Martin Sichel- A simple analysis of the blast initiation of detonations

8/3/2019 Martin Sichel- A simple analysis of the blast initiation of detonations

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A c t a A s t r o n a u l i c a . Vol. 4, pp. 409-424. Pergamo n Press 1977. Printed in Great Britain

A s i m p l e a n a l y s i s o f t h e b l a s t i n i t i a t i o n o f d e t o n a t i o n sM A R T I N S I C H E L ~ f

Department of Aerospace En#neering, The University of Michigan, Ann Arbor, MI 48109, U.S.A.(Received 4 March 1976; revised 21 July 1976)

Abstraet--A detonation can be initiated in a fuel-oxidizer cloud by a blast wave of sufficient strength.The Zel'dovich criterion for initiation is that the blast energy must be such that when the pressureratio across the shock front has decayed to the Chapman-Jouguet value, the distance traveled by theshock must be at least of the order of the reaction zone thickness. A modified version of thiscriterion has been expressed in quantitative form, and used to determine a dimensionless initiationenergy using the stoichiometric value as a reference. Good agreement was found between the theoryand experimentally measured initiation energies for MAPP-air, and C2H2-O2 mixtures. The theoryalso predicts the variation of initiation-energy with ambient pressure.

I n t r o d u c t i o nA DETONATION can be initiated in a fue l oxi dizer clo ud b y a b last wa ve ofsufficient strength. The determination of the minimum or critical blast energyfor initiation has been the subject of many experimental and theoreticalinvestigations, and also is the subject of the present paper.

Instantaneous combustion at the blast front would be the simplest model touse as a basis for an initiation theory. However, in this case it can be shown thattransit ion from blast wave to detonation behavior always occurs regardless ofthe blast energy (Levin, 1967; Korobeinikov, 1969). The observed existence of acritical blast energy below which no detonation occurs must therefore be tied tothe ignition delay, characteristic of most combustion. This relation between theignition delay or induction zone and the critical initiation energy was firstrecognized by Zel 'dovitch e t a l . (1956), in their qualitative initiation criterion.Thus, direct initiation of spherical detonations requires that when the pressurebehind the initiating blast has decayed to the Chapman-Jouguet value, the shockradius must be at least of the order of the reaction zone thickness of the mixture.

Bach e t a l . (1971) dealt with this problem by retaining the discontinuouscombustion front; however, with a reduced heat release Qe which is only afraction of the total heat release Q per unit mass of mixture. The ratio Q d Q waschosen as an empirical function of both the shock strength and the inductiondistance, i.e. the distance between the shock front and the start of combustion.This theory, although semi-empirical, appeared to predict the main phenomenawhich arise during blast initiation. The critical energy, however, can only bedetermined by the cumbersome computation of a series of shock trajectories foreach set of parameters.

tProfessor of Aerospace Engineering. 409


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410 M. Sichel

Nicholls et al. (1974a) used an energy balance to develop a direct analyticalrelation for the critical initiation energy. The energy of the fluid in the inductionregion behind the blast front decreases as the wave decays. Nicholls et al.(1974a) argue that this decay is opposed by the energy released by combustion,and for initiation to occur the theory requires that the rate of induction zoneenergy decay equal the rate of combustion heat release before the blast Machnumber Ms has dropped below a critical auto-ignition value MA. M,, is a Machnumber below which the Chapman-Jouguet (C-J) detonations fail to propagatebecause the excessive growth of the induction zone causes uncoupling betweenthe shock and combustion fronts. The choice of the energy in the induction zoneessentially ignores the cumulative effect of combustion on the attenuation of theblast wave. During sample computations the theory resulted in induction zonelengths, A, which were larger than the blast radius Rs. While there are, thus,some difficulties, the theory contains the main idea that initiation depends on abalance between blast decay and the opposing effect of combustion heat release.

Lee and Ramamurthi (1975) have introduced the concept of a criticaldetonation kernel to explain initiation. The integrated form of the energyequation is used to show the combined effects of blast energy release andcombu stion upon the Mach numb er Ms of the initiating blast wave. Only the heatof combustion released between the blast center and the edge of the inductionzone that is to radius R, -A is taken into accoun t. A critical kernel radius R ** isdefined by equating the effects of blast energy and combustion upon Ms, theMach number of propagation. M**, the Mach number corresponding to R**must exceed the auto-ignition limit for initiation to occur, and this conditionleads to an expression for the critical initiation energy.Shock waves are freque ntly used to detona te liquid and solid explosives.Typically, the shock is applied to the ex plosive in the form o f a pulse of a certainstrength and duration, and Walker (1975), among others, has proposed a theoryrelating the critical energy to the shock pressure and duration. This techniquediffers considerably from blast ignition so that Walker's theory is not directlyapplicable.The theory developed below is based on a modified form of the Zel'dovitchcriterion expressed in a quantitative form. The main idea is still that the criticalblast energy for initiation depends on a balance bet ween the driving effect of thecombustion energy and the induction zone growth due to the decay of theinitiating blast.F o rm u l a t i o n

The Zel'dovitch criterion (1956) is that: for initiation the blast energy must besuch that when the pressure ratio across the shock front has decayed to theChapman-Jouguet (C-J) value of the fuel oxidizer mixture, the distance traveledby the shock must be at least of the order of the reaction zone thickness of themixture. In order to express this criterion in quantitative form the self-similarsolution of Taylor (1950), and Sedov (1959) is used to describe the initiatingblast. The self-similar point blast theory also was used by Lee et al. (1966) inorder to compute reaction zone thickness from measured initiation energies viathe Zel'dovitch criterion.


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A simple analysis of the blast initiation of detonations 411

T h e p o i n t b la s t t h e o r y d o e s n o t a c c o u n t f o r t h e e f fe c t o f c o m b u s t i o n ;t h e r e f o r e , th e p r e s e n t t h e o r y c a n b e c o n s i d e r e d a s a z e r o t h o r d e r a p p r o x i m a t i o nt o t h e i n i t i a t i o n p r o b l e m . A c c o r d i n g t o t h e s e l f - s i m i l a r t h e o r y , t h e b l a s t r a d i u s ,b l a s t M a c h n u m b e r , a n d th e t e m p e r a t u r e , i n d u c e d v e l o c i t y a n d p r e s s u r ei m m e d i a t e l y b e h i n d t h e b l a s t c a n b e e x p r e s s e d i n t h e f o r m :

R_z = p,~.+2 , (1 )Ro

2 / ( ~ ) / - _ . , , . + ~ ,M =- (v + 2) (2)T2 83 '(3' - 1) k. ?-t2.,.+2,~ (3)

= O , + 2 ) ' ( 3 " + 1 ) ~ , ~( 4 )Co (v +2 ) (3 " + 1) V \ a ~ /

P__2 83' k-"-r-~ -t2""~+2" (5)po ( v + 2 ) ( 3" + 1) a ~H e r e t h e s u b s c r i p t z e r o r e f e r s to th e u n d i s t u r b e d m e d i u m a n d s u b s c r i p t tw or e f e r s to c o n d i t i o n s i m m e d i a t e l y d o w n s t r e a m o f th e s h o c k . T h e d i m e n s i o n l e s st i m e { i s d e f i n e d b y

"[ = t /T o ; T o = R o / C o ~ v / ( k d a ~ ) (6 )a n d R o , t h e c h a r a c t e r i s t i c e x p l o s i o n r a d iu s , i s r e l a t e d t o th e b l a s t e n e r g y E b y

R o = ( E / k ~ p o C o 2 ) " " . (7)T h e g e o m e t r i c f a c t o r s v a n d k~ a r e 1 , 2 , 3 a n d 1 , 2 ~ r , 4 r, r e s p e c t i v e l y f o r p l a n e ,c y l i n d r i c a l a n d s p h e r i c a l w a v e s . T h e p a r a m e t e r a~ , w h i c h i s g i v e n i n T a b l e 1 ,d e p e n d s o n b o t h v a n d t h e r a t i o o f s p e c i f i c h e a t s 3". p o a n d C o a r e t h e d e n s i t y , a n d t h es p e e d o f s o u n d i n th e u n d i s t u r b e d f u e l - o x i d i z e r m i x t u r e .

T h e b l a s t d e c a y i s e s s e n t i a ll y c o m p l e t e b y t h e t i m e R s - R o o r t - 0 (T o) w h e r eTo c a n b e c o n s i d e r e d a c h a r a c t e r i s t ic e x p l o s i o n t im e . T h e s t r o n g b l a s t t h e o r y f ai ls

Table 1. Variation of the blast wave parametera. with the ratio of specific heats, 3"3' v = 1.0 V = 2.0 v = 3.0

1.1 2.9383 2.6449 2.29971.2 1.9114 1.7622 1.53831.3 1.3620 1.2661 1.10051.4 1.0264 0.9666 0.84281.5 0. 8163 0.7711 0.67431.6 0.6695 0.6346 0.55721.7 0 .5619 0.5356 0.4716


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412 M. Sichel

b e f o r e t h i s p o i n t i s r e a c h e d ; h o w e v e r , t h e p h e n o m e n a r e l e v a n t t o i n i t i a t i o n o c c u ra t m u c h s m a l l e r v a l u e s o f R s .

T h e Z e l ' d o v i t c h c r i t e r i o n i s n o w m o d i f i e d b y a p p l y i n g i t a t t h e c r i t i c a l b l a s tr a d iu s R * , d e f in e d b y K o r o b e i n k o v e t a l . ( 1 9 7 2 ) a s t h a t r a d i u s w h e r e t h e b l a s te n e r g y a n d t h e e n e r g y o f c o m b u s t i o n i n s id e t h e b l as t f r o n t a r e e q u a l . T h e h e a tr e l e a s e d b y c o m b u s t i o n w i l l b e g i n t o s i g n i f i c a n t l y i n f l u e n c e p r o p a g a t i o n a t t h i sr a d i u s , a n d t h e t r a n s i t io n f r o m b l a s t w a v e t o C - J b e h a v i o r , i f i t o c c u r s , t a k e sp l a c e n e a r t h e c r i t i c a l r a d i u s ( B a c h e t a l . , 1 9 71 ; L e v i n , 1 96 7 ). U s i n g a n a s t e r i s k t od e n o t e b l a s t w a v e p r o p e r t i e s a t t h e c r i t i c a l r a d i u s , t h e m o d i f i e d Z e l ' d o v i t c hc r i t e r i o n b e c o m e s A* -< 8. (8)A * i s t h e r e a c t i o n l e n g t h a n d t h e p a r a m e t e r 8 , w h i c h m u s t b e l e s s t h a n o n e ,d e p e n d s o n t h e d e t a i l s o f th e i n i t i a ti o n p r o c e s s . T h e e q u a l i t y h o l d s i n (8 ) w h e nt h e b l a s t e n e r g y e q u a l s t h e c r i t i c a l v a l u e E , .

B y c o m b i n i n g e q n s ( 1 ) , ( 2 ) , ( 7 ) a n d ( 8 ) , i t i s n o w r e a d i l y s h o w n t h a tC ~ M , 2 / A * ' V / v + 2 \ 2 (9)

a n d ( 9 ) p r o v i d e s a q u a n t i t a t i v e s t a t e m e n t o f t h e Z e l ' d o v i t c h c r i t e r i o n . T h ed i ff i cu l ty i s t h a t t h e p a r a m e t e r 8 d e p e n d s o n t he c o u p l i n g b e t w e e n c o m b u s t i o na n d t h e b l a s t f r o n t f o r w h i c h a d e t a i l e d a n a l y s i s i s n o t y e t a v a i l a b l e . T h e c r i t i c a li n i t i a t i o n e n e r g y w i l l v a r y w i t h t h e f u e l o x i d i z e r m i x t u r e r a t i o . I f t h e c r i t i c a le n e r g y i s n o r m a l i z e d u s in g t h e s t o ic h i o m e t r i c v a lu e E , a s r e f e r e n c e t h en o r m a l i z e d c r i t i c a l i n i t i a t i o n e n e r g y

( lO )E,~-- - ~ \A ~! \ p , , ~ C , , ~ a w , )w i ll b e i n d e p e n d e n t o f 8 p r o v i d e d t h a t t h is p a r a m e t e r d o e s n o t v a r y a p p r e c i a b l yw i t h t h e m i x t u r e r a ti o . S t o i c h i o m e t r i c q u a n t i ti e s a r e d e f i o t e d b y t h e s u b s c r i p tS. T h e n o r m a l i z e d i n i t i a t i o n c r i t e r i o n ( 1 0 ) h a s t h e a d v a n t a g e t h a t i t d o e s n o td e p e n d o n 8 , a n d h e n c e o n a d e t a il e d c o n s i d e r a t i o n o f c o m b u s t i o n - b l a s t f r o n tc o u p l in g . I f th e c r i t ic a l e n e r g y o f i n i t i at i o n is k n o w n f o r a n y o n e m i x t u r e r a t i o ,( 1 0 ) c a n b e u s e d t o d e t e r m i n e E , f o r a n y o t h e r f u e l - o x i d i z e r m i x t u r e . F i r s t ,h o w e v e r , ( 1 0 ) m u s t b e e x p r e s s e d i n t e r m s o f p a r a m e t e r s r e l a t e d t o t h ep r o p a g a t i o n p r o p e r t i e s o f C - J d e t o n a t i o n s a n d t o th e k i n e ti c s o f t h e c o m b u s t i o nr e a c t i o n . F o r t h i s p u r p o s e i t i s c o n v e n i e n t t o e x p r e s s M * i n t e r m s o f t h ed i m e n s i o n l e s s t i m e [ * u s i n g (2 ). A t t h e c r i ti c a l b l a s t r a d i u s t h e b l a s t e n e r g y Ee q u a l s t h e c o m b u s t i o n e n e r g y c o n t a i n e d i n s i d e t h e r a d iu s R * . F r o m t h isc o n d i t i o n a n d ( 1 ) i t f o l l o w s t h a t

"{* = (v C o: /Q ) ~+~/2~ (11 )


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A simple analysis o f the blast initiation of detonations 413

w h e r e Q i s t h e h e a t o f c o m b u s t i o n p e r u n i t m a s s o f f u e l o x i d i z e r m i x t u r e , t * c a nb e r e l a t e d t o t h e p r o p e r t i e s o f a C - J d e t o n a t i o n p r o p a g a t i n g t h r o u g h t h e m i x t u r eu n d e r c o n s i d e r a t i o n i f i t i s a s s u m e d t h a t Q is c o n s t a n t a n d e q u a l to t h e C - Jv a l u e . F o r m o s t C - J d e t o n a t i o n s M e 2> > 1 , w h e r e M c is th e M a c h n u m b e r o fp r o p a g a t i o n , a n d t h e n i t c a n b e s h o w n ( N i c h o ll s e t a l . , 1974b) t ha t Q i s r e l a t ed t oM c b y

C o ~ M /Q - 2(y22 - 1) (12)w h e r e y 2 is t h e r a t io o f s p e c i fi c h e a t s o f t h e p r o d u c t s o f c o m b u s t i o n . E q u a t i o n(1 2 ) i s d e r i v e d b y a s s u m i n g t h a t t h e f lu i d b e h a v e s a s a p e r f e c t g a s b o t h u p s t r e a ma n d d o w n s t r e a m o f th e d e t o n a t io n . T h e n , c o m b i n i n g ( l l ) a n d ( 12 ) r e s u lt s i n

/ - , [2 v (y 2 2 - 1)] (~2~2"= [ M~2 (13)

t h e d e s i r e d r e l a t i o n b e t w e e n t-* a n d C - J p r o p e r t i e s .I m m e d i a t e l y b e h i n d t h e l e a d i n g s h o c k t h e f lu i d m o v e s w i t h v e l o c i t y (/~ , - v2)

r e l a ti v e t o t h e s h o c k f r o n t . I f T, i s a n i n d u c t i o n o r c h a r a c t e r i s t ic r e a c t i o n t i m e ,t h e i n d u c t i o n d is t a n c e A c a n b e a p p r o x i m a t e d b y

A = ~ ' ,( /~, - v2) = 1",C oM (y - 1) /(y + 1). (14)

T h e r e l a t iv e v e l o c i t y , a n d ~ -,, w i ll c h a n g e a s t h e f l u i d m o v e s a w a y f r o m t h es h o c k ; h o w e v e r , i n t h e e a r l y s t a g e s o f b l a s t in i t ia t i o n , w h i l e t h e t e m p e r a t u r e T2b e h i n d t h e s h o c k i s h i g h , ~, a n d h e n c e A / R , w i l l b e s m a l l s o t h a t (1 4) s h o u l d b e ar e a s o n a b l e a p p r o x i m a t i o n . N o w c o m b i n i n g ( 2 ) a n d ( 1 4 ) w i t h ( 1 0 ) r e s u l t s i n

Ec -* * "t s~ - ~ : ~ I ~ ,~ - ~T , ) (15)w i t h

E q u a t i o n (1 5) p r o v i d e s a r e l a ti o n f o r t h e n o r m a l i z e d i n i t ia t i o n e n e r g y i n t e r m s o ft-* , w h i c h i s r e l a t e d t o C - J p r o p e r t i e s b y ( 13 ), a n d ~ -* , w h i c h i s g o v e r n e d b y t h ek i n e t i c s o f t h e c o m b u s t i o n r e a c t i o n .

A k i n e ti c m o d e l f o r t h e i n d u c t i o n c h e m i s t r y o r a n e m p i r ic a l fo r m u l a i s n e e d e dt o e v a l u a t e t h e i n d u c t i o n ti m e T ,. T h e s i m p l e s t k i n e t ic s c h e m e w h i c h t a k e s t h ei n f lu e n c e o f a m b i e n t d e n s i t y p o i n t o a c c o u n t i s t h e s e c o n d o r d e r i r r e v e r s ib l er e a c t i o n

A + B k / ) C (16)


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4 1 4 M . S i c h e l

w h e r e A is t h e f u e l , B t h e o x i d i z e r , C t h e c o m b u s t i o n p r o d u c t s , a n d k~ t h ef o r w a r d r a t e c o n s t a n t . T h i s m o d e l h a s b e e n a d o p t e d h e r e . W i t h ( A o) a n d ( B )o t h ei n it ia l c o n c e n t r a t i o n s o f f u e l a n d o x i d i z e r , a c h a r a c t e r i s t i c r e a c t i o n t i m e ~-, c a nb e d e f i n e d b y

' T I = I ( A ) o l [ d ( A ) l d t ] o l = l lk~ (B)o. (17)N o w , a s s u m i n g t h a t k s f o l l o w s a n A r r h e n i u s l a w s o t h a t

k~ = ( I /K ) ex p ( - E , I ~ T ) (18)~- , w i l l be g iven by

~-, = [M (B)o l ex p ( E , / ~ T ) . (19)S i n c e m a n y o f th e i n d u c t i o n r e a c t io n s i n c o m b u s t i o n a r e s e c o n d o r d e r , th em o d e l (1 6 ), t h o u g h e x t r e m e l y s i m p l e , is n o t u n r e a s o n a b l e . A n u m b e r o f e m p i r i c a lf o r m u l a s f o r t h e i n d u c t i o n ti m e h a v e t h e f o r m ( S c h o t t a n d K i n s e y , 1 95 8; W h i t e , 1 9 67 )

I n [ ( A y" ( B ) I " 7 , ] = A + B / T (20)w h i c h a g r e e s w i t h ( 1 9 ) .

W h e n t h e o x i d i z e r is a m i x t u r e o f n i t r o g e n a n d o x y g e n , ( B )o , t h e i n it ia lc o n c e n t r a t i o n o f o x y g e n b e h i n d t h e s h o c k is g i v e n b y

(B)o = [3 ' + 1~ po[32(1 + rN)(v~b + 1)] -* (21)\ ~ / - - 1 /w i t h rN t h e n i t r o g e n o x y g e n r a ti o b y m a s s , v , t h e s t o i c h i o m e t r i c f u e l - o x i d i z e rm a s s r a t i o a n d ~b t h e e q u i v a l e n c e r a t i o , v ,~ b i s t h e a c t u a l f u e l - o x i d i z e r m a s s r a t i oo f t h e m i x t u r e .T a k i n g T i n (1 9) a s T 2, t h e t e m p e r a t u r e i m m e d i a t e l y d o w n s t r e a m o f th es h o c k , a n d c o m b i n i n g ( 3) , ( 15 ) , ( 19 ) a n d ( 21 ) y ie l d s t h e f o l l o w i n g e x p r e s s i o n f o rt h e n o r m a l i z e d i n i t i a t i o n e n e r g y :

= ( t , 2 /~ + 2 )E , . / E . 7 1 { ( t * / t * ) [ ( l + v s & ) / ( l + v , ) ] y e x p v B ~ ( E , / ~ T o ) - ~ ~ - i .2~/(~+~)) (2 2)w h e r e

B~ = [(v + 2)2(,,/+ 1)2/8-,/ (-,/ - 1)] (adk~) .T h e d i r e c t e x p r e s s i o n ( 9) f o r t h e c r i t ic a l in i t ia t i o n e n e r g y E , a l s o c a n b e r e l a t e dt o t h e k i n e t i c a n d C - J p a r a m e t e r s u s i n g ( 2 ) , ( 1 4 ) a n d ( 1 9 ) s o t h a t

- ( v B o E ~E,. = po' "Co"+ 2t* v8 VD~ e x p \ ~ { *2v/'v+2)) (23 )


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A simple analysis of the blast initiation of detonations 4 1 5

with2 v [ v - l ~ [kv~

Dr = 32r(1+ rN)(v, th +l ) ~ ~-~--~] av~-~]Equat ion (23) shows how the critical initiation energy Ec varies with the ambientdensity po. Thus, assuming that the C-J Mach number Me, and the parameterare independent of po, it follows that E~ oo po1-~Comparison with experiment

Critical initiation energies for MAPP gas-air mixtures have been measuredby Collins and Parsons (1973) and by Fry and Nicholls (1974). Lee and Matsui(1976) have measured initiation energies in acetylene oxygen mixtures.

The Collins and Parsons (1973) experiments were made by placing anexplosive charge at one end of a plastic bag 4 x 4 x 20 ft long. Detonat ion wasobserved before any appreciable interaction with the walls of the bag couldoccur; hence, these tests corresponding closely to initiation by a point chargewith v = 3. The Fry-Ni cholls (1974) experiments were made in a pie-shapedshock tube designed to simulate cylindrical clouds. This shock tube has anincluded angle of 20 and is 2.05 in. high. The initiating charge was placed at thevertex of the pie-shaped tube, and these tests thus corresponded closely tocylindrical sym met ry with v = 2. The Collins -Par sons (1973) data, which has beentaken from the plot of critical threshold energy vs % MAPP in air by volume, isreproduced in Table 2 below. The Fry-Nicholls (1974) data, reproduced fromNicholls et al. (1973) is shown in Table 3. In each case, E d E c , was computedfrom the data and tabulated as a function of the % MAPP by volume of thefuel-air mixture.

Lee and Matsui (1976) used an electric spark d ischarge to initiatecylindrical detonations in a chamber 13cm in alia. and 3 .7cm wide. Onlythe spark energy released during the first quarter cycle of the discharge iseffective in producing initiation, and this energy is reported as the criticalinitiation energy in Table 4. Here E d E , is also tabulated as a function of the %C2H2 by volume of the C~H2-O2 mixture.

T a b l e 2 . C r i ti c al i n i ti a t io n e n e r g y M A P P - a i r m i x t u r e s( C o l l i n s - P a r s o n s ( 1 9 7 3 ) b a g t e s t s )

% M A P P % M A P P E q u iv a le n ce E c E cb y v o l . b y w t . r a t i o ( ~b ) ( J ) Ec,

3 . 4 8 5 . 1 2 0 . 7 7 7 2 . 2 5 1 06 2 5 . 4 03 . 6 5 5 . 3 6 0 . 8 1 6 3 . 4 4 x 1 07 3 . 8 94 . 2 5 6 . 2 3 0 . 9 5 7 8 . 3 8 x 1 04 0 . 9 55 .00 7 .3 0 1 .134 1 .01 x 10? 1 .146 .93 10 .03 1 .605 1 .80 x 10? 2 .048 . 2 5 1 1 . 8 6 1 . 9 3 8 4 . 4 0 1 0? 4 . 9 89 .45 13 .51 2 .24 9 9 .39 107 10 .629 . 8 5 1 4 . 0 6 2 . 3 5 5 2 . 2 5 x 1 06 2 5 . 4 0


8/16

416 M. Siche l

Table 3. Critical initiation energy MA PP -a ir mixtur es(Fry-Nicholls (1974) shock tube tests)

% MAPP % MAPP Equivalence E, E,by v o l . by wt. ratio (~b) (J/c m) E,2 . 9 4 . 2 8 0 . 6 4 3 1 9 , 4 9 0 6 . 0 8 63.3 4.86 0.736 9,552 2 . 9 8 03.8 5.58 0.851 5,359 1.6724.3 6.30 0.968 3,035 0.9476.0 8.72 1.376 3,716 1.1596.8 9.85 1.572 4,740 1.4798.5 12.21 2.002 10,506 3.2789.7 13.85 2.315 18,178 5.663

Table 4. Critical initiation energy acet ylen e-ox ygen m ixtures,po=a mbi ent pre ssu re= 100torr (from Lee-Matsui (1976) spark

discharge tests)% C2H2 % C2H 2 Equiv alenc e Ec E.by vol. by wt. ratio Ok) (J/cm) E.

20.0 16.88 0.625 3.6 x 10 ~ 3.0828.6 24.55 1.000 1.17 x 10- ' 1.0040.0 35.14 1.667 8. 4x 10 2 0.71850.0 44.83 2.500 8.4 x 10 2 0.71860.0 54.93 3.750 2.5 21.4

Table 5. Critical initiation energy, theoretical results for MAPP-air mixturesCyn ica l wave Spheficalwave~

v = 2 u = 3Equivalence % MA PP t

rati o (~k) by vol. Mc 3'2 [* Ec/E. [* EdE.0.400 1.831 4.13 1. 2661 1.41 10 -t 73.9 0.275 133.10.600 2.722 4.71 1.2218 8.89 x 10 -2 4.61 0.186 14.530.660 2.984 4.84 1.2068 7.78 x 10 -2 2.76 0.167 6.140.888 3.974 5.20 1. 76 68 5.34 10 -2 1.070 0.122 1.1641.000 4.456 5.30 1 . 1 65 1 5.09x 10 -2 1.000 0.117 1.0001.122 4.974 5.41 1.1632 4.83 x 10 -2 0.950 0.112 0.8961.361 5.969 5 .51 1.1868 5.38 x 10 -2 1.152 0.123 1.3051.605 6.964 5.53 1.2196 6. 37 x 10 -2 1.713 0.141 2.641.854 7.951 5.50 1.2434 7.21 x 10 2 2.50 0.157 5.022.109 8.955 5.46 1. 26 03 7.91 x 10 -2 3.52 0.169 8.95

?MA PP composition: methyl acetylene (C3I-L,), 35.91%; propadien e (Cffh) , 24.48%;pr opan e (C3Hs), 18.58%; (C,H~o), 13.74%; propy len e (C3H6), 7.29%. M olecula r wt. = 43.37.Stoichiometric mixture: 4.434% by vol.; v, = stoich, fuel-air mass ratio = 0.06946.

~Other pa ra me te rs use d in the theory : 3' = 1.4; a2 = 0.9666; Be = 3.165; To = 298.16K;a~ = 0.8428; B3 = 2.156.


9/16

A simple analysis of the blast initiation of detonations 417

Equation (22) was used to compute EdEcs from the theory developed above.The results of the theoretical calculations are shown in Table 5 for MAPP-air,and in Table 6 for C2H2-oxygen. Mc and 3,2 which are needed to compute t-*from (13) were obtained using the Gordon- McBride P rogram (1971) fordetonation properties.

The author is not aware of measurements of activation energies for theinduction reaction in MAPP-air mixtures. Hence Ed~ for MAPP-air wasdetermined, here, by requiring the computed curve of EriE, vs % MAPP andthe curve drawn using the Fry-Nicholls (1974) data to match at one point. In thiscase, the point where EriE, = 2.5 and the mixture contains 7.95% MA PP byvolume was arbitrarily chosen. Then E,/~ was found to have the value of3350K. This value does not appear unreasonable although considerably lowerthan the values of 8700 and 9130K reported for acetylene oxygen (White, 1967)and diluted hydrogen oxygen mixtures (Schott and Kinsey, 1958), respectively.In the case of MAPP-air the variation of 3/, the ratio of specific heats of theunburned mixture is ve ry small; hence, it was possible to take 71 --- 1 and to useconstant values for B. and a. in the calculations.

There are small disparities between the MAPP compositions quoted byCollins and Parsons (1973), Fry and Nicholls (1974) and used here. The quotedcompositions and the mixture properties are summarized in Table 7, and it canbe seen that the differences in molecular weight and stoichiometric mixturecomposition are small. The composition and MAPP C-J properties used herewere taken from Sichel and Hu (1973). The measured and theoretical values of(EdE,) are all plotted vs the % MAP P in air by volume without adjustment for theslight differences in composition for, as Table 7 indicates, these differences shouldnot have large effects on the results.

The experimental and theoretical values of EriE, for MAP P-air are compared inFig. 1. The theory and experiment are in remarkable agreement. The theory appearsto confirm that the difference between the Collins-P arsons (1973) and Fry-Ni cholls(I 974) results are mainly due to the d ifferen tgeometries of the te st apparatus used, i.e.v = 2 for the Fry-Nicholls (1974) tests and v = 3 for the Collins-P arsons (1973) tests.The the ory also appears to predict detonation limits reasonably well and reproducesthe skewness of the EriE, curves relative to the stoichiometric mixture composition.Agreement appears to be best for rich mixtures; for lean mixtures the theoryunderestimates EdE,.

From Table 6 it is evident that the variation of 3' is appreciable in the case ofC2H~O2 so that variations in 71, a~ and B~ must be taken into account in thecomputations. Soot fo rmation becomes appreciable when the proportion of C2H2exceeds 50% by volume, and then eqn (13) for i-* will no longer be valid. Hence,theoretical calculations were only made for mixtures up to 50%C2H2.Computations were made for E,/~ =8700 and 13,880K. The first valuecorresponds to that repor ted by White (1967) for very lean C2Hz--O2 mixtures;the second value was determined by equating theoretical and experimentalvalues of EdE, for the 20%C2Hz--O2 mixture.

Theory and exper iment for C2Hz--O: are compared in Fig. 2, and are in goodagreement. The theo ry reproduces the skewness of the EdE, curve with respect


10/16

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11/16

A simple analysis of the blas t initiation of detonations 419

50.0

I0.0E.

5.0

I 0

0 CoIhnsond Parsons 1975) dataNtcholl~eto l . (]974) data

Cylindrical

~~ theory

~ l Spherical~ ' ~ / theory 0! / /

. o / / o 'A

) ,\ \ I / o / /[ % ] ~ t / E , , R = 3 3 4 7 " K

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Stoichiometric~/~--- mixture

t 1 I IO0 20 40 60 80 I00% MAPP by volume

Fig. l. Experimental and theoretical variations of normalized initiation energy forMAPP--air mixtures.

to the stoichiometric mixture and essentially appears to reproduce thedetonation limits. The theory also reproduces the initial drop in initiation energyas the mixture ratio exceeds the stoichiometric value. With E , [ ~ = 8700K thetheoretical value of E J E , is below experiment for lean mixtures and issomewhat above the measured value for rich mixtures. With E , / ~ t = 13,880K,the theory produces the correct rich detonation limit but yields low values ofE d E , between 30 and 50%C2H2 by volume. Optimum agreement between theoryand experiment would probably be obtained for an activation temperature E ~ / ~lying between the two values above. What is significant is the essential agreementbetween the value E , / ~ suggested by the initiation theory presented here and thevalue reported by White (1967).Equation (23) suggests that Ecoc po~-~, or for fixed ambient t emperature ,that Ecoc po -~. Measured initiation energies for C2H2 at reduced pressures havebeen reported by Bach e t a l . (1971), Matsui (1973), Lee and Ramamurthi (1975)and Lee and Matsui (1976). Some of this da ta which is discussed in detail by Lee


12/16

4 2 0 M . S i c h e l

10.0

e~

I 0

" - 0 - - 0 - " L e e o n d M o l s u i ( 1 9 7 6 ) d o toT h e o r y :

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1 I I 1 I I20 30 4O 50 60C 2 H 2 % by v o l u m e

F i g . 2 . E x p e r i m e n t a l a n d t h e o r e t i c a l v a r i a t i o n o f n o r m a l i z e d i n i t i a t i o n e n e r g y f o ra c e t y l e n e - o x y g e n m i x t u r e s .

a n d M a t s u i ( 1 9 7 6 ), i s p l o t t e d i n F ig . 3 a l o n g w i t h c u r v e s c o r r e s p o n d i n g t oc y l i n d r i c a l a n d s p h e r i c a l s y m m e t r y w i t h E c oc p o ' , p o 2.

T h e e l e c t r i c s p a r k i n it i a ti o n e x p e r i m e n t s r e p o r t e d b y L e e a n d M a t s u i ( 1 9 7 6)w e r e d e s i g n e d t o p r o d u c e w a v e s w i t h c y li n d r ic a l s y m m e t r y a n d , i t c a n b e s e e nf r o m F i g . 3 , t h at t h e v a r i a t io n o f E c w i t h p o c l o s e l y f o l l o w s E c ~ p o -' a sp r e d i c t e d b y t h e p r e s e n t t h e o r y . T h e e a r l i e r l a s e r s p a r k m e a s u r e m e n t s b y B a c he t a l . ( 1 97 1 ) a l s o a p p e a r t o v a r y i n v e r s e l y w i t h p o , a g a in s u g g e s t i n g c y l in d r i c a ls y m m e t r y . T h e a c t u a l v a l u e o f E c i s v e r y s e n s i t i v e t o th e e n e r g y s o u r c e u s e d f o ri n it i a ti o n , a s d i s c u s s e d i n d et a il b y L e e a n d M a t s u i ( 1 9 7 6) . T h e d i f f e r e n t p r o p e r t i e so f l a s e r a n d e l e c t r i c s p a r k s t h u s r e s u l t in th e d i f f e r e n c e i n t h e a b s o l u t e v a l u e s o fE c s h o w n i n F i g . 3 .

M a t s u i ( 1 9 7 3 ) u s e d a p l a n a r d e t o n a t i o n w a v e t r a v e li n g i n a t u b e a s t h ei n it i a ti o n s o u r c e a n d d e t e r m i n e d t h e m i n i m u m t u b e d i a m e t e r n e e d e d t o i n it ia t e ad e t o n a t i o n i n a l ar g e r s u r r o u n d in g v o l u m e . T h i s m e t h o d o f i n i ti a t io n c l o s e l ys i m u l a t e s t h e i n it i a ti o n o f a s p h e r i c a l d e t o n a t i o n w i t h a p o in t s o u r c e o f e n e r g y .


13/16

A s imp le aanalysis ot the M ast initiation of detonations 421

oc

m id ~ . , ~ , . . - 1 ~ ( 1 9 7 S )Electric ~ ~ ~ ~ '2CzHz 50e-J/m \ %%i.eQ o n d Mo Ilkll (19"r6) " % % % %

~ ~ % % % % 2C2He 50Z d/cm~ '~ . , _ \ \ B~k e t ol . (nit1)- ~ x . % L a se r ~ l ' k

" , ~ k MtSul 19?o3)flo

. . . . . i . . . . . i ' ' ' . . . . '0 0 0 ~ O0 iO O0C r i t i c a l e n e r g y , d

Fig. 3. The variation of critical initiation energy with pressure.

L e e and Matsui (1976) used the critical tube diameter data to estimate the criticalinitiation energies plotted in Fig. 3. There is close agreement between the Matsui(1973) data and the variation Ec 0: po-2 predicted for spherical s ymmet ry by thepresent theory.Lee and Ramamurthi (1975) also measured the variation of Ec with po for astoichiometric hydrogen oxygen mixture. Figure 3 shows that while the 2Hr-O2initiation energy increases with decreasing po, the rate of increase appears to bemuch less than indicated by (20), possible because (16) is not a goodapproximation for H2--O2 kinetics. An interesting consequence of (23) is that E cshould be independent of po for plane waves.

Bach e t a l . (1971) expressed their critical initiation energy in terms of theratio of Ac, the induction distance of the C-J detonation to the explosion radiusRo. The modified Zel'dovich criterion presented here can be expressed in thesame form by combining eqn (8) with the expressions (14) and (19) for the inductiondistance A and the ignition delay time ~, with the result

A c < ~ M - * / v + 2 \ (24)

Here T2c is the temperature downstream of the shock but upstream of thereaction zone for the C-J detonation. Bach et a l . (1971) computed a critical valueof ( A d R o ) = 0.091 for the initiation of a spherical detonation in a stoichiometricC2Hz--O2 mixture with po = 100 torr. By using (24), this value of (Ac /Ro) and thedetonation parameters of CzHz--O2, it now is possible to determine thecorresponding value of the parameter 8. With Mc =7.013, E , / ~ =8700K,


14/16

422 M. Sichel

a3 = 1.0, B3 = 3.08 5, {* = 6.272 x 10 -2, Tzc/To = 8 . 887 , t he r e s u l t i s ~ = 0 . 282 . T h i sc e r t a i n l y i s n o t a n u n r e a s o n a b l e v a l u e .D i s c u s s i o n

T h e t h e o r y d e v e l o p e d h e r e v e r if ie s th a t t h e m i n i m u m i n it ia t io n e n e r g y isd e t e r m i n e d b y a b a la n c e b e t w e e n t h e c o m b u s t i o n h e a t r e l e a se a n d t h e g r o w t h o ft h e i n d u c t i o n z o n e . T h e s i m p l e r e l a t i o n ( 2 2 ) f o r t h e v a r i a t i o n o f t h e i n i t i a t i o ne n e r g y w i th t h e f u e l - o x i d i z e r m i x t u r e r a t io , w h i c h i s d e v e l o p e d o n t h e b a s is o ft h e Z e l ' d o v i c h c r i t e ri o n , is in e x c e l l e n t a g r e e m e n t w i t h e x p e r i m e n t a l m e a s u r e -m e n t s f o r t w o q u i t e d if f er e n t , f u e l - o x i d i z e r c o m b i n a t i o n s . T h e t h e o r y c a n b eu s e d t o c o m p u t e t h e i n i t i a t i o n e n e r g y E c a t a n y f u e l - o x i d i z e r r a t i o , g i v e n am e a s u r e d v a l u e o f E~ f o r o n e p a r t ic u l a r m i x t u r e a n d a n e s t i m a t e o f t h ea c t i v a t i o n e n e r g y o f t h e i n d u c t i o n r e a c t i o n . T h e e x p e r i m e n t a t i o n r e q u i r e d t oe s t a b l i s h t h e i n i t i a t i o n c h a r a c t e r i s t i c s o f a g i v e n f u e l c a n t h e r e b y b e r e d u c e da p p r e c i a b l y . W i t h s u ff ic ie n t i n f o r m a t i o n a b o u t t h e c h e m i s t r y o f t h e i n d u c t i o nz o n e , t h e t h e o r y w i ll al s o p r e d i c t t h e e f f e c t o f a m b i e n t d e n s i t y u p o n t h e i n i t i at i o ne n e r g y .T h e s i m p l i c i t y o f th e t h e o r y c o m e s f r o m t h e u s e o f s e l f- s im i l a r s t ro n g b l a s tt h e o r y t o d e s c r i b e t h e i n i ti a ti n g b la s t . T h u s , t h e t h e o r y r e a l l y is t h e z e r o t h o r d e rs o l u ti o n o f a n e x p a n s i o n s c h e m e i n w h i c h t h e f ir st o r d e r s o l u ti o n w o u l d a c c o u n tf o r t h e i n f l u e n c e o f c o m b u s t i o n o n t h e i n i t i a t i n g b l a s t . T h e s u c c e s s o f t h i s s i m p l et h e o r y i n t h e c a s e o f e x p l o s i v e l y an d s p a r k i n i t i at e d d e t o n a t i o n w a v e s s u g g e s t st h a t n o n - i d e a l b l a s t b e h a v i o r d u r i n g t h e i n i t i a l p h a s e s o f b l a s t f o r m a t i o n w i l l n o th a v e a l ar g e e f f e c t o n i n i t ia t io n . A t t h e s a m e t i m e , t h e c r i ti c a l p h e n o m e n a w h i c hd e t e r m i n e w h e t h e r o r n o t t h e i n i t i a t i n g b l a s t w a v e q u e n c h e s a p p e a r t o o c c u rn e a r R s = R * w h e r e t h e i n f l u e n c e o f c o m b u s t i o n o n w a v e p r o p a g a t i o n i s s ti llr e l a t i v e l y s m a l l .

T h e p r e s e n t t h e o r y b y n o m e a n s r e p r e s e n t s a c o m p l e t e s o l u ti o n o f t h ei n i t i a t i o n p r o b l e m . N o e x p l i c i t m e a n s o f d e t e r m i n i n g t h e p a r a m e t e r ~ i sp r o v i d e d . A k e y f e a t u r e o f th e p r e s e n t d e v e l o p m e n t is t h a t th e p a r a m e t e r 8 isn o t r e q u i r e d f o r t h e c o m p u t a t i o n o f E d E , . H o w e v e r , a n a b s o l u t e t h e o r y f o rc o m p u t i n g E ~ d e p e n d s c r u c ia l l y u p o n 8, a n d h e r e m a n y p r o b l e m s r e m a i n .D e t e r m i n a t i o n o f ~ w i l l r e q u i r e a d e t a i l e d c o n s i d e r a t i o n o f t h e p r o c e s s e s i n t h ei n d u c t i o n z o n e l e a d i n g t o u n c o u p l i n g o f c o m b u s t i o n f r o m t h e b l a s t f r o n t . L i f i a n( 19 7 5) h a s a t t e m p t e d t o r e l a t e th i s c o u p l i n g t o t h e r a t e o f i n c r e a s e o f t h ei n d u c t i o n d i s t a n c e i n r e a c t i n g b l a st w a v e s . I n t h e B a c h et a l . ( 1 9 7 1 ) a n d L e e a n dR a m a m u r t h i (1 9 75 ) t h e o r i e s , u n c o u p l i n g o c c u r s w h e n e v e r t h e w a v e M a c hn u m b e r d r o p s b e l o w t h e a u t o - i g n i t i o n l i m i t . T h e w o r k o f S t r e h l o w ( 1 9 6 8 ) ,a m o n g o t h e r s , i n d i c a te s t h a t t r a n s v e r s e w a v e s i n t h e r e a c t io n z o n e o f d e t o n a t i o nw a v e s m a y a l s o p l a y a n i m p o r t a n t r o l e i n t h e c o u p l i n g p r o c e s s . A g r e a t e ru n d e r s t a n d i n g o f c o m b u s t i o n - s h o c k c o u p l in g is n e e d e d t o d e t e r m i n e s u c hp a r a m e t e r s a s ~ a n d t h e a u t o - i g n i t i o n M a c h n u m b e r a pr ior i .

A n o t h e r s h o r t c o m i n g o f t h e p r e s e n t t h e o r y i s t h a t c o n d i t io n s i m m e d i a t e lyb e h i n d t h e b l a s t w a v e a r e u s e d t o c o m p u t e t h e i n d u c t i o n l e n g t h A . I n a m o r ec o m p l e t e t h e o r y t h e v a r i a t i o n o f t h e t e m p e r a t u r e a n d v e l o c i t y o f a f lu i d p a r t i c l ea s it t r a v e r s e s t h e i n d u c t i o n z o n e s h o u l d b e t a k e n i n t o a c c o u n t .


15/16

Asimple analysis of the blast initiation of detonations 423

The ra te of energy re lease a t the blas t source can have a la rge inf luence onthe in i t ia t ion energy i f the ra te fa l ls be low a cer ta in minimum value . Thisp rob lem has been s tud ied ex tens ive ly by Bach et al. (1971) , Knys tautas and Lee(1975) and Matsu i and L ee (1975) using spark init i ators . The ef fect of ene rgyre lease ra te has been ignored in the present t heory ; henc e , th is theor y wi l l onlybe va l id for in i t ia t ion sources wi th high ra tes of energy depos i t ion.

T he theory deve loped he re appea rs to take the impor tan t phys ica l pa rame te rsinvolved in in i t ia t ion in to account , and provides a s imple means of es t imat ingminimum ini t ia t ion energies .Acknowledgements--This work was supported by the Air Force Armaments Laboratory, Eglin AirForce Base, Florida, under contract F08635-74-C-0123. The many useful suggestions of Prof. J. A.Nicholls, the project director, are also gratefully acknowledged.The author would also like to thank Prof. J. H. Lee of McGill University who provided theC2H2-O2 data and pointed out the connection between the initiation criterion presented here and theoriginal Zel'dovich criterion.

R e f e r e n c e sBach, G. G., Knystautas, R. and Lee, J. H. (1971) Initiation criteria for diverging gaseousdetonations. In Thirteenth Symposium (International) on Combustion, p. 1097. The CombustionInstitute, Pittsburgh, Penn.Collins, P. M. and Parsons, G. H. (1973) Detonation initiation in unconfined fuel-air mixtures. In

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Lifian, A. (1975) Theoretical modeling of the structure of the flow behind blast waves propagating inunconfined reactive mixtures. Paper presented at 1975 AFOSR Contractor's Meeting on Fuel-Air Explosion Research, Eglin AFB, Florida.Matsui, H. and Lee, J. H. (1976) Influence of electrode geometry and spacing on the critical energyfor direct initiation of spherical gaseous detonation. Comb. and Flame 27, 217.Nicholls, J. A., Sichel, M., Fry, R. S., Hu, C., Glass, D. R., DeSaro, K. and Kearney, K. (1973)Fundamental aspects of unconfined explosions. Technical Report AFATL-TR-73-125, Air ForceArmament Laboratory, Air Force Systems Command, U.S. Air Force, Eglin AFB, Florida.Nicholls, J. A., Pierce, G. H., Dye, L., Fry, R., Kearney, K. and Liou, M. (1974a) Mechanism in thedetonation regime. Final Report UM-012118-F, Dept. of Aerospace Engrg., The Univ. ofMichigan for Fast Reaction Group, Picatinny Arsenal.


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