240 I'ropellant5. Explosires. 'kiotechnlcs 16. 210-244 (1991)
Modeling Hydrodynamic Behaviors in Detonation
P. K. 'rang
Applied Theoretical Physics Division. Los Alamos National Laboratory. Los Alamos, NM 87 545 (USA)
Modellierung des hydrodynamischen Verhaltens der Detonation Ein niehrstufiges Reaktionsmodell. das entwickelt wurdc zur
allgeineinen Anwendung von der Ziindung tiis f u r Detonation bei heterogeneo brisanten Sprengstoffen. wird eingesctrl speziell /u r Sirnulierung der Grenztl2chengeschwindigkcit und dcr StoBplat- tcnexpcrirnente bei der Detonation von trianiinotrinitrobenzolhal- tigen Sprengstoffen. Einc Vereinfachung de\ Kombinalionsmod- ells fuhrt ZLI Geschwindigkcitsrclaticlnen. die nur zwei dominante Stufen umfassen: eine schncllc Rclation, die den Hauptanteil der Reaktionen darstellt und durch Ausbreitung und Zcrset/ung bestimmt ist. und eine langsanie. die wahrschcinlich dic Bildung grol3er Kohlenstoffmolekule widerspiegelt. Das schcinbar hohcr cnergetische Verhalten der Zustandsgleichung nahe der Detona- tionsfront ist tatsichlich auf den langsamen ReaktionsprozeB zuriickzufuhren.
Modelisation du cnmportement hydrodynamique de la dbtona- tion
Un moddc de reaction en plusieurs phases. mis au point pour une application generalc allant dc I'amorpge h la detonation d'explo%ifs hkterogenes delicats. cst utilisk specialement pour siinuler la vitcsse interfaciale et les experiences avcc plaques de choc lors de la detonation d'explosifs 3 base de triamino- trinitrobenzol. Une simplification d u modble de combinaison conduit 5 des i-elations dc ITitesses q i i i n'englobent que deux phases dominantes: unc relation rapide qui constitue I'essentiel des reactions et est determinee par la propagation ct la dkcompositiori. ct unc phase lente. qui reflkte vraisemblablement la formation de gl-andes molecules dc carbone. Le comporteinent energktique apparemment plus eleve de I'Cquation d'etat au voisinage du front de detonalion est effectivement dd au processus de reaction l en t .
A multistagc reaction model developcd for general use from initiation through detonation of heterogeneous high explosivcs is used to specifically siniulate the interface velocinietry and plate push experiments of a triamino-trinitrobenzene-based explosive in detonation. A simplification of the unified model leads to a rate relation that includcs only two dominant stages: a fast one that represents the major portion of reaction dictated by propagation and decomposition. and a slow one that reflects probably the formation of large carbon molecules.The apparent more energetic behavior of the equation of state near the detonation front is actually due to the slow reaction process.
I t is known that in simulating the cylinder tests, the initial motion of the metal tube is very sensitive to the Chapman- Jouguet (CJ) pressure used for calculation('); and also that thc equation of state (EOS) based on the cylinder test cannot be applied to another system in general. The problem can be demonstrated by the inability o f the same EOS to predict plate push experiments, noticeably the early motion behavior(2). This dilemma leads many to believe that the EOS depends on system geometry, thus forcing researchers to "normalize" the EOS to fit the empirical result instead of making predictions. The difficul- t ywe believe, is not entirely from the EOS, but rather from the simplification we make of the reaction process in detonation, namely, the usual assumption of very fast reaction.
I t has been observed that some explosives exhibit nonideal or nonsteady dctonation behavior, showing increasing effective CJ pressure with respect lo increasing HE charge length, contrary to the simple detonation theory with unique CJ pressure("-'). This condition arises from the presence of an extended reaction zone, the consequence o f a slow reaction (or dclaycd energy release) found in some HE. The presence of a long time scale in reaction renders more intimate interaction between reaction and hydrody-
namics, and therefore, the effect of detonation seems more sensitive to the system configuration. This sensitivity to local hydrodynamic conditions explains why we have to manipulate the EOS for different applications when we use programmed burn in which the reaction rate is infinite.The subject of nonideal detonation, however. remains contro- versial(h), but with more recent and better experimental evidence. we d o believe in the existence of the phenome- non. For some HE, the origin of the slow reaction process is believed to be caused by solid carbon coagulation(7). In the presence o f a slow process following a fast one, the apparent CJ pressure reflects the condition of the partially reacted (although almost completed) statc of the explosives rather than the final product. This paper concerns the significance of the slow reaction in contributing to the overall hydrodynamic effect in detonation as demonstrated in two types o f experiments and related simulations, without invoking the detailed chemical compositions or kinetics.
2. Reaction Model
We summarize a unified reaction model(8.") in this section. This model consists of three rate equations that govern the energy release in the hot spot. the bulk reaction, and the slow reaction that are assumed to be present in the initiation and detonation of heterogeneous HE. Each of these reactions is characterized by a process time for that particular stage.
The total reaction (or product) fraction A is given by
5, = q h h + (1 - 11 - q)& + 41k, (1) where q is the hot-spot mass fraction and VJ the slow process mass fraction; both are constants. The hot-spot reaction fraction hl, is determined by
0721-31 15/91/OSl0-0240$3.50 + ,2510 0 VCH Verlags_pesell~chaft mhH. D-6910 Weinheirn. 1991
Propellants. Explosives. Pyrotechnics 16. 240 -244 ( lY91)
TI, is the hot-spot process time and is related to thc shock state. the current state, and the chemical kinetic paramet- ers of the HE!",. The bulk reaction ELb is initially controlled by the energy transfer between the hot-spot products and the reniaining reactants: its rate is
Modeling Hydrodynamic Behaviors in Dctonat ion 23 1
In Eq. ( 3 ) , fo is the threshold condition that the hot-spot reaction fraction h h must exceed in order to support bulk reaction. The bulk process time T,, can be written as
which defines a relationship hetween Th and an energy transfer time tc; 5 , is a monotonic decreasing function of pressure("). After the hot spot burn is completed, h h = 1. Eq. (3) has the form of first-order reaction. From then on. the reaction may not he controlled by the energy transfer process; in fact, we find that it is necessary to weaken the dependence of the reaction time on pressure("') and to force ~h having a lower limit. 'Therefore, the following trcatmcnt is proposed:
Tb - maX(Tc, T ' h ) ( 5 )
where T, is a constant repretenting the controlling procew, most likely a condition of chemical reaction. Finally, the rate of the slow process is
where T, is the slow process tinie.The slow process time T, is taken as a constant sincc it is thought to be insensitive t o the thermodynamic state.
For detonation. the hot-spot process can be considered instantaneous. In this case. the rate equations become simpler and the process contains essentially two stages:
11 + (1 - ?I - I$) A,, + ,$A, (7)
Tn the current work to look for the detonation behavior only, Eqs. (7): (S), and (9) are actuallyused in modeling the experiments presented in the next two sections. Also.we do not assume constant ~b because of the rapid change of pressure in the reaction zone; t h e decreasing value o f pressure results in some increase of TI, through T~ as seen in Eq. (5). It should be remembered that the original rate equations are more useful in system applications because they encompass the full range of processes from initiation to detonation. Earlier modeling work on interface veloci- rnetry and plate push used the complete rate equa- tions(' 1.12).
3. Modeling Interface Velocity Experiments
Direct experiments technique with high precision is not yet available to investigate the detonation behavior interior of the HE charge without some degree of interference. We use alternatives such as interface vclocity record to infer the reaction process. Experiments werc performed using a Fabry-Pcrot interferometer to measure interface velocity between explosive and transparent window(l3,.The sy tcms were driven with a plane-wave lens, 25 mm of Cornposi- tion R, and a layer of 10-mm aluminum. Figure l(a) shows the setup. Increasing interface velocity histories with increasing explosive charge lengths were observed for triamino-trinitrobenzene(TATl3)-based explosives. indi- cating nonsteady detonation. The result of such experi- ments forms the basis of an earlier modeling work, suggesting the existence of a slow process near the end of reaction in detonation("). The simulations of these carly experiments were given in Ref. 9. I n the calculations. cases with arid without slow processes are presented, but the simulations with s l o w processes match the cxpcriments much better in general except that the calculations show sharper peak at the detonation front. The peaks actually correspond to the von Neumann spikes that have been detected in separatc experiments for cyclotetramethylene tetranitramine(HMX)-based explosives(lJ) and in HMX- and TATB-based cxplosives(I5). The von Neumann spikes are difficult to calculate accurately using finite difference scheme and artificial viscosity to handle the shock condi- tion. Nevertheless, their existence is observed; the lack of such evide