The paper introduces a method for determining kinetic parameters for the pressed explosive tetranitropentaerytrite. This method includes analytical dependence for the period of induction, which is used to process the experimental data, and the ANSYS CFD finite element analysis software package. The found values of the kinetic parameters made it possible to obtain a satisfactory agreement between the calculated and experimental ignition distances for pressed tetranitropentaerytrite

Предложен способ определения кинетических параметров для прессованного взрывчатого вещества тетранитропентаэритрита. Данный способ включает в себя применение аналитической зависимости для периода индукции, с помощью которой проводится обработка экспериментальных данных, и пакета программ конечно-элементного анализа ANSYS CFD. При использовании найденных значений кинетических параметров удалось получить удовлетворительное согласие между расчетным и экспериментальным временем до воспламенения для прессованного тетранитропентаэритрита

It is known that when products containing explosives get into the thermal field of a fire, an explosion is possible. For the purposes of safe handling of explosion-hazardous products it is crucial to be able to determine the time from the start of thermal field action to the moment of ignition and a safe temperature of the external thermal field, as well as the site of reaction, for subsequent prediction of the character of process development (explosive agent burning out under conditions of layer-by-layer or convective combustion, low-velocity explosion, or detonation). For that, it is necessary to know kinetic characteristics of the explosive.

The kinetic parameters are normally determined by the results of experiments that imply heating up small samples of explosives. For example, in the US. the ODTX method is used for calibration of the kinetic parameters [1, 2]. It essentially comes down to the following (Fig. 1): a spherical explosive sample, ∅1.27 cm (r = 0.635), is placed in massive cylindrical aluminium blooms having hemispherical concaves and heated up to a certain temperature (100...300 °С). After that, time is measured, from the moment of explosive sample placement in the blooms till the ignition moment, which is registered by the signal from a microphone arranged near the blooms. It is only the temperature of the blooms that varies in the experiments.

To describe the experimental data obtained according to the ODTX method, models of multistage (34 stages) decomposition of explosive can be used [1, 2]. However, in thermal calculation of real articles, it is preferable to use simple models, in which the most significant points of thermal decomposition of explosive would be taken into account. As is known, during thermal decomposition and in development of thermal explosion in condensed explosives, an extremely important role is played by autocatalytic reactions [

A + Z → 2Ζ,

where A - initial substance (explosive);

Z - substance (catalysing agent) accumulating in the course of reaction and accelerating decomposition A. The rate of reaction will be written in this case [

where [A] - current relative mass concentration of the initial substance A, i. e. [A] = mA (t)/ mA (t = 0);

mA(t) - mass of initial substance A at a random moment of time;

mA(t = 0) - mass of initial substance A at the initial moment of time;

t - time;

z - pre-exponential factor;

E - activation energy;

R - universal gas constant;

T - temperature;

[Z]0 - initial relative mass concentration of catalysing agent Z , [Z]о

mz(t = 0) - mass of substance z at the initial moment of time.

In expression (1), the magnitude of [Z]0 characterises starting capabilities of the reaction [

For numerical determination of time before ignition in the ODTX experiments, it is necessary, in addition to equation (1), to solve the equation of thermal conductivity with the Arrhenius heat source [

where ρ - density;

c - heat capacity;

λ - thermal conductivity coefficient;

Q - heat of reaction (J/m3).

The initial and boundary conditions for equations (1) and (2), as applies to the ODTX experiments, appear as follows:

where Tt=0 - explosive sample temperature at the moment of time t = 0;

Tr=b - temperature at the boundary of spherical explosive sample (on radius r = b = 0.635 cm);

Tб - temperature of the blooms;

- temperature gradient in the sample centre (on radius r = 0);

- initial substance concentration gradients at sample boundary and in its centre.

Equations (1) and (2) are solved numerically. The values of thermophysical parameters, e.g., for pressed tetranitropentaerytrite (PETN), can be found in [

Fig. 2 shows experimental data with the values of time before ignition obtained using the ODTX method [

The combustion theory employs a widely used notion of an induction period, i. e. time for reaching the maximum rate of chemical reaction in an explosive, or time to the occurrence of ignition [

γ - Todes number (γ ~ 0,001), characterising exothermic degree of explosive decomposition reaction [

The strong autocatalysis condition is written as follows: [Z]0 4]. Formula (4) can be used for approximation of the experimental results (Fig. 2) by the least-squares method, determining parameters E, z, which can be used subsequently in solving equations (1), (2). It should be pointed out that the values of time to ignition in the experiments differ from one another by the orders of magnitude (10...104 s) within a narrow range of blooms temperature values variation (140...200 °С). In order to reduce excessive influence of large time-to-ignition values (10...104 s) when processing data by the least- squares method, it should be practical to use a modified expression for the induction period instead of formula (4), namely:

For estimation of parameters E, z by formula (5), one should fix upon the values of magnitudes included in this formula. Paper [

where Hf - PETN specific melting heat. Further, to take into account the ingress of heat during melting, we assume that adding to heat capacity c = 1250 Дж/кг °С in the temperature range of 20....170 °c is additional heat capacity спл = Hf /(170 °С - 20 °С), i. e. heat capacity value с + спл will be used in formula (4). We also assume that the initial relative mass concentration of substance [Z ]0 = 10-5. For such value, the strong autocatalysis condition [Z ]0

Next, we should elaborate on the values of kinematic parameters obtained. Indeed, in reality, apart from thermal decomposition occurring in PETN while heating up, the processes of melting and convective heat transfer take place as well. These processes may influence the calculated time-to-ignition values. For this reason, calculations in the ANSYS CFD finite-element analysis software package (licence No. 774-2013-ША dated 01.10.2013) were performed to determine time to ignition for all the values of blooms temperatures. The calculations took into account thermal decomposition of explosive, melting, and convective heat transfer. In addition to equation (1), the calculations were performed with consideration of the laminar heat convection equations (Boussinesq equations), same as in the paper [

where Cf – heat capacity associated with PETN melting in the temperature range of 136…141 °С, Cf = Hf / 5 °С = 25,6 кДж/кг ⋅ °С;

u – liquid PETN velocity;

p – pressure;

μ (T) – coefficient of dynamic viscosity, depending on temperature;

F – buoyancy force (Archimedes force),

dρ / dT – coefficient taking into account explosive density dependence on temperature [

T0 – initial temperature, T0 = 20 °С;

g – gravity acceleration.

The dependence of viscosity μ(Τ) , Pa⋅s on temperature Т, °С is as follows:

The dependence of PETN viscosity on temperature is unknown as of present. It is selected such that before the temperature of 136 °С the PETN would behave nearly as a solid substance, where convection is insignificant, and at temperatures above 141 °С, as a low-viscous fluid. Such technique is applied in the paper [

The calculations were performed in the axisymmetric Eulerian formulation. Cell size - max. 0.5 mm. The calculation geometry is shown in Fig. 4. The boundary conditions are given there as well. At that, given the axisym- metricity, only half of the explosive sample was considered. The calculation would be stopped at the moment when calculation step became less than 10-6 s. That moment corresponded to the ignition start. For the best compliance between experimental and calculated time to ignition, the magnitude of z varied in a range of 4,293-1014...4,293-1015, while the magnitude of Е = 142,803 J/mol remained unchanged. Taken as a measure of the best compliance was the least value of the sum of squares of experimental and calculated time-to-ignition deviation logarithms The result obtained shows that best suiting for description of the experimental data is the value of preexponential factor z = 3,293 1015 1/c.

Fig. 5 shows temperature profiles for two moments of time and for the temperature of blooms equal to 149 °С. As can be seen from Fig. 5, the non-melted explosive sediments down. Due to convection, the ignition region is formed at the top. It should be noted that taking into account the convective heat transfer and melting does not much affect the time to ignition, as, ultimately, for the best compliance between the calculated and experimental results we did not have to greatly vary the value of pre-exponential factor obtained at the stage of experimental data processing by formula (5). Similar results were obtained in [

Fig. 6 shows the results of numerical calculations in comparison with the experimental data from [

The approach described above allows to determine the values of kinetic parameters for pressed PETN. Subsequently, using the obtained values of kinetic parameters it will be possible to make calculations to determine time to ignition for complex products containing explosives on the base of PETN and for different heating-up modes.

Summing up all of the above, the paper offers a method for finding optimal values of kinetic parameters for PETN explosive. The method rests on the results of experiments for determining time to ignition, conducted on explosive samples of ∅1.27 cm. At the first stage, by means of analytical expression for the induction period, preliminary values of kinetic parameters are determined, using which it can be possible to satisfactorily calculate time to ignition. At the second stage, using calculations made in the ANSYS CFD finite-element analysis software package, fine calibration of the preexponential factor is performed. Calculations were made with consideration of melting and convection in the explosive. Computations run with the most optimal set of kinetic parameters lead to satisfactory match between the calculated and experimental data.

The authors declare that there are no conflicts of interest present.