3.3. Burn Rate
For the measurement of the burning rate, delay compositions with an Sb content from 30 to 60 wt% were prepared. The burn rate of the compositions containing more than 60 wt% of the fuel was not investigated due to technological difficulties related to the granulation of the samples. The samples were loaded into tubes using different pressures. Five tests were performed for each composition and loading pressure. The dependence of the delay composition density on the loading pressure is shown in
Figure 4. The figure also shows the standard deviations of the measured densities. As expected, this dependence is not linear. Generally, the increase in density decreases with the loading pressure.
The dependencies of the burning rate of the tested compositions on the loading pressure and the composition density are shown in
Figure 5 and
Figure 6, respectively. The figures also show the standard deviations of the measured burning rates. The combustion of the Sb/KMnO
4 composition with 30% Sb could not be initiated for a loading pressure greater than 400 MPa.
Figure 5 shows that in the tested range of the Sb content in the composition, a clear increase in the burning rate is visible. However, for a fuel content above 50%, this increase is less. These findings correlate nicely with the dependance of the delay time on the antimony content presented in [
4]. The smallest changes in the dependence of the burning rate on the Sb content are visible for high loading pressures (over 400 MPa). Hence, for Sb/KMnO
4 compositions in which more than half of the mass is fuel and which have been loaded under high pressure, no changes in the burn rate are observed. This is a great advantage of these compositions in the technology of delay detonators’ production, as it eliminates the loss of accuracy of detonators due to the possible non-homogeneous mixing of the components. Due to the high softness of the oxidant, the tested compositions can be dry granulated without the use of additional binder. However, the lack of a mixing step in the binder solution may hinder the homogenization of the production batch and necessitate the extension of the mixing process. Therefore, the occurrence of a plateau in the curves in
Figure 5 is extremely important.
The dependences of the burn rate on the loading density of the Sb/KMnO
4 composition in detonators was approximated by a linear function (
Figure 6). A good correlation between the experimental data and the approximation function was obtained. The increase in the density of the composition is accompanied by a decrease in the burning rate. It is justified by the diffusion control of reactions taking place in the system, mainly antimony oxidation with gaseous oxygen resulting from the thermal decomposition of KMnO
4. The increase in density makes it difficult for oxygen to penetrate the sample. The effect of density is especially visible for compositions with an antimony content above 45%, for which the total differences in the burning rate reach even 6 mm s
−1. For compositions containing antimony in the range of 30–35%, the dependence of the burning rate on the density is much weaker, however, it cannot be ignored in the technology of detonators.
It should be emphasized that apart from the quantitative composition and the loading pressure, there are also other factors that strongly affect the burning rate. Beck [
2] investigated the effect of antimony grain size (<8 µm and <53 µm) on the burning rate of the Sb/KMnO
4 composition with 30% Sb loaded with a pressure of 222 MPa into a detonator.
Figure 7 presents the results given in Ref. [
2] and the measured burning rates for the composition containing antimony particles with a diameter smaller than 34 µm and loaded with a pressure of 229 MPa. The determined dependence of the burning rate of this composition on the antimony content correlates well with the dependencies given in Ref. [
2].
3.4. Heat of Combustion
In Ref. [
2] pressure profiles were measured after the ignition of the Sb/KMnO
4 compositions loaded into closed aluminum tubes. During the combustion of the Sb/KMnO
4 compositions (with a 50 wt% antimony content), the pressure in the tube after the initial ignition pulse was constant at 1.4 MPa until the entire sample was burned, and then it gradually decreased. In the case of the compositions containing 36% and 30% Sb, the pressure initially stabilized at 2 MPa and then increased rapidly due to the evolution of oxygen and gaseous reaction products. To create combustion conditions similar to those in detonators, the calorimetric bomb was filled with argon at a pressure of 2 MPa.
The heat of combustion of the Sb/KMnO
4 compositions containing 30 to 70% antimony was measured. The solid products of combustion were generally shaped in a similar way as the original samples (
Figure 8a), but in the case of high antimony contents, a metallic Sb ball appeared on the bottom of the quartz crucible (
Figure 8b). Moreover, samples with a high antimony content melted and filled the quartz crucibles (
Figure 8c).
The results of the measurement of the heat of combustion of the tested compositions are presented in
Table 1. The masses of the samples before and after combustion are also given. The masses of the metallic Sb balls are shown in the parentheses. The highest values of the heat of combustion were measured for samples containing 40–45% Sb, and the lowest values were determined for samples of 65–70% Sb.
Figure 9 shows the percentage mass loss of the Sb/KMnO
4 compositions burned in the bomb filled with argon. According to Ref. [
8], the term “gasless” relates to compositions that generate up to 10 mL of gases per g of composition. In the case of the tested Sb/KMnO
4 compositions, O
2 may be the lightest gaseous product of combustion. A volume of 10 mL of O
2 corresponds to a mass of approx. 14 mg. For the other gaseous combustion products of the compositions tested, the mass corresponding to 10 mL would be much higher. Therefore, it can be assumed that for the tested compositions, the criterion of 10 mL g
−1 can be replaced with the criterion of 14 mg g
−1. This value has been marked in
Figure 9. All compositions with 40% Sb and more can certainly be treated as gasless compositions. The assessment of the compositions containing 30% and 35% Sb in this respect requires the determination of the composition of gaseous combustion products.
As the composition burns in the detonator, the pressure in the time-delay element may vary. Therefore, the combustion heat of two selected Sb/KMnO
4 compositions for different initial pressures in a calorimetric bomb was investigated. The measurement results are presented in
Table 2.
Taking into account a certain dispersion in measured heat shown in
Table 1, it can be concluded from the analysis of the results presented in
Table 2 that the combustion heat of the tested compositions does not depend on the argon pressure in the range of 0.2–1 MPa. Slightly higher average heat values were obtained for the pressure of 2 MPa.
3.6. Thermochemical Calculations
An attempt was made to calculate the theoretical heat of the combustion of the Sb/KMnO
4 compositions. The qualitative composition of the combustion products was determined using the CHEETAH thermochemical calculation program [
9]. The CHEETAH computer program performs thermodynamic calculations for heterogeneous chemical systems containing gases, liquids or solid particles. The thermochemical equilibrium state of the reaction composition is determined by minimizing the Gibbs free energy under the condition of a mass balance of the elements present in the composition and other constraints, for example, a constant volume or constant pressure. The values of the thermodynamic functions of the individual compounds of the reacting composition necessary to perform the calculations are determined from the dependence of the specific heat
Cp on the temperature and the appropriate thermodynamic relationships. The dependence of
Cp on temperature is described by a polynomial.
where
R is the gas constant,
C1,
C2, …,
C7 are constant coefficients,
θ =
T/1000 and
T is the absolute temperature.
Before performing the calculations, the library of chemical reaction products was extended to include the potential components of the combustion products. It was not possible to find the dependence of
Cp on the temperature for non-stoichiometric K-Mn-O compounds, as well as some of the other compounds, which occur in solid combustion products (
Table 3) or were identified as products of KMnO
4 decomposition in Ref. [
1]. However, the formation of the following gaseous substances was predicted: K, K
2, KO, Mn, Sb, Sb
2, Sb
4, SbO, Sb
4O
6; and condensed ones: K
2O, KO
2, K
2O
2, Mn, MnO, Mn
2O
3, Mn
3O
4, MnO
2, Sb, Sb
2O
3, Sb
2O
4, Sb
2O
5, MnSb, Mn
2Sb. The temperature dependencies of specific heat and the standard enthalpy of formation of these substances necessary for the calculations were taken from the thermochemical data tables [
10,
11]. The constants of the relationship (Equation (1)) were determined by approximating the table data using the least squares method. The ideal gas equation was used to describe the physical properties of gaseous combustion products. Solid products were assumed to be incompressible.
In order to determine the adiabatic combustion temperature
Ta, it was assumed that the tested system is isolated and that the combustion process takes place under constant pressure. After determining the isobar in a given temperature range and determining the state of the thermochemical equilibrium at selected points of this range, the difference between the full enthalpy of the equilibrium state at temperature
T and the full enthalpy of the initial composition in the standard state (
T0 = 298.15 K) was calculated.
where
is the full enthalpy of the
i-th combustion product,
ni is the number of moles of the
i-th combustion product,
k is the number of combustion products,
is the full enthalpy of the
j-th component of the composition,
nj is the moles of the
j component and
l is the number of pyrotechnic composition components. The full enthalpy of a substance is defined as follows.
where
is the enthalpy of formation at temperature
T0, and
and
are the enthalpies of the substance at
T and
T0, respectively.
In calculations performed with the CHEETAH code, it was assumed that burning takes place under conditions of constant pressure (
p = 2 MPa). The temperature for which the difference in Equation (2) was close to zero was treated as the adiabatic combustion temperature
Ta. The composition of combustion products at this temperature was the basis for determining the heat of combustion using Hess’s law. The heat of combustion was calculated as the difference between the standard enthalpy of formation of the combustion products and the standard enthalpy of formation of the initial pyrotechnic composition.
The dependence of the adiabatic combustion temperature and the heat of combustion on the antimony content in the Sb/KMnO
4 composition is shown in
Figure 12. The figure also shows the experimental combustion heats from
Table 1.
The large difference between the calculated heat of combustion and the measured heat for almost the entire tested range of Sb content in the composition is surprising. It also means that the calculated adiabatic combustion temperature may differ significantly from the actual temperature in the combustion wave. In order to answer the question of whether the omission in the thermochemical calculations of the solid combustion products identified in the XRD analysis could be the reason for such a difference, the hypothetical combustion reactions of the 2 Sb + 2 KMnO4 composition were considered. The antimony content in this composition is 43.6 wt%. This value is close to the antimony content in the composition with 45 wt% Sb, for which the highest heat of combustion of 1570 J g−1 was obtained.
In order to calculate the heat of reaction using Hess’s law, it is necessary to know the enthalpy of formation of the substrates and products. The enthalpy of formation of the components of the 2 Sb + 2 KMnO
4 composition and simple combustion products was taken from [
10,
11]. The enthalpy of formation of K
2MnO
4, K
2MnO
2, K
3MnO
4 and KSbO
3 was calculated on the basis of data included in [
12]. The hypothetical combustion reactions and reaction enthalpies are as follows.
The assumption of different reaction products means that the heat of combustion can differ even twice. The lowest thermal effects are obtained by assuming only simple metal oxides in the products (Equations (6) and (7)). Similarly, the presence of simple oxides is assumed in thermochemical calculations using the CHEETAH code. For example, for a composition containing 45% Sb, the calculated combustion products are the solids K
2O, Mn
3O
4, Sb
2O
4 and small amounts of O
2, K and KO gases. The calculated heat of combustion for this composition is 1085 J g
−1. The measured value is much higher (1570 J g
−1). However, the XRD analysis shows that KSbO
3 is the dominant component of the combustion products of compositions with a similar composition (40% Sb—
Table 3). Its presence in reaction products significantly increases the heat effect (Equation (10)). Thus, the reason for the higher values of the measured heat of combustion of the tested compositions is the presence of KSbO
3 and non-stoichiometric K-Mn-O compounds. The above analysis shows that the only way to determine the heat of combustion for Sb/KMnO
4 compositions are calorimetric measurements in an inert gas. The heat determined in this way can be used in modeling the combustion process of these compositions, for example, with the use of models [
13,
14,
15].