D. Desbordes- Critical Initiation Conditions for Gaseous Diverging Spherical Detonations

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JOURNALDE PHYSIQUE TVColloque C 4, supplkment au Journal de Physique III,Volume 5, mai 1995Critical Initiation Conditions for Gaseous Diverging Spherical DetonationsD. DesbordesLaboratoire dlEnergktique et de Dktonique, URA 193 du CNRS, ENSMA, Site du Futuroscope, BP . 109,8696 0 Futuroscope cedex, France

Abs t rac t : he diverging spherical detonation wave in gaseous explosive s is obtained eitherwith a point source of explosion of energy E or through the transmission of a plane detonationfrom a cylindrical tube of diameter d into a large volume. The mechanism of detonationinitiation in both cases is based on the shock to detonation transition.The experimen tal criticalconditions lead to an initiation criterion for detonation resulting from the competition betweenthe expansion behind the leading shock wave on one hand and the shock-induced chemical heatrelease on the other. Whatever the type of ignition source, the detonation is obtained when theradius of curvature of the wave overcom es a particular critical value& whose si ze includes aconstant and large number of cell width hcj (R, 20 hcj) and then can be considered asintrinsic to the detonative mixture used.kj, hich is the m ean size of the cellular structure of aCJ detonation, is proportional to the global chemical induction length Li, calculated in the ZNDscheme, by also a large factor (generally more than 10). Two o ther criteria define the criticalinitiation energy Ec and the critical tube diameter dc for ob taining detonation with respect to thisintrinsic length& .

1. I N T R O D U C T I O NTh e direct initiation of the diverging spherical detonation in gaseous explosives, is based on the

so-called shock to detonation transition and has been generally studied via two modes of ignition. Thefirst mode cons ists in the diffraction of a planar C J or overdriven detonation wave from a tube of i.d. d(or from orifices of different shapes) into a large volume [ l] to [7].Th e second mode is the deposition inthe medium , by a sufficiently powerful1 point source of explosion, of an energy E creating quasi-instantaneously a strong decaying spherical shock wav e [l],[g] to [ ll].The critical onset of the spherical detonation has been widely studied in both cases essentially inorder to define the detonability limits of a mixture. These limits must be regarded as intrinsic limitsbecause depending on the reactive mixture for given initial conditions (of pressure p and temperatureTo) and only on one w ell-controlled external parameter, as the critical diameter of transmission dc of thetube, or on the critical initiation energy Ec of the source.In this work are summarized all the important features concerning the problem of the criticalinitiation of a detonation in free space in situations d e f ie d above.2. I N I T I A T I O N BY D I F F RA C T I O N O F A P L A N A R D E T O N A T I O N - M O D E 1

The first idea of Laffitte in 1925, in order to produce the spherical diverging detonation in agaseous reactive mixture (spherical detonation had never been observed at that time), was to use a planedetonation propagating in a rigid tube, wh ich is genera lly easy to initiate because of the confining, and totransmit it in a larger volume [12].Lafitte never succeeded to obtain spherical detona tion because of thetoo sma ller size of the diameter of the tube he used with respect to the reactive mixture. Nevertheless,Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1995413
http://www.edpsciences.org/http://dx.doi.org/10.1051/jp4:1995413http://dx.doi.org/10.1051/jp4:1995413http://www.edpsciences.org/


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C4-156 J O U R N A L DE P H Y S I Q U E I V

later on, after the experimental precursor w ork of Zeldovich et al. (1956) [l], his mode of production ofspherical detonation wave w as clearly demonstrated.In this mode, the detonation regime exists in the medium contained in the tube w here detonationproducts expand in constant cross section area. This detonation is submitted to a large perturbation whensuddenly the confining disappears. Then, the detonation may :1) go on, propagating in diverging spherical geometry or,2) be quenched.At the ex it of the tube of diamete r d, when the plane detonation diffracts, the two-dim ensionallateral expansion propag ates from the edge towards the axis of the tube and under subcritical conditionsdestroys completely the plane detonation wave as it penetrates into the larger volume. The planedetonation wave turns into a decaying two-dimensional cu rved shock wav e followed by a graduallydecoupling flame front.At critical conditions, when the mean radius of curvature R of the diffracted shock wave hasincreased to a threshold value, the onset of detonation suddenly occurs on the ax is of the tube and goesthrought the whole volume before to propagate close to the CJ conditions. A quasi sphericalpredetonation sphere [7] is then observed with a radius R =R, (as displayed in Fig. 1).

Fig. 1 Smok ed-foil record of predetonation sphere of mean radius R, for critical transmission ofa plane detonation into a large vo lume in C2H2+ 2.5 0 2 mixture at To = 293 K.


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For most of the gaseous detonation mixtures currently used (CnHI11/02and H2 /02 or /Air)observations give :

So, at that mean radius, independently of the mixture used and of the siz e d of the tube, th e shock wavestrength is qua si identical if referred to the initial conditions for a plane-CJ diffracted de tonation wave.Moreover, many experimental data provide a noteworthy correlation between R, and the detonation cellsize kJ f the mixture [13], as :

Thu s, considering Eqs. (1) and (2), the classical notion of critica l tube d iam eter d, of transmission of aC J plane detonation wave into a large volume can be defined and appears as a direct consequence of theRCE 20 hcj requirement, i.e. :

If now we con sider the diffraction of a plane overdriven detonation w ave which turns into a C Jspherical detonation wave, experiment shows that at critical conditions, the RCr 20 ?q~ule remainsvalid 1141. Nevertheless, size of the critical diam eter dc decreases depen ding on t he strength of the planedetonation wave. For not too high values of D/DcJ and depending o n the reactive mixture [14], dc isfound a decreasing function of D/Dcj which follows the dc = 13 h rule. Detonation c ell size h dependson D/Dcj as the following :

As D/DcJ increases from 1, this kind of source tends progressively tow ards the ide al point source ofexplosion (at least in the decreasing part of the relationship).3. INITIATION BY AN EXPLOSION POINT SOURCE - MODE 2

In his experiment in 1925 [12], Laffitte succeeded for the first time to obtain a sphericaldiverging detonation wave by the action of high explosive point source. At that time, this divergingpropagation of the wave was rejected since 1917 [l51 by Jouguet. Neverteheless around the 401s,independ antly Taylor [ l61 and Zeldovich [l71 gave a particular solution of the expansion of thedetonation products behind suc h a C J wave.In a general way, the critical initiation of spherical detonation does not depend on the type ofsource if it can be considered as an ideal strong point source of explosion (cf. Taylor [l81 and Sedov[19]) as for instance, laser spark, exploding w ire or detonation of high exp losive. In su ch conditions,the energy deposition shows the sa me trend and the resulting shock wave produced and its decay m ay be

scaled by only one parameter, i.e the energy of the source E (for fixed p,). The intense decayingspherical shock wa ve produced in the reac tive mixture turns into an overdriven expand ing detonationwave. Th is detonation presents a very large curvature of its front and is subm itted to the action of theintense rear expansion wave and then decays rapidly as R increases. For criti cal initiation, a t a radius R= R,, the detonation is totally quenched. T he very fine cellular system observed fo r R < R,, with cellsize h R,),the sudden onset of detonation occurs in the whole volume at different locations in the unburnedcompresse d lay er located between the sh ock front and the flame (cf. Fig. 3). As a remarkab le result, thisradius R, is correlated to the cell width hrr, as in the mode 1 of initiation. bv a factor of about 20 for.


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Fig. 2 Smoked-foil record of the detonation cellular structure near the initiation sou rce and itsfailure in critically initiated spherical detonation in C2H2 + 2.5 0 2 mixture at To = 293 Kand p, = 30 tom (hcj= 5mm - R* 20 mm).

Fig. 3 Smoked-foil record of the predetonation sphere of mean radius R, for critical detonationinitiation in C2H2 + 2.5 02mixture at To = 293 K and p, = 40 Torr ( ~ C J 4mm).


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Fig. 4 Critical predetonation radius RC versus cell size h c j in various reactive mixtures fordifferent initiation sources. (FE : Exploding wire - L A : Laser spark - E S : HighExplosive) + C2H2+ 0 2 - C2H2+ 2.5 0 2 A C2H2 + 2.5 0 2 + 3.5 Ar (50% Ar)A C 2 H 2 + 2 .5 0 2 + 9 Ar (72% Ar) t C 2 H 2 + 2 .5 0 2 + 10 Ar (75% Ar)V C 2 H 2 + 2.5 0 2 + 13 Ar (79% Ar) - W C 2 H 2 + 2.5 0 2 + 5 N2 -

H2 + Air ($=l) - C2H4 + Air ($=1.3) - C3Hg+ Air (@ 1)(white point = spherical propagation - black point = hemispherical propagation).

M ore recently, som e experiments [20] [21] have show n that, for som e reactive mixtures dilutedby a large amount of a monoatomic inert as He, Ar or Kr, the critical radius R,= k h c J presents aconstant k dep ending on the amoun t of diluent and is, for a dilution of 7 5 to SO%, abou t twice thecommon value, i.e. k z 40-44. In such mixtures the RC= 1.6 d, rule holds, meaning that similarityconcerning thermodynamical characteristics required for initiation is m aintained.In a general w ay, instabilities that generally arise during the process of chemical energy release inthe detonation front depend deeply on the amplitude of the reduced global activation energy E A I R T ~of the chem ical system and undoubtely at the origin of the cellular and subcellular structure of thedetonation wave [22]. For the systems mentionned above of particularly low values of E A / R T ~E )in comparaison with values of com mo n reactive systems, the cellular structure is very regular (based onone particular frequency of instability). Moreover, detonation in su ch syste ms propagating in tubesexhibits a three-dimensional structure depending strongly on the rigid boundaries and its failure(initiation) because of lack of instrinsic instabilities of higher frequencies is more easy (difficult) toobtain [23] (1241) when submitted to an external perturbation. In connexion with that, in divergingpropagation the selfsustained detonation appears not so resistant to expansion and requires a largercritical radius of curvature RC if refer red to ~ C J )nd then larger critical tube diameter dc and criticalinitiation energy E, [l31 [21].


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JOURNAL D E PHYSIQUE IV4. DISCUSSION

In the precursor work of Zeldovich et al. (1956), who studied carefully these two modes ofinitiation of detonation, is established wh at we called the "Z eldovich criterio n" for initiation. It may besummarized into necessary conditions concerning two parameters (intensive and extensive) of theinitiation problem, respe ctively :

1") the strength p, of the shock wave m ust be larger than a threshold value (to insure a t least theself ignition of th e reactive m ixture), i.e. ps - p m > p, and,2") the radius of curvature R of the w ave is at least equal to the chemical induction length L$') ofthe mixture, Li remaining the sole charac teristic length of the problem .So the critical initiation energy may be written as

Experim ents on critical initiations performed after that work confirm qualitatively such a criterion[25], but give no quan titative estim ation of energy E,.In fact, experim ents [7]-[l l ] reported above s how clearly that onset of spherical detonationoccurs at criticallity for a very large radius of curvature of the wave (at least one order of m agnitudehigher than the cell size k j ) .Th e physical mechanism for the onset of detonation considering a progressive distance from thesource is the following in the two m odes [l31 :1") at the beginning of the process the detonation regime exists (CJ or overdriven planedetonation in m ode 1 and overdriven spherical detonation in m ode 2).2") this detonation is destroyed by the strong negative g radient of the expan sion wave be hind thecurved detonation front by respectively :

- the lateral expansion starting from the corner and converging to the axis of the tube in the caseof the sudden disappearing of confining,- the central spherical expansion in the case of initiation by the point source.3") a s the wave rec overs a small curvature with regard to the cell size Acj, the onset of the self-sustained detonation occurs.For simplicity of analysis, we have considered only the second initiation mode because of pointsymm etry (notice that for m ode 1, diffraction of a highly overdriven plane detonation wave by a verysm all tube diameter is quite representative of the case considered because in such situation the shape ofthe source may be ignored). In that case, the curved multiheaded d etonation exists for very sm all R,because of the large strength of the wave (D/Dcj > 1 and p, >> PZND).Inside the sphere of radius R *,where R* is the radius of the spherical shock wave w here the energy of the source E is balanced by thechem ical energy release , i.e. :

the motion of the wave is governed by the energy deposition and the combustion observed is the"induced" detonation regime. At R = R*, the detonation regime still exists, because cellular structuresare alm ost recorded. S o, the chemical heat release has played a role at that radius equivalent for thepropagation of the sho ck, from the point of view of ener gy, to the external source. For R (=R,) slightlylarger than R*, the three-dimensional structure of the detonation vanishes. Most of the experimentsshow [l3 1 clearly that in first approxim ation :

( l ) The chemical induction length used by Zeldovich and coworkers was certainly a kind of hydrodynamic mean scaleseparating the shock front and the flam e in the detonation structure considered in the ZND frame. A proof of that is reportedin the same work where critical tube diameter is correlated to Li by the relationship dc = 15 Li which is very close to thedc = 13 hc j rule. So, Li in that work was representative approximately of the size of the cell spacing of the reactivemixture even if at that time the intrinsic detonation cellular suucture had not been yet discovered.


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1") R, sR* and2") R s is equ al to 0.2 RC, so

Considering E qs (2) (6) (7) and the approximate relationship = 2(y2 - 1)Q, the criticalenergy of initiation can be ex pressed by :

where

Th is expression gives generally a reasonably good estimation of the energy of real sour ces [13]-[21] and is very close to those given by Vasiliev and Grigoriev [26], Lee [27] and Knystautas e t al[28].In other respects, the small curvature of the wave front for the existence of the selfsustaineddetonation is highlighted in the exp erimental stud y of Murray e t a1 [29]. Detonations pro pag ate in tubewith removing wall (plastic film) of different thicknesses w hich perm it to vary the rear ex pans ion rate.Th e authors demonstrated c learly that a too strong rear expansion destroys the detonation w ave and acriterion for the critical existence of the self-sustained propagation of the "curved" detonation in differentchemical sy stem s has been given concerning the radius of the wave front, i.e. RC> 16 k j .Recently, in correlation with this problem of existence of curved self-sustained detonation wave,He and Clavin [30] bring an interesting contribution to the problem of initiation of divergingdetonations. The y dem onstrate from simulation that existence of curved g eneralized CJ detonation wavenee ds a minimu m radius of curvature R, of the wave of amplitude, at least two or three order ofmagnitude larger than Li, the only scale of the problem, in agreement with observations [l l] givenabove.5. CONCLUSIONS

Critical direct initiation of spherical gaseous d etonations in different gaseous reactive systems hasbeen investigated.Tw o modes of initiation are studied :l") transm ission into a large volume of a plane CJ or overdriven detonation wave propagating ina d i.d. tube and2") deposition of an energy E by an ideal strong point sou rce of exp losion.As generally observed :1") Th e onset of spherical detonation occurs after a predetonation sphere of radius R =RCwhich appears

to be an intrinsic parameter of the explosive mixture linked to the cell spacing hcj by a large andconstant factor (generally 20).2") as a consequ ence of 1):a) the detonation w ave w hich exists generaIly near the "source" cannot resist strong curvatureeffects, even in CJ conditions. So, the self-sustained CJ detonation wave needs a maximumcurvature in comparison to hcJ. When radius of detonation takes the value R, roughly, thedetonation may ignore the rear expansion wave. Th is can be sum marized by a competitionbetween the rea r expansion effects and the chemical production in the R, = 2 0h elationship.b) R, may vary from few milIimeters for common C nHm - H2/02 reactive mixtures at ambiantconditions to few me ters fo r less detonable mixtures as C,Hm - H2IAir mixtures.c) the critical diameter of transmission d, is linked to h by the clas sical relationship d, = 13 h,h = h a or the CJ wave and results from the RC= 20 hcJ requirement.d) the critical initiation energy Ec of the strong point source of explosion v aries as :


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and may be very large, owing to the fact that R, contains a large number of cells (20), eachcell includes itself generally -a arge number of chemical induction length Li (generally 30 forCnH m/A ir systems), i.e. RC 600kJ.Discrepancy from classical rules is observed in detonation presenting a very regular cellularsystems. I n these systems, the c ritical radius of curvature R C of a detonation is larger ifcompared to LCJ (or L i ~ j )han in common reactive mixtures and propagation in rigid tubesmore dependen t on confinement. Th is emphasizes the fact that the three-dimensiona l structure ofdetonation plays a spec ific role in the propagation, the initiation and failure of the self-sustaineddetonation wave, which cannot be explained on the simple basis of the ZND structure of thewave.

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D. Desbordes- Critical Initiation Conditions for Gaseous Diverging Spherical Detonations

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