GRUPOS BENEFICIARIOS

The reaction-rate equations above have been incorporated into the new heat conduction and chemistry coding in Peruse and will be used to represent the chemical reactions of HMX and the binder throughout this work. Note that Arrhenius kinetics are not yet available in Corvus or Petra. For each reaction ratei, three parameters are needed: the frequency factorZi, the activation energyEi and the heat of reactionqi.

HMX

Several different Arrhenius reaction rates for HMX are available in the literature. Hen- son [82,178] collected together a wide variety of data for the time to explosion in HMX, including one-dimensional time-to-explosion (ODTX) and detonation data, and fitted them with a simple formula corresponding to a single step Arrhenius rate:

lnt= E1

RT −lnZ1.

Unfortunately, this approach produces explosion times that disagree with the results of thermochemical calculations in ReactDiff[115] and with Hubbard & Johnson’s approxi- mate formula for the time to explosion from a single step Arrhenius rate [179]:

t= cv,sT 2_R Z1q1E1 exp _E 1 RT . (3.5)

Figure 3.3: Arrhenius plot giving the time to explosion versus initial temperature for HMX. Explosion times calculated using Henson’s rate parameters in ReactDiff simula- tions with uniform heating (0D sims) and using equation3.5(0D calcs) do not agree with his compilation of experimental data [82,178]. McGuire & Tarver’s rate parameters [86] give good agreement to the experimental data in ODTX simulations, but do not extrapo- late well to high temperatures in 0D simulations. Therefore, both sets of parameters are unsuitable for use in this work.

The red line in figure3.3represents explosion times from both ReactDiffand equation3.5 since they give similar results in 0D geometry, i.e. a single HMX region with uniform initial temperature. Figure3.3shows that Henson’s reaction rate parameters (the red line) produce explosion times several orders of magnitude faster than Henson’s experimental data (the black line). Although Henson’s data are an excellent source of information on the chemistry of HMX, his reaction rate coefficients are not useful.

McGuire & Tarver [86] advocate a three-step scheme that has been fitted, using a heat transfer code, to data from ODTX experiments. The ODTX geometry is a 1.27 cm- diameter sphere of explosive, initially at room temperature, surrounded by anvils that are heated to a constant temperature [123]. Thermal conduction from the anvils into the sam- ple raises its temperature, causing chemical reactions to begin. The time at which the anvils are forced apart by the accumulating high-pressure reaction products is recorded as the time to explosion. Although there are several options for deriving the corresponding time to explosion from ReactDiffsimulations [115], the time at which there is a sudden increase in the concentration of the final reaction products is used in this work (see sec-

Figure 3.4: Arrhenius plot giving the time to explosion versus initial temperature for HMX. Uniform heating (0D sims & calcs) and ODTX simulations using Menikoff’s re- action rate [83] give a good match to Henson’s compilation of experimental data [82,178] and were chosen for use in this work.

tion5.1). Their dependence on heat conduction as well as Arrhenius chemistry makes ODTX simulations rather different in character to 0D simulations, but both sets of data are often represented on a single Arrhenius plot [e.g.,82]. Figure3.3shows that McGuire & Tarver’s three-step reaction rate matches the experimental data in ODTX simulations (green pluses) but does not extrapolate well to high temperatures (green dashed line). It also fails to produce a self-sustaining detonation in one-dimensional hydrodynamics cal- culations. Henson et al. recently published a complex multi-step Arrhenius scheme for HMX [85] and Tarver has proposed a reaction scheme that accounts for cross-reactions between HMX and binder [180]. Although these have not been investigated in this work, they could be tried in future.

Menikoff [83] uses a single step Arrhenius reaction-rate based on Henson’s data to model detonation in PBX9501. This has parametersE1 = 149 kJ/mol and lnZ1 = 12.5, for Z in µs−1_{. Using equation 3.5, Meniko}ff_{’s rate produces explosion times (the blue} dashed line in figure3.4) which agree well with Henson’s data at high temperatures where bulk heating is the most significant effect. ODTX simulations in ReactDiff show that Menikoff’s parameters produce explosion times of the same order of magnitude as the experimental data, although with the wrong gradient. Using a single-step Arrhenius rate,

it is difficult to match both the gradient of the low-temperature ODTX data and the high- temperature explosion times. Menikoff’s parameters are a good compromise and will be used in this work (table 3.8). They allow detonation to propagate in one-dimensional hydrodynamics calculations, an essential requirement for a mesoscale model that, it is hoped, will simulate the shock-to-detonation transition.

Maximum and minimum values for lnZ1andE1were estimated by making reasonable changes to the Arrhenius plot in figure3.4 using equation3.5. The explosion time curve with the minimum gradient has lnZ1 = 11.6 andE1 = 140 kJ/mol, while the curve with the maximum gradient has lnZ1 = 13.8 and E1 = 160 kJ/mol, forZ inµs−1. Menikoff quotesq1 ∼ 5 kJ/g [83]. In their three-step scheme, McGuire and Tarver [86] useq =

−100+300+ 1200 cal/g = 5.8618 kJ/g, which is not dissimilar to Menikoff’s value. The value ofq1 = 5.8618 kJ/g was chosen for use in this work, since it has been used in deriving the JWL reaction products EOS for HMX and it is close to the value of 5.96 kJ/g from Cheetah [136].

lnZ1 E1 q1

12.5 1.49 0.058618

(Zinµs−1) Mbar cm3/mol Mbar cm3/g

Table 3.8: Single-step Arrhenius reaction-rate parameters for HMX in hydrocode units.

Binders

Only a limited data set is available for the reaction rates of either of the binders. For PBX9501’s binder, some ODTX experimental data are available and are plotted in fig- ure3.5. Even accounting for the slower heat conduction expected in the ODTX geometry due to its lower thermal conductivity, this seems to indicate that the binder is more reac- tive than HMX at certain temperatures, which is surprising because PBX9501’s binder is often described as inert. Extrapolating these data into the high temperature regime would be fraught with danger, but a simple approach is to assume that the binders behave in the same way as HMX. Appropriate values of lnZ and E1 were determined by fitting equation3.5to the HMX data (the black line) in figure3.5. Therefore,E1 = 149 kJ/mol is used for both binders, with lnZ1 = 13.2 for the binder in PBX9501 and 13.3 for the binder in EDC37 (table3.9). The slight differences in lnZ1 arise because of the different thermal properties of the two binders.

Figure 3.5: Arrhenius plot showing that ODTX experimental data for the binder in PBX9501 [135] are similar to Henson’s compilation of HMX data [82, 178]. Reaction parameters derived from Menikoff’s rate for HMX are used for the binders in PBX9501 and EDC37 in this work.

lnZ1 E1 q1

PBX9501’s binder 13.2 1.49 0.044959

EDC37’s binder 13.3 1.49 0.039083

(Z inµs−1_{) Mbar cm}3_{/mol Mbar cm}3_/g

Table 3.9: Single-step Arrhenius reaction-rate parameters for the binders in PBX9501 and EDC37.

Although there is considerably more uncertainty in the Arrhenius reaction rate param- eters for the binders than for HMX, a first guess for the maximum and minimum reaction rates is to use a similar range as for HMX. The time to explosion curve with minimum gradient would therefore correspond to lnZ1 = 12.4 and E1 = 140 kJ/mol, while the curve with the maximum gradient has lnZ1 = 14.6 and E1 = 160 kJ/mol. Values for the heat of reactionq1are calculated from the JWL EOS as 3.9083 kJ/g for the binder in EDC37 and 4.4959 kJ/g for the binder in PBX9501. Maximum and minimum values are not quoted owing to the difficulty in estimating them, and the requirement that the JWL reaction products equation of state is consistent with the value ofq1.