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Classical Theory of Detonation W Classical Theory of Detonation W. Davis, C. Fauquignon To cite this version: W. Davis, C. Fauquignon. Classical Theory of Detonation. Journal de Physique IV Proceedings, EDP Sciences, 1995, 05 (C4), pp.C4-3-C4-21. 10.1051/jp4:1995401. jpa-00253700 HAL Id: jpa-00253700 https://hal.archives-ouvertes.fr/jpa-00253700 Submitted on 1 Jan 1995 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE IV Colloque C4, suppltment au Journal de Physique In, Volume 5, mai 1995 Classical Theory of Detonation W.C. Davis and C. Fauquignon* Energetic Dynamics Los Alamos, 693 46th Street, Los Alamos, NM 87544, U.S.A. * French-German Research Institute, 5 rue du Gin&ral Cassagnou, 68301 Saint-Louis cedex, France Abstract In a first part is presented the model of the Ideal Detonation. Emphasis is placed on the physical assumptions made in the setting and resolution of the continuum mechanics equations to be used. The basic elements of computations of ideal detonation parameters will be described. The experiments performed to check the predictions, and their results, will be reported. Some discussion of explosives in which chemical equilibrium is not reached, due to slow diffusive mixing of the reactants or to conditions where some of the reactions are very slow, will also be given. These explosives are often called non-ideal explosives. Obviously, when the reaction zone is not very thin relative to system dimensions, or when it is not very short relative to system times, the Ideal Detonation model, restricted to plane, steady flow, is inadequate. The third part of the paper is concerned with curved detonation fronts, and with time dependent processes. The steady detonation of small-diameter cylindrical cartridges will be discussed as one example, and the initiation of detonation in the shock-to-detonation transition as another. Detonation theory is not a closed subject, and much effort is currently being spent to extend classical theory. Some of the newer ideas will be introduced in a concluding part. 1. THERMO-HYDRODYNAMIC DESCRIPTION OF DETONATION (C. Fauquignon) 1.1 Introduction An explosive is defined as a substance which can deliver very quickly a large amount of energy and produce a large amount of gas. The energy has generally a chemical origin and the reactions progress in the surrounding material by drawing a part of the energy previously produced. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1995401 C4-4 JOURNAL DE PHYSIQUE IV According to the processes of the energy transfer in the non reacted material, two regimes are defined, deflagration and detonation, which are differentiated from the usual combustion by their velocities of propagation (some mm/s for the usual combustion, several hundred rn/s for the deflagration and several thousand m/s for the detonation). It follows that the flow of the high pressure reaction products is the acting agent of the energy transfer and of the self sustaining of the combustion. As a consequence, the description of the process calls in the fluids mechanics associated with the thermo-chemistry for the setting of the energy balance. In this presentation, we will only consider the case of the detonation in which, contrary to the deflagration, the energy transfer in the non reacted material is supersonic. The detonation is characterised by the existence of a shock wave in the front of which the reactions start. The model of the Ideal thermo-hydrodynamic Detonation which will be presented does not take into account the physico-chemical processes in the shock front or in its neighbourhood and which may require revision of the assumption of the discontinuity of the shock front considered as the frontier between two states at thermodynamic equilibrium. 1.2 Model of the Ideal Detonation The model considers (Figure l a) a reactive supersonic flow, self sustained by the shock wave, steady as far as the end of the reactions : all the exo-energetic reactions are assumed to reach an equilibrium at the same time, this equilibrium being a function of the temperature and of the pressure. Beyond this point, an unstationnary rarefaction takes place being controlled by the boundary conditions of the explosive charge. In this downstream flow,the composition of the products may change as a function of p, V, T, but no energy can be transfered upstream and influence the characteristics of the detonation. The set of the conservation equations to be solved in building the model neglect the conductive and radiation effects and assume a lamina non viscous flow. The steady state condition means that the shock front and the plane of the end of reaction propagate at the same velocity D in a plane semi-infinite geometry. The indexes 0 and 1 correspond respectively to the non reacted and totally reacted material. If we consider that the plane of the end of reaction is the head of an unstationnary rarefaction, the condition of a steady state upstream is fulfilled if this plane is sonic, D = Q + 6, where Q and fi are the material velocity and the sound speed at this plane. This condition was found by Chapman [l] and Jouguet [2] who have shown its consequences in the thermodynamic (p, v) plane (Figure lb). In the (p, v) plane, 0 represents the fresh explosive and the locus of the possible final states of the reaction products is the curve (H) obtained by solving the conservation equations : D D v EXPLOSIVE REACT1ON (NON-STEADY (STEADY) I PRODUCTS ZONE I EXPANSION) (1) SHOCK EQUILIBRIUM FRONT OF REACTIONS FIG. la : SPATIAL SCHEME PRESSURE P - PO D* (D - u)~ .>RAYLEIGH LINE (R) ----p =- p l '\\ ,. v-v, v,' v* CHAPMAN-JOUGUET POINT (CJA) I I I '.l0 POl A VOLUME v v0 FIG. lb : CHAPMAN-JOUGUET MODEL (p, V) VOLUME v* v0 FIG. 1c : COMPLETE DESCRIPTION (p, V) FIG. 1 : IDEAL THERMO-HYDRODYNAMIC DETONATION C4-6 JOURNAL DE PHYSIQUE IV The basic difference from an Hugoniot for non-reacted explosive is in the meaning of E = E, = E', (internal energy of the fresh material + qo (energy of formation of the explosive molecule) E =E' (internal energy of the reaction products + q (energy of formation of the products) As a consequence (po, vo) is not on (H) and the ordinate on (H) for V = V. is a point B such that pg - p, is the increase of the pressure for a constant volume reaction at V = V,. By solving the conservation equations it is possible to express the velocities D, u as functions of p and V: As a consequence, any final state on (H) corresponds a velocity D as a result of the intersection of (H) and of (R) called the Rayleigh line. It is shown that the steady state condition for the Ideal Detonation is fulfilled when (R) is tangent with H; At the tangent point, called CJ point, (R) is also tangent with the isentrope S a2 the slope of which is - p As a consequence, at the CJ point It is also interesting to note that the CJ detonation velocity is the minimal velocity which can be reached. Until now, the reaction zone has been neglected or assumed to have no thickness. It will be seen later (see part 2) that the present description of the Ideal Detonation is sufficient for computation of the detonation parameters. 1.3 A more complete description : The ZND model A description of the complete structure of detonation with propagation of the shock in the fresh explosive has been proposed quasi-simultaneously in 1942 by Doering, Zeldovitch and von Neumann [3], [4], [5]. The construction in the (p, v) diagram needs to know the Hugoniot H, of the non reacted explosive (Figure lc). The steady state condition and the use of the Rayleigh line (R) show that (po, v0), @*,V*) and D2 (p, d) are aligned on a straight line of slope equal to -vIn the reaction zone the pressure drops steeply from a maximum value p*, called the ZND spike, to the CJ pressure p.A The simplest interpretation of the model would be to calibrate the p* -$segment in the extent of h reaction I, from I = 0 at p* to L= 1 at p. However, we enter here into the most controversial part of the model as we will see now. 1.4 First consideration on the limits of the model The following considerations deal with the simplicity of the physical assumptions and with the reality of one dimensional plane and semi-infinite geometry. a) The existence of a non-reacted Hugoniot is questionnable if out of equilibrium effects and early decomposition occur. The main consequence would be to modify the reaction scheme and would have a major influence in the build-up to detonation following a shock of a short duration. b) The validity of the assumption of a single reaction rate and of the absence of diffusion effects between the species has been roughly considered in the comparison between computations and experiments and is used to divide the explosives in non-ideal and ideal compositions. This point will be examined in part 3 at the same time as the coyncidence of a sonic state with the end of the exo-energetic reactions.
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