Flame propagation simulations

Temperature histories from a typical diamond anvil cell simulation are shown in fig- ure 6.20. The results are similar in character to super-critical hotspot simulations, but the focus is on the behaviour of the reactive wave rather than the hotspot. The defla- gration rate decreases as the wave propagates further into the HMX (i.e. the separation between the gauge traces increases). This is because the system pressurises as the HMX is converted into gaseous reaction products, causing the temperature of the reaction prod- ucts to fall, which reduces the rate of heat conduction. The unreacted HMX takes longer to heat up before it can react, so the reactive wave speed is lower. The initial deflagra- tion rate in this simulation is 2.9 m/s, somewhat lower than the experimental data and Reaugh’s simulations. To investigate the reasons for the low deflagration rate in Peruse, the sensitivity of the simulations to material properties and geometry will be explored below. Coarse meshing (5 zones/µm) was used for many of these initial calculations to reduce the computational run-time; the effect of mesh density is investigated later.

Since the reactive wave is driven by thermal conduction (see section6.1), any mate- rial properties that affect the temperature of the unreacted HMX and the reaction prod- ucts could affect the flame speed. The temperature of the gaseous reaction products is controlled by the CJ temperature TCJ and specific heat capacity cv,CJ. Nominal values ofTCJ = 3000 K and cv,CJ = 2 J/g K from chapter 3 give an initial deflagration rate of 1.1 m/s in a diamond anvil cell simulation. A similar calculation with TCJ = 4000 K had an initial deflagration rate of 1.7 m/s owing to higher temperatures in the gaseous reaction products. Ifcv,CJ is increased to 3 J/g K instead, the initial deflagration rate is unchanged but the deflagration rate falls less quickly as the system pressurises, because the gas temperature is higher. The temperature of the unreacted HMX depends on spe- cific heat capacitycv,s. Reducing cv,s from its baseline value of 1.1 J/g K to 0.8 J/g K increases the deflagration rate from 1.1 to 1.4 m/s.

Other parameters that could affect the flame speed are the thermal conductivitykand the Arrhenius reaction rate parameters lnZ1 and E1. Increasing k from 0.4 W/m K to 0.5 W/m K modifies the deflagration rate by 0.2 m/s. Changing the Arrhenius param- eters to the maximum and minimum values in section 3.4 affects the deflagration rate

Figure 6.20: Temperature histories from a diamond anvil cell flame propagation simula- tion at 10 GPa withr1 = 1µm, r2 = 9µm, r3 = 0µm and 40 zones/µm meshing. The hotspot, with an initial temperature of 1600 K, reacts promptly and its temperature rises to∼3500 K (black, red and green gauges). A heat-conduction-driven reactive wave prop- agates outwards into the surrounding HMX. When it arrives at each gauge location (blue, yellow, brown, etc.), the temperature rises from 295 to ∼2500 K as chemical reaction converts the unreacted HMX into gaseous reaction products.

by less than 0.1 m/s. In his flame propagation simulations, Reaugh uses a four-step re- action scheme for HMX from reference 135but with the frequency factor Z4 increased by a factor of 80, in order to match the results of molecular dynamics simulations of HMX decomposition [26, 87]. These rate parameters can be used with a single-step Arrhenius rate to give an over-estimate of the behaviour that would be obtained using Reaugh’s reaction scheme. Peruse simulations with lnZ1 = 18.67 (for Z in µs−1) and

E1 =117.7 kJ/mol give a deflagration rate of 4.0 m/s for these reaction-rate parameters, compared to 1.1 m/s for the baseline values.

These initial simulations show that the effect of changing material propertiesTCJ,cv,CJ,

cv,s,k, lnZ1andE1is relatively small. It does not change the character of the simulations from that in figure6.20and, in particular, the deflagration rate always falls as the system pressurises.

Effect of meshing and geometry

Mesh resolution is important for flame propagation simulations. Figure 6.21 shows the deflagration rate as a function of mesh resolution for simulations with r1 = 1µm,

r2= 9µm andr3 =0µm. The maximum deflagration rate increases from 1 m/s to almost 3 m/s as the meshing is raised from 5 to 40 zones/µm. The close agreement between the results of the 40 and 100 zones/µm calculations shows that mesh convergence has been achieved. Figure6.21 also shows temperature histories from the simulations with 5 zones/µm and 40 zones/µm meshing. The temperature overshoots at 5 zones/µm re- duce in magnitude as the mesh density is increased, and disappear altogether once mesh convergence has been achieved at 40 zones/µm. Reaugh’s simulations gave similar be- haviour [87]: “My general observation was that if the temperature profiles were smooth curves, then the resolution was adequate. If they were very steep, or the temperature wiggled around, then a finer mesh was called for.” The resolution required to achieve mesh convergence depends on the simulation, but either 40 or 100 zones/µm was used to produce the final results in section6.4. For comparison, the critical hotspot simulations used 20 zones/µm meshing because: “with slower flame speed, one can get away with a coarser mesh because the heat is spread out” [87].

As has already been mentioned, the size of the computational domain has a significant effect on the character of flame propagation simulations because it controls the rate at which the system pressurises. For example, a simulation similar to that in figure6.20was

Figure 6.21: The effect of mesh resolution on flame propagation simulations in Peruse. The temperature overshoots that occur after the steep temperature rise indicate that the results are not converged at 5 zones/µm. Also, the maximum deflagration rate (which oc- curs at∼2µm radius) increases significantly with mesh resolution from 5 to 40 zones/µm. Convergence is achieved at 40 zones/µm and above, where the temperature histories are smooth and there is no further increase in the maximum deflagration rate.

run withr2reduced from 9 to 1µm. The reactive wave fails to propagate in this geometry because the high pressures generated when the hotspot reacts reduce the temperature of the reaction products, and therefore the rate of thermal conduction into the unreacted HMX. This strong dependence on the size of the computational domain means that a geometry as close as possible to the experimental configuration should be used. In the experiments,r1 = 2.5µm and 150µm ≤ r2 ≤ 400µm. Unfortunately, a computational geometry this big would be prohibitively expensive to run. A compromise geometry with

r1 = 1µm, r2 = 9µm andr3 = 10µm was used to generate results for comparison with the experimental data.