Steady-state model of detonation

Under the assumption of a steady-state model of detonation, the EXPL05 program allows the calculation of the detonation parameters and also the chemical composition of the system at the C-J point. For the calculations, the BKW equation of state is applied for gases, where X, is the mole fraction of the i-th gaseous component and kj is the molar covolume of the i-th gaseous detonation products ... 

The Becker-Kistiakowsky-Wilson (BKW) equation of state described in Appendix E is the most used and best calibrated of those used to calculate detonation properties assuming steady-state and chemical equilibrium. Comparison of the calculated and experimental detonation properties permits evaluation of the errors to be expected from steady-state modeling of detonation products. Table 2.1 lists the calculated and experimental C-J properties of various explosives and mixtures. The calculated detonation product compositions of some of the explosives are given in Table 2.2. 

As previously stated, this discussion is valid for homogeneous explosives, such as the ones used in the military, since their reactions are predominantly intramolecular. Such explosives are often referred to as ideal explosives, in particular when they can be described using the steady state model of Chapman and Jouguet. In heterogeneous explosives (non-ideal), which are currently used in civil applications, intermolecular (diffusion controlled) mechanisms are predominant for the air bubbles, cavities or cracks (etc.). As a general rule detonation velocities increase proportional to the diameter. 

Structure of a steady-state plane detonation wave with finite reaction zone) (It is an analytical elaboration of von Neumann s model of detonation wave)... 

In this chapter we will only consider the ideal detonation case. We shall start by examining a simple model of detonation. We will then go on to see methods for estimating steady-state detonation parameters and from these to estimating the Hugoniots of detonation product gases. We will then look at interactions of detonation waves with other materials with which the explosive is in contact. 

According to the ZND model of detonation, a steady-state plane detonation wave has the pressure-time profile shown in Figure 4.32. 

The assumption of stable, one-dimensional detonations is not valid when one considers small-scale details, as Chapter 1 shows. However, although steady-state theory is invalid on a microscale, it does provide an excellent first approximation and a very useful aid in detonation performance calculations. Assumptions of chemical equilibrium in the steady-state model are incorrect. One of the interesting problems in modeling the equation of state of detonation products is finding reasonable changes in the detonation product composition which will reproduce the experimentally observed explosive and propellant performance. 

Also, because initiation of homogeneous explosives results in overdriven detonations in the practical case, they will not exhibit build-up. The nitromethane experimental data seem to scale and to be adequately described by a steady-state model. The nitromethane overdriven detonation may decay to a steady-state detonation or may decay to a flow that continues to be time dependent (oscillates), perhaps requiring greater experimental resolution to detect. 

The velocity of advance of the front is super sonic in a detonation and subsonic in a deflagration. In view of the importance of a shock process in initiating detonation, it has seemed difficult to explain how the transition to it could occur from the smooth combustion wave in laminar burning. Actually the one-dimensional steady-state combustion or deflagration wave, while convenient for discussion, is not easily achieved in practice. The familiar model in which the flame-front advances at uniform subsonic velocity (v) into the unburnt mixture, has Po> Po> an[Pg.249]

Pugh et al. In confined charges, the steady-state detonation head, should, in this model, be somewhat larger because confinement would lower at least the initial velocity of... 

Reynolds number, p 46), etc 61-72 (Shock relationships and formulas) 73-98 (Shock wave interactions formulas) 99-102 (The Rayleigh and Fanno lines) Ibid (1958) 159-6l(Thermal theory of initiation) 168-69 (One-dimensional steady-state process) 169-72 (The Chapman-Jouguet condition) 172-76 (The von Neumann spike) 181-84 (Equations of state and covolume) 184-87 (Polytropic law) 188, 210 212 (Curved front theory of Eyring) 191-94 (The Rayleigh transformation in deton) 210-12 (Nozzle thepry of H. Jones) 285-88 (The deton head model) ... 

The model used in Ref 36a is a compressible, non-dissipative, adiabatic, reactive medium into which a detonation wave is propagated in steady state. The medium is a cylinder of indefinite length ... 

In confined chges the steady-state deton head should, in this model, be somewhat larger because confinement would lower at least the initial velocity of the release Waves from the side. The detonation-head development and its steady-state fotm in confined and unconfined chges are illustrated in Fig 5.1 of Ref 52, p 92 (which is also reproduced here) taking into account the spherical shape of wave front... 

Preceding discussions (in this chapter and Chapter 8) relate to problems attending the measurement of propagation rates in heavy-metal azides. The alternative of calculating the maximum, steady-state detonation rates in these azides from first principles is not well established. Nevertheless, a one-dimensional thermohydrodynamic model does exist which can yield reasonable values for detonation properties (velocity, pressure, product density and composition, particle velocity, etc.) [113-119]. 

A classic example of observed explosive properties which can not be described by the usual steady-state chemical equilibrium models is the detonation velocity of TNT as a function of density reported by Urizar, James, and Smith. The velocity is plotted as a function of density in Figure 2.1. It has a sharp change of slope of 3163 to 1700 m/sec/g/cc at 1.55 g/cc. This problem is discussed in more detail in the section on carbon condensation. 

There are clues, to be discussed later in this chapter, indicating the processes that result in failure of the steady-state detonation model. They are currently of little value to the numerical engineer who wishes to treat an explosive as realistically as practicable. He would like a description of the explosive behavior that would work in divergent and convergent geometry as well as in plane geometry. The study is limited to one-dimensional flow. 

The build-up model assumes that a real nonsteady-state detonation can be approximated adequately by a series of steady-state detonations with instantaneous reaction whose effective C-J pressures vary with the distance of run. This empirical model depends completely upon experimental data for its calibration. If the magnitude or duration of the initiating pulse is changed, or the the explosive is one for which experimental data are not available, new experimental data must be generated and the model must be calibrated for the new system. 

To obtain a rough physical understanding of the mechanism of the instability, attention may be focused first on a planar detonation subjected to a one-dimensional, time-dependent perturbation. Since the instability depends on the wave structure, a model for the steady detonation structure is needed to proceed with a stability analysis. As the simplest structure model, assume that properties remain constant at their Neumann-spike values for an induction distance after which all of the heat of combustion is released instantaneously. If v is the gas velocity with respect to the shock at the Neumann condition, then may be expressed approximately in terms of the explosion time given by equation (B-57) as Z = vt. From normal-shock relations for an ideal gas with constant specific heats in the strong-shock limit, the Neumann-state conditions are expressible by v/vq = po/p —... 

Clavin [5] performed quasi-steady analysis of the direct initiation process. They developed the critical curvature model, which states that the failure mechanism of the detonation is mainly caused by the nonlinear curvature effect of the wave front. Eckett et al. [6] proposed the critical decay-rate model and pointed out that the critical mechanism of a failed detonation initiation process is due to the unsteadiness of the reacting flow. Their theory for spherical detonation initiation has been supported by numerical simulation and experimental data. 

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