which rate is derived from the kinetics of the reaction. The heat balance equation can be now solved from r = 0 (centre of cylinder) to r = R (surface of the cylinder) with AKTS-Thermal Safety Software [1]. The temperature profiles have to be considered for all layers. In each layer the initial temperatures at t = 0 have to be introduced. At the centre and if a layer is perfectly isolated on its left or right side, the boundary condition (see Fig. 8) is derived from the symmetrical properties of the temperature profile at the wall surface. The other boundary conditions (II, Fig. 8) are derived from comparison of the heat transfers at the interface between the different layers. We have for two layers in contact:

- Boundary (I): symmetry axis
 
(if perfect isolation)

- Boundary (II): interface:


Consideration of the decomposition kinetics and application of the boundary conditions enable the calculation of the heat transfer, the temperature distribution and the reaction progress in a larger amount of material as encountered in the cook-off experiments. The slow cook-off simulation and experiments of the rocket motor are presented in figure 9. The ignition temperature of the slow cook-off was 126?C (Fig. 9A). The simulation of the cook-off behaviour and the predicted ignition temperature of 124?C (Fig. 9B) are in good agreement with the experiments. In the simulation the following parameters were used: TRocket initial = 40?C during 4 hours followed by a heating rate of 3.3?C/h; rocket motor diameter = 125 mm, thickness of the boost propellant layer = 31.5 mm, thickness of the sustain propellant layer = 31 mm; thermal diffusivity of boost and sustain propellant l/(rCp) = 0.02 cm2/s. Knowledge of the decomposition kinetics, thermal diffusivity and specific heat combined with FEA can be used to determine the ignition temperature very precisely. More generally, applications of FEA and accurate kinetic description enable the determination of the effect of scale, geometry, heat transfer (insulation), thermal conductivity and ambient temperature on the heat accumulation conditions. In fact, the assumption that it is safe to handle an energetic material at any temperature below the first appearance of an exothermic signal on the DSC curve can be false and even dangerous. The highest safe temperature for handling any energetic material depends on several factors such as its size, shape, and prior thermal history. Therefore, safe storage or transport conditions with tailored safety margins can be defined using numerical simulation.


Conclusions

With at least three DSC experiments done under isothermal or non-isothermal conditions it is possible to compute the kinetics of decomposition of the products of interest. For these calculations the results obtained by thermoanalytical techniques recording the heat generation (DSC) were used. In general, the same approach can be applied to other thermoanalytical signals such as the change of the mass (TG), the rate of evolution of the gaseous products (TG-MS, TG-FTIR) as well as the change of the pressure under isochoric conditions (isoperibolic calorimetry). Employing advanced mathematical modeling and kinetics, it is possible to calculate the progress of decomposition reactions under temperature conditions different from those at which the experiments were carried out. In general, accurate kinetics enables calculation of the reaction progress in extended temperature ranges and under temperature conditions for which the experimental collection of the data is difficult. These difficulties are prevalent at low temperatures (requiring very long investigation times), as well as under specific temperature fluctuations.

Thermal safety simulation of self-reactive chemicals depends on the properties of the energetic substance (decomposition kinetics, heat conductivity, specific heat, loading density), properties of the container (e.g. size, geometry, the rate of the heat transfer to the environment) and surrounding temperature. Finite Element Analysis (FEA) and advanced kinetic description enable determination of the effect of scale, geometry, heat transfer, thermal conductivity and ambient temperature on the heat accumulation conditions. Influence of complex thermal environment such as stepwise temperature profile of cook-off experiments can be used for verification of the numerical computations. It is then possible to cover in detail several different situations, problem definitions and results interpretation for thermal stability and safety analysis. In general, the heat accumulation conditions can be calculated for any surrounding temperature profile such as isothermal, non-isothermal, stepwise, modulated, shock and additionally temperature profiles reflecting real atmospheric temperature changes (yearly temperature profiles of different climates with daily minimal and maximal fluctuations). This analysis can then be applied for the determination of the critical design parameters such as critical radius of a cylinder or sphere, the thickness of the isolation, influence of the surrounding temperature for safe storage or transport conditions. In fact, numerical simulations can be used to replace experiments in situations, which are not directly accessible to the experiment for timing reasons. The examples of such modeling analysis can be helpful for guiding screening and development activities of candidate energetic materials. If modeling proceeds in parallel with experimental studies, then it should result in lower costs in the development phase of a project.

The proposed method has several advantages:
- It is convenient: Adiabatic devices of the required complexity are not available in every safety laboratory; DSC devices are however more widely available.
- It is versatile: With one set of measurements different adiabatic and non-adiabatic situations can be calculated. The method can be used to predict the rate of the reaction progress da/dt and temperature rise dT/dt for any temperature. If the pressure rise is measured, the rate of the pressure rise dP/dt can be determined for any temperature as well.
- It is secure and economical: The calculation of adiabatic reaction progress and/or explosion conditions requires only a small amount of material.