Introduction

Deflagration to detonation transition (DDT) is an important character of explosives. Bernecker et al. [1] summarized former studies and separated DDT process into seven stages. They are preignition, conductive burning, convective burning, hot spot burning, shock formation, compressive burning and detonation. The whole seven stages do not appear in all DDT process. Depending on the explosive type, binding strength or ignition condition, some stages might be dominant and the others not obvious. So far, the numerical studies of DDT are mainly based on the two-phase mixture model [2–6]. The model allows for unbalanced phase velocities and phase temperature and includes source terms for drag and convective heat transfer. These studies mostly focused their attention on the transition process from convective burning to detonation by presuming that the combustion starts with strong ignition. However, the combustion of explosives mostly starts with conductive burning under thermal ignition condition. In this paper, we intend to study this DDT process in granular explosives by numerical simulation.

The main challenge in the numerical simulation of two-phase detonation models is the method of capturing the strong discontinuity in the field. The space–time conservation element and solution element (CE/SE) method [7] is applied to solve the governing equations. This method provides a new way to solve the hyperbolic conservation equation. It has many features differing from the existing well-established schemes. The CE/SE method satisfies the local and global flux conservation in space and time and needs simple treatments of boundary conditions. It has been successfully applied to the calculation of mathematical model of DDT in our previous work [8–10].

The paper is organized as follows. In Sect. 2, we give the model equations and perform an analysis to simplify the simulation. Section 3 is devoted to briefly describing the CE/SE scheme and further validating the numerical code. In Sect. 4, we investigate the DDT process which starts with conductive burning under thermal ignition. Based on the simulation, we then study the effect of particle diameter on the transition behavior. In Sect. 5, we give the summary and conclusion of this paper.

Physical model

Governing equations

The governing equations are based on the two-phase flow model of Refs. [3–5]. It is assumed that the gas and solid phases occupy a given spatial position simultaneously. The corresponding local volume fraction sums to unity, that is

$$\varphi_{1} + \varphi_{ 2} = 1$$
(1)

where the subscripts 1 and 2 denote the gas and the solid, respectively.

The conservation equations of mass, momentum and energy are described according to the associated interphase transfers. The mass conservation equations of gas and solid are:

$$\frac{{\partial (\rho_{1} \varphi_{1} )}}{\partial t} + \frac{{\partial (\rho_{1} \varphi_{1} u_{1} )}}{\partial x} =\Gamma$$
(2)
$$\frac{{\partial (\rho_{2} \varphi_{2} )}}{\partial t} + \frac{{\partial (\rho_{2} \varphi_{2} u_{2} )}}{\partial x} = - \varGamma$$
(3)

where ρ is the density and u the velocity. Γ is the rate of gas generation due to solid combustion and given by Γ = ρ 2 φ 2 rA sp h(T 2 − T ig). Here r is the burn rate of solid explosive, and A sp is the total specific surface of combustion. In conductive burning process, the flame remains at the surface of the explosive and proceeds layer-by-layer, and A sp is equal to the cross-sectional area. In convective burning process, the reaction rate is defined by the individual explosive grains, and A sp is equal to 6/d for the spherical particles with diameter of d. h(T 2 − T ig) is an on–off function, if T 2 < T ig h = 0, otherwise h = 1. T ig is the ignition temperature of explosive.

The momentum conservation equations of gas and solid are:

$$\frac{{\partial (\rho_{1} \varphi_{1} u_{1} )}}{\partial t} + \frac{{\partial (\rho_{1} \varphi_{1} u_{1}^{2} + p_{1} \varphi_{1} )}}{\partial x} =\Gamma u_{2} - D$$
(4)
$$\frac{{\partial (\rho_{2} \varphi_{2} u_{2} )}}{\partial t} + \frac{{\partial (\rho_{2} \varphi_{2} u_{2}^{2} + p_{2} \varphi_{2} )}}{\partial x} = - \varGamma u_{2} + D$$
(5)

where p is the pressure and D is the drag force due to relative motion of gas and solid phase. In conductive burning process, the drag force is ignored. In convective burning process, D = 4μ 1(u 1 − u 2)f pg/d 2 is used. Here μ 1 is the gas viscosity, and f pg = φ 2/φ 21 (276 + 5(Re/φ 2)0.87) is a dimensional drag coefficient in which the Reynolds number is defined as Re = ρ 1|u 1 − u 2|d/μ 1.

The energy conservation equations of gas and solid are:

$$\frac{{\partial (\rho_{1} \varphi_{1} E_{1} )}}{\partial t} + \frac{{\partial [\rho_{1} \varphi_{1} u_{1} (E_{1} + {{p_{1} } \mathord{\left/ {\vphantom {{p_{1} } {\rho_{1} }}} \right. \kern-0pt} {\rho_{1} }})]}}{\partial x} = \frac{\partial }{\partial x}\left( {\lambda_{1} \frac{{\partial T_{1} }}{\partial x}} \right) + \varGamma (E_{2} + q_{\text{chem}} ) - Du_{2} - Q$$
(6)
$$\frac{{\partial (\rho_{2} \varphi_{2} E_{2} )}}{\partial t} + \frac{{\partial [\rho_{2} \varphi_{2} u_{2} (E_{2} + {{p_{2} } \mathord{\left/ {\vphantom {{p_{2} } {\rho_{2} }}} \right. \kern-0pt} {\rho_{2} }})]}}{\partial x} = \frac{\partial }{\partial x}\left( {\lambda_{2} \frac{{\partial T_{2} }}{\partial x}} \right) - \varGamma E_{2} + Du_{2} + Q$$
(7)

where E (=e + u 2/2) is the total specific energy, T the temperature, λ the thermal conductivity, q chem the heat of combustion. Q is the convective heat transfer between gas and solid given by Q = 6φ 2(T 1 − T 2)h pg/d. Here h pg = 0.65 kgRe0.7Pr0.33/d is a heat transfer coefficient, k g is the gas thermal conductivity, and Pr is the Prandtl number.

The following equation is used to determine the volume fraction change:

$$\frac{{\partial \varphi_{2} }}{\partial t} + u_{2} \frac{{\partial \varphi_{2} }}{\partial x} = \frac{{\varphi_{1} \varphi_{2} }}{{\mu_{\text{c}} }}\left[ {p_{2} - p_{1} - \frac{{\varphi_{2} }}{{\varphi_{20} }}(p_{20} - p_{10} )} \right] - \frac{\varGamma }{{\rho_{2} }}$$
(8)

where the subscript 0 denotes the initial state, μ c the compaction viscosity. The equation shows that the volume fraction is affected by the compression and combustion process.

A particle number density evolution equation is used to assure that the total number of particle in the system is conserved.

$$\frac{\partial n}{\partial t} + \frac{{\partial (nu_{2} )}}{\partial x} = 0$$
(9)

where n is the particle number density. This equation denotes that no particle agglomerates or breaks up.

Equations of states (EOS) for gas phase in JWL form and for solid phase deduced from Helmholtz free energy are given below:

$$p_{1} = { A\text e}^{{ - \text R_{1} \rho_{10} /\rho_{1} }} + { B}{\text e}^{{ - \text R_{2} \rho_{10} /\rho_{1} }} + \omega \rho_{1} c_{{{\text{v}}1}} T_{1}$$
(10)
$${\text e_{1}} = \frac{A}{{R_{1} \rho_{10} }}{\text e^{{ - \text R_{1}} \rho_{10} /\rho_{1} }} + \frac{B}{{R_{2} \rho_{10} }}{\text e^{{ - \text R_{1} \rho_{10} /\rho_{1} }}} + c_{{{\text{v}}1}} T_{1}$$
(11)
$$p_{2} = C_{2} (T_{2} - T_{20} )\frac{{\gamma \rho_{2}^{2} }}{{\rho_{20} }} + \frac{{K_{\text{T}} }}{N}\left[ {\left( {\frac{{\rho_{2} }}{{\rho_{20} }}} \right)^{\text N} - 1} \right]$$
(12)
$$e_{2} = C_{2} (T_{2} - T_{20} ) - C_{2} T_{20} \gamma \left[ {\left( {\frac{{\rho_{2} }}{{\rho_{20} }}} \right) - 1} \right] + \frac{{K_{\text{T}} }}{{\rho_{20} N(N - 1)}}\left[ {\left( {\frac{{\rho_{2} }}{{\rho_{20} }}} \right)^{\text N - 1} - (N - 1)\left( {1 - \frac{{\rho_{20} }}{{\rho_{2} }}} \right) - 1} \right]$$
(13)

where C v1 and C 2 are the specific heat of gas and solid phase, respectively. A, B, R 1, R 2, ω, γ, K T and N are the constant parameter of the EOS.

Under thermal ignition, the combustion of explosives starts with conductive burning. The criterion of transition from conductive to convective burning is very important. Margolin and Chuiko [11] derived a simplified and approximate criterion based on previous works to study whether penetration of a single pore by hot gas will result in ignition.

$${\text{An}} = \frac{{\rho_{2} C_{2} }}{{\lambda_{2} }}rd_{\text{pore}} \ge {\text{An}}^{*}$$
(14)

where the d pore is the pore diameter. The convective burning forms when An exceeds a critical value An*. From a collection of experimental data for various explosive powders, Margolin and Chuiko determined An* = 6. For the porous bed fill with spherical gains, we set the hydrodynamic diameter be d pore = 2dφ 1/3(1 − φ 1).

Conductive heating is much slower than convective heating in the combustion of explosives. The time scale of numerical simulation might conflict between conductive and convective burning. So we separated the DDT process into two steps. Before the onset of convective burning, we solved the conductive burning process. And after that we solve the convective burning to detonation transition process.

Simplification of conductive burning

For the conductive burning, experiments [12, 13] showed that the burn rate of solid explosive obeys the empirical formula r = ap m well, where a and m are constant parameters. So we can catch the conductive burn front instead of solving the complex model equations.

A level set function ψ(x) is introduced tracing the conductive burning. The ψ(x) defines the signed distance to the burn front. That is ψ(x) = 0 on the front, ψ(x) = −|δ| in the gas production zone and ψ(x) = |δ| in the unreacted zone. Here |δ| is the distance from point x to the burn front. The evolution equation could be written as

$$\frac{\partial \psi }{\partial t} + r\frac{\partial \psi }{\partial x} = 0$$
(15)

The change rates of volume, mass and total energy of the gas phase behind burning front can be given by:

$$\frac{{{\text{d}}V_{\text{g}} }}{{{\text{d}}t}} = rA$$
(16)
$$\frac{{{\text{d}}M_{\text{g}} }}{{{\text{d}}t}} = \rho_{2} \varphi_{2} rA$$
(17)
$$\frac{{{\text{d}}E_{\text{g}} }}{{{\text{d}}t}} = \rho_{2} \varphi_{2} rA(E_{2} + q_{\text{chem}} )$$
(18)

where V g denotes the volume, M g the mass, E g the total energy, and A the cross-section area of the DDT tube. The evolutions of V g(t), M g(t) and E g(t) can be easily solved by integrating the equations with given initial conditions.

A further estimate of the run-distance of conductive burning in DDT is performed. For granular HMX explosive with 100 μm diameter and 85 % loading density, the thermal diffusivity λ 2/ρ 2 C 2 is about 10−7 m2s−1 and the pore diameter d pore is on the order of:

$$d_{\text{pore}} = \frac{{2d\varphi_{1} }}{{3(1 - \varphi_{1} )}} = \frac{{2 \times 100 \times 10^{ - 6} \times 15\% }}{3 \times 85\% } \approx 10^{ - 5}\; {\text{m}}$$

According to the criterion Eq. (14), the burn rate while the transition to convective burning is

$$r_{\text{trans}} \ge {\text{An}}^{*} \frac{{\lambda_{2} }}{{\rho_{2} C_{2} }}\frac{1}{{d_{\text{pore}} }} \approx 6 \times 10^{ - 7} \frac{1}{{10^{ - 5} }} = 0.06\;{\text{m s}^{-1}}$$

The time and spatial scales of DDT experiments [14, 15] are on the order of 50 ms and 500 mm. Multiply r trans by the time, we can obtain that the run-distance of conductive burning is <3 mm, which is very short compared with the spatial scale. So the gas production is assumed in equilibrium in conductive burning process, and the velocity, density and specific internal energy of gas are given by

$$u_{1} (t) = 0$$
(19)
$$\rho_{1} (t) = M_{\text{g}} (t)/V_{\text{g}} (t)$$
(20)
$$e_{1} (t) = E_{\text{g}} (t)/M_{\text{g}} (t)$$
(21)

The gas-phase evolution can be obtained by calculating Eqs. (16–21) coupling with EOS (10–11).

The compaction effect of conductive burning on the unreacted area of the explosive is ignored due to the low pressure and short run-distance. Thus, only heat conduction equation is used in the solid phase.

$$\rho_{2} C_{ 2} \frac{{\partial T_{2} }}{\partial t} + \rho_{2} C_{ 2} r\frac{{\partial T_{2} }}{\partial x} = \frac{\partial }{\partial x}\left( {\lambda_{2} \frac{{\partial T_{2} }}{\partial x}} \right)$$
(22)

The boundary conditions are:

$$T_{2} = T_{\text{s}} ,\,{\text{at}}\quad \psi (x) = 0$$
(23)
$$T_{2} = T_{ 2 0} ,{\text{at}}\quad \psi (x) = \infty$$
(24)

Here T 20 is the initial temperature and T s the burning surface temperature of explosive. In principle, the temperature evolution can be obtained by solving Eqs. (22–24). For simplicity, the analytical solution of temperature at steady state is given below.

$$T_{2} = T_{ 20} + (T_{\text{s}} - T_{ 20} )\exp \left[ { - \frac{r}{{{{\lambda_{2} } \mathord{\left/ {\vphantom {{\lambda_{2} } {\rho_{2} C_{2} }}} \right. \kern-0pt} {\rho_{2} C_{2} }}}}\psi (x)} \right]$$

As both the melting and boiling point temperatures of HMX are below 600 K, we supposed T s = 600 K to investigate the temperature penetration depth. According to the experiment of Son et al. [12] and the model of Ward et al. [16], the burn rate of HMX at 2 atm is about 0.0007 m s−1. Considering the above transition burn rate (r trans ≈ 0.06 m s−1) additionally, we calculated the temperature distribution with the burn rates 0.0006, 0.006 and 0.06 m s−1. Figure 1 shows the temperature profile of steady deflagration in the explosive. Along with the decreasing burn rate, the penetration depth of heat conduction decreases from 0.8 to 0.1 mm and even to 0.01 mm. Outside the “depth,” the temperature is T 20. Under unsteady deflagration condition, the temperature penetration depth will decrease with the development of burning and be much shorter than that of steady deflagration. So the temperature in unreacted area is assumed to be T 20. That also accords with the common sense of slow heat conduction in explosives.

Fig. 1
figure 1

Temperature profiles of steady deflagration in explosive with various burn rates

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CE/SE scheme and code validation

The CE/SE scheme is a unique centered scheme. Take the convection equation \({{\partial U} \mathord{\left/ {\vphantom {{\partial U} {\partial t}}} \right. \kern-0pt} {\partial t}} + {{\partial F(U)} \mathord{\left/ {\vphantom {{\partial F(U)} {\partial x}}} \right. \kern-0pt} {\partial x}} = 0\), for example, the CE/SE scheme treats space and time as one entity. By considering t as a coordinate with equal status with x, the above equation can be written in divergence form as \(\nabla \cdot \vec{h} = 0\), where \(\vec{h} = (F,U)\) is the space–time flux vector. Utilizing Gauss divergence theorem, the equation can be written in the integral form \(\oint\nolimits_{S(V)} {\vec{h} \cdot {\text{d}}\vec{s}} = 0\), where S(V) is the boundary of V. The integral equation can be applied to any closed space–time region. The concept of the solution element (SE) and the conservation element (CE) is introduced. In the SE, variables are continuous. The integral equation is applied over the CE. Details are illustrated in Ref. [7]. Figure 2 is the staggered space–time mesh of the computational region. The CE/SE scheme reads

$$\begin{aligned} U_{\text{j}}^{\text{n}} & = \frac{1}{2}\left[ {U_{{{\text{j}} - 1/2}}^{{{\text{n}} - 1/2}} + \frac{\Delta x}{4}(U_{\text{x}} )_{{{\text{j}} - 1/2}}^{{{\text{n}} - 1/2}} + U_{{{\text{j}} + 1/2}}^{{{\text{n}} - 1/2}} - \frac{\Delta x}{4}(U_{\text{x}} )_{{{\text{j}} + 1/2}}^{{{\text{n}} - 1/2}} } \right] \\ & \quad + \,\frac{1}{2}\frac{\Delta t}{\Delta x}\left[ {F_{{{\text{j}} - 1/2}}^{{{\text{n}} - 1/2}} + \frac{\Delta t}{4}(F_{\text{t}} )_{{{\text{j}} - 1/2}}^{{{\text{n}} - 1/2}} - F_{{{\text{j}} + 1/2}}^{{{\text{n}} - 1/2}} - \frac{\Delta t}{4}(F_{\text{t}} )_{{{\text{j}} + 1/2}}^{{{\text{n}} - 1/2}} } \right] \\ \end{aligned}$$

where U x and F t denote spatial and temporal gradients of the variables.

Fig. 2
figure 2

Staggered space–time mesh of the CE/SE scheme

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The spatial gradients of the variables can be obtained simultaneously.

$$(U_{\text{x}} )_{\text{j}}^{\text{n}} = \frac{{\left| {U_{\text{x}}^{ + } } \right|^{\upalpha} U_{\text{x}}^{ - } + \left| {U_{\text{x}}^{ - } } \right|^{\upalpha} U_{\text{x}}^{ + } }}{{\left| {U_{\text{x}}^{ + } } \right|^{\upalpha} + \left| {U_{\text{x}}^{ - } } \right|^{\upalpha} }}$$

where \(U_{\text{x}}^{ \pm } = \pm \frac{{U_{{{\text{j}} \pm 1/2}}^{{{\text{n}} - 1/2}} + \frac{\Delta t}{2}(U_{\text{t}} )_{{{\text{j}} \pm 1/2}}^{{{\text{n}} - 1/2}} - U_{\text{j}}^{\text{n}} }}{\Delta x/2}\) and \(\alpha = 1\sim2\). The other gradients, such as F x, U t and F t, can be calculated by the chain rule and conservation equations.

To validate our numerical code, the DDT phenomenon in which numerical results are available in open literature [17, 18] is simulated. For the sake of similarity, all the physical parameters are the same with that of literature. In the study, a 0.5-m long HMX explosive bed packed to 73 % of theoretical maximum density (TMD) with uniform particles of 200 μm diameter is considered. The transition from deflagration to detonation occurs due to the compression of a moving piston with velocity of 100 m s−1. In present calculation, a mesh resolution of 2000 nodes is used. Figure 3 shows the gas phase velocity histories. Initially, a compaction wave propagates away from the piston. At about t = 25 μs, ignition occurs near the piston surface, and a sharp increase in the gas velocity from 100 to 750 m s−1 is observed. The ignition time agrees well with the 26 μs of Narin et al [17]. As the combustion of particles, a high-pressure region forms. The combustion then accelerates due to the increasing gas pressure and undergoes transition to detonation. The detonation wave propagates at a faster speed and overtakes the compaction wave near x = 0.16 m at t = 50 μs. The result is in a very good agreement with that of Gonthier and Powers [18]. Finally, a steady detonation with a speed of 7496 m s−1 is observed. All the results are listed in Table 1.

Fig. 3
figure 3

Numerical results of the gas velocity

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Table 1 Comparison of present results with those of literature
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Numerical results

Using above physical model and numerical method, the DDT process in a 200-mm long HMX bed packed to 85 %TMD with uniform particles of 100 μm diameter is calculated. The numerical mesh consists of 500 uniform nodes. The physical parameters are ρ 1 = 1.0 kg m−3, φ 1 = 0.15, T 1 = 300 K, ρ 2 = 1890 kg m−3, φ 2 = 0.85, T 2 = 300 K, d = 100 μm, a = 2.9 × 10−9 mPa−1 s−1, m = 1, λ 2 = 0.345 W m−1 K−1, μ c = 1000 kgm−1 s−1, q chem = 5.84 × 106 J kg−1, T ig = 450 K. The EOS parameters are A = 220.3 GPa, B = 15.4 GPa, R 1 = 4.5, R 2 = 1.5, ω = 0.28, C v1 = 2400 J kg−1 K−1, N = 9.8, γρ 2 = 2100 kg m−3, K t = 13.5 GPa, C 2 = 1050 J kg−1 K−1. A high gas temperature zone over a short distance of 0 ≤ x ≤ 20 mm is assumed to ignite the explosive. That is φ 1 = 1.0, T 1 = 1000 K and p 1 = 0.8 MPa when 0 ≤ x ≤ 20 mm.

Figure 4 shows the gas pressure histories in the DDT process. After ignition at left, the pressure slowly increases due to the slow conductive burning. This process retains a relatively long time. After about 8.16 ms, the onset of convective burning occurs when the gas pressure exceeds about 30 MPa. The explosive porosity is penetrated by flame, and the burn rate becomes significantly fast. It causes a sharp rise in pressure near the burn front. Following the onset of convective burning, the burn rate increases with gas pressure, continued pressurization occurs, and a steady detonation with the peak pressure of 38 GPa quickly forms within 0.01 ms. Figure 5 shows the propagation of burn front during the DDT process. In the early stage, the combustion wave moves very slow, no more than 0.2 mm within 8.16 ms. After onset of convective burning, the flame moves faster and faster and quickly forms a steady detonation with the speed of 8165 m s−1.

Fig. 4
figure 4

Gas pressure during the transition from conductive to convective burning

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Fig. 5
figure 5

Propagation of burn front during the DDT process

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To study the effects of porosity, the process with different particle diameters is further calculated. Figure 6 shows the burn front propagation and peak gas pressure with different particle diameters. The run-time to detonation increases from 6.6 to 13.5 ms as the particle diameter decreases from 200 to 10 μm. From the Andreev criterion Eq. (9), the burn rate formula, the experiential pore diameter formula, the critical transition pressure transition from conductive to convective burning can be obtained, that is \(p_{\text{trans}} = \sqrt[{\text{m}}]{{3\lambda_{2} {\text{An}}^{*} (1 - \varphi_{1} )/2a\rho_{2} C_{2} \varphi_{1} }} \cdot d^{{ - 1/{\text{m}}}}\). As the particle diameter decreases, the decreased pore is more difficult to penetration. It can be seen from Fig. 6b that the critical transition pressure increases from 15 to 300 MPa. In our simulation, it is supposed that the explosive is confined to a rigid tube. In this case, the tube wall can completely supply the necessary confinement for the DDT process. So the DDT can always occur. However, there will be a limiting confinement in actual tube. For slow pressurization, the yield strength of a steel tube may be on the order of 100 MPa. In that case, if the rupture pressure of the wall is less than the critical transition pressure, the tube wall will rupture before the onset of convective burning, the combustion will slow down, or even quench due to the sudden pressure drop. As a result, the DDT cannot occur because there is insufficient confinement to support the rapid combustion process.

Fig. 6
figure 6

a Burn front propagation and b peak gas pressure with various particle diameters

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Conclusions

This paper has studied the DDT process starting with conductive burning in granular explosives. A homemade code based on the CE/SE scheme is used to simulate the mathematical model. A piston-driven DDT phenomenon is first calculated to validate the code. The present results are in a very good agreement with those of the references. The DDT process under thermal ignition is then simulated. The development of conductive burning, convective burning and detonation is analyzed. The effects of particle diameter on the transition behavior are discussed also. The run-time to detonation increases with the decrease in particle diameter.