An experimental and theoretical study of deflagration-to-detonation transition (DDT) in the granular explosive, CP

C O M B U S T I O N A N D F L A M E 65: 15-30 (1986)

15

An Experimental and Theoretical Study of Deflagration-to-Detonation Transition (DDT) in the Granular Explosive, CP* MELVIN R. BAER, ROBERT J. GROSS, and JACE W. NUNZIATO Fluid and Thermal Sciences Department, Sandia National Laboratories, Albuquerque, New Mexico 87185

and EUGENE A. IGEL Electrical Engineering Department, Texas Tech University, Lubbock, Texas 79409

In this paper, we present results of an experimental and theoretical study of the combustion processes associated with deflagration-to-detonation transition in the granular explosive, CP. Image-enhanced, highspeed streak photography recorded the various stages of flame spread within heavily confined charges as viewed through a Lexan window in a thick-wall, stainless steel tube. To characterize the phenomena, we applied a multiphase reactive flow model based on the theory of mixtures. This nonequilibrium model treated each phase as fully compressible and incorporated a compaction model for the granular reactant. Formulation of the constitutive models included a pressure-dependent burn rate and experimentally determined porous bed permeability. Predictions of this model were in good agreement with experimental observations.

1. INTRODUCTION It is generally accepted that the rapid combustion of confined, gas-permeable reactants is due to a combination of thermal, mechanical, and chemical processes that can self-sustain accelerated burning from deflagration to detonation. The earliest stage of this complex sequence of processes is one controlled by the effect of conduction heat transfer. As combustion progresses, hot product gases are forced into the unreacted material, causing rapid flame spread far exceeding the layer-to-layer deflagration

* This work was performed at Sandia National Laboratories supported by the U. S. Department of Energy under Contract Number DE-AC04-76DP00789. Copyright © 1986 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017

rate driven solely by conduction. Often this burn velocity in porous explosives is observed to be of the order of 0.1-1.0 mm/~s [1]. In addition to the increased energy transport, burning is initiated over the granular surface area as hot gases preheat regions connected by pores. This convective burning is inherently unsteady, and confinement of the combustion gases causes rapid pressure buildup as a result of the transport and interphase transfer of mass, momentum, and energy. During this stage of burning, the combustion zone acts as a permeable "pist o n " which is driven into the unreacted material by the action of the high-pressure combustion gases. Mechanical effects such as pore collapse and/or configuration changes of the connected porosity of the material may occur to impede or

0010-2180/86/$03.50

16 restrict the gas flow. In turn, mechanical load transfer between the gas and solid phases causes hydrodynamic steepening of the pressure (or stress) associated with combustion. With sufficient confinement, pressures can ultimately form a shock wave of sufficient amplitude to induce significant compression of the solid phase reactant. The latter stage is dominated by mechanical compressive heating effects associated with the growth of a shock wave [2]. Furthermore, local "hot spots" form as a result of microscale shock focusing during pore collapse, plastic work heating, and intergranular friction at grain boundaries or in regions of high shear (see, e.g., Nunziato [3]), enhancing chemical energy release and wave growth to detonation. As is evident, DDT (Deflagration-to-Detonation Transition) is a complex combustion process consisting of four regimes: conductive burning, convective burning, compressive burning with shock formation, and detonation. Although DDT physics has been studied experimentally and theoretically since the early 1960s, (beginning with the pioneering work of Griffith and Groocock [4]), much remains to be clarified. For the sake of brevity, a literature survey of the subject is omitted in this work; for that, the reader is referred to a companion article [5] or to the bibliography of Ref. [6]. In the present study we purposely interfaced experiment with theoretical modeling to demonstrate that DDT can be qualitatively and quantitatively predicted. The theoretical model follows closely prior work which provided the foundations of continuum mechanics for reactive multiphase flow [5, 7, 8]. Consequently, we shall focus on the constitutive models necessary to describe the explosive of interest. To study DDT phenomena experimentally, we used a granulated form of the explosive with the acronym CP-(2-(5-cyanotetrazolato)pentaaminecobalt(III) perchlorate), C2HIsNIoCoC1208. CP has been used in very limited applications and has not been completely characterized (much of the available data can be found in Refs. [9-12]). Upon thermal initiation, this explosive is highly reactive and the growth to detonation can occur within a run distance of several

MELVIN R. BAER ET AL. millimeters. Since a relatively small amount of material is required for growth to detonation, this explosive is ideally suited for studying DDT physics in less restrictive laboratory conditions and optical studies can be conducted with little damage potential. In the next section, we present experimental observations achieved through high-speed streak photography of the flame spread within heavily confined charges. We then review the theoretical modeling and describe the major aspects of the model. The model predictions are shown to agree reasonably well with the experimental observations. This modeling attempts to define the role of gas permeation during burning, and demonstrates how the compressional effects in both gas and solid phases combine with compaction effects to produce the transition to detonation.

2. EXPERIMENTAL OBSERVATION OF FLAME SPREAD IN A POROUS COLUMN

In this section, we describe an experimental technique for the optical observation of flame spread in a porous column of granular CP. As in prior DDT studies [1, 13], we recorded the trajectory of the burn front with a high-~speed streak camera. Our initial work examined 2.08-mm diameter columns of granular explosive confined in transparent Pyrex tubes of 3-mm wall thickness. These columns were uniformly packed to a desired partial density by incremental loading of the granular CP [crystal density (p~*) of 1.97 g/ cm3]. Sixteen pressings were normally used to form a 16-mm column. Optical microscopy of the unpressed explosive revealed a particle size distribution with a geometric mean diameter between 135 and 144 #m, and 0.6% of the material were fines of diameter less than 10 #m. However, after pressing, the crystalline structure changed dramatically and reexamination of the particle distribution revealed a markedly smaller particle size as a result of crystal

DEFLAGRATION-TO-DETONATION TRANSITION fragmentation [mean particle diameter (dp) was reduced to 10-15 #m] [11].1 At one end of the granular column, a bridgewire ignition source was placed. The alumina header contained two 0.5-mm diameter Kovar pins which were connected by a straight 0.05mm diameter, 1.25-mm long, 1-fl Tophet C bridgewire. In all tests the column was ignited by supplying a square-wave constant voltage pulse (100 /zs) to the bridgewire. Oscilloscope records indicated 70 A were typically delivered to the bridgewire during a 4-5-/xs interval. This relatively large current provided prompt ignition which was sufficiently reproducible to synchronize the flame spread process within the 35-/~s recording time of a streak camera. All optical observations of the flame spread were conducted with an Imacon image-converter camera. Briefly, an image of the housing was optically relayed to an S-20 photocathode that converted the photon signal to a photoelectron image amenable to electrostatic deflection and focusing at a P- 11 phosphor. This phosphor converted the photoelectron image back to a photon image that was coupled by a fiber optic faceplate to photographic emulsion. The inherent sensitivity of this camera was fully utilized in capturing low-radiance deflagration phenomena. The image sweep speed of 2 mm//xs was calibrated to within + 0 . 5 % by recording a certified megahertz-driven light-emitting diode in real time on the test records. The lateral magnification of the entire system was obtained before each experiment by photographing a grid attached to the experiment housing. The spatial and temporal axes of the camera were also recorded to complete certification [14]. Recordings were made on Kodak Royal-X Pan film that was processed in DK-50 for 10 min at 20°C, and the negatives were analyzed with a coordinate comparator to give flame position vs.

CP is a brittle material and in a unstressed state consists of crystal clusters. Crystal fragmentation occurs upon pressing in a column, even with low applied loading pressures.

17

time data. In all cases, the range of object radiance outstripped the system's dynamic recording range and thus appropriately located filters were often used to prevent overexposing the detonation region without blocking the low radiance of the flame front during deflagration. In early observations of the flame spread in Pyrex-confined columns of granular CP, we noted a lack of continuity between deflagration and detonation similar to the observations reported by other experimenters [1, 4, 13]. In particular, the loss of light intensity during deflagration suggested the possibility that a compacted "impermeable p l u g " formed before the onset to detonation. However, rather than reflecting an aspect of DDT physics, one could also conclude that the loss of light output was an optical aberration induced by the gross fractur= ing of the Pyrex as a stress wave was communicated ahead of the flame front. This concern motivated us to redesign the test columns to observe flame spread better during the transition to detonation. Heavily confined charges of CP were held in a steel housing slotted to accommodate a polycarbonate (Lexan) window for optical observation. With this technique, the recording difficulties were circumvented and a well-defined flame front trajectory was produced. Figure 1 depicts the stainless steel housing used to confine a porous column of CP within a cylindrical bore of diameter 2.08 mm. A longitudinal slot of 0.25-mm width and 1.27-mm depth gave optical access to the CP column. This slot was filled by bolting a " T " - s h a p e d Lexan window into the steel housing. To prevent a leakage path, the Lexan protruded into the 2.08-mm bore (by a few hundredths of a millimeter) and the excess material was reamed and honed to give a smooth contiguous periphery. Figure 2 shows an example of a streak record of flame spread in a CP column of an initial density 1.6 g/cm 3 confined in the stainless steel housing. Image-enhancement techniques were used to reveal details of the flame spread. Streak records were first digitized with a Perkin-Elmer microdensitometer which scanned 25-/~m square sampling areas located on 30-~tm centers pro-

18

MELVIN R. BAER ET AL.

L~ L~--[__

SECTION A-A (mm) (Rotated 90 Omg.CW)

Fig, 1. Steel confinement device with Lexan window.

ducing a 512 × 512 pixel array which was stored on a magnetic disk. Each pixel in this image array was assigned a gray level range between 0 to 255 for image reconstruction with or without enhancement using a Comtal Vision 120/Image-Processor [15, 16]. In this example two identical images were created. Then, the entire images were offset seven pixels to give a visually appealing image enhancement after the gray levels from one image were subtracted from the other (pixel by pixel). Clearly, in Fig. 2, the flame spread is continuous and the transition is a short region of rapid acceleration that culminates in a steady detonation. Traveling ahead of the intense combustion is a region of lower radiance which may be the leading compaction front. This region was most pronounced at the higher CP densities, and below an initial density of 1.4 g/cm 3 its existence was often difficult to ascertain. Flame front trajectories in porous columns of

10 mm

v

5/.ts Fig. 2. A streak photograph displaying the flame front trajectory during combustion of granular CP with initial density of 1.6 g/cm 3 (~b* = 0.8).

DEFLAGRATION-TO-DETONATION TRANSITION

/'

15 14

~13 S_~2 z

9

g z

7 6

2 1 0

0

1 2 3 4 5 6 7 8 RELATIVE TIME (ps)

Fig. 3. Temporal variation of the combustion front in CP columns of varied density obtained from the streak photographs.

CP are shown in Fig. 3 for the range of densities from 1.2 to 1.6 g/cm 3. To clarify detail, each plot was temporally offset from the other by 1 /zs. At densities greater than 1.4 g/cm 3, three distinct stages of combustion are clearly delineated by changes in the slope of the burn front trajectory. The first stage of convective burning was characterized by a flame velocity of about 1 mm//zs and lasted 1-2/xs. The combustion wave was seen to change abruptly into a different stage of combustion, burning at a higher constant rate of 1.35-1.60 mm//~s prior to the onset of transition--suggestive of compressive burning. This convective/compressive flame consumed 3-4 mm of column length and its velocity was a weak function of initial density (estimated velocity increase was 0.075-0.1 mm//zs per 0.1 g/cm3). As the flame spread continued, rapid acceleration occurred within a column range of 0.5-1.0 mm, immediately followed by steadystate detonation. At the low densities, there was often an absence of two-stage deflagration and both stages coalesced to give a limited run of high-speed, compressive burning which finally evolved into steady-state detonation. A common characteristic of the observed streak records was the occurrence of sudden change in object-radiance (brightness) from the combustion zone prior to the transition to detonation. This quantum jump in object-radiance (typically by a factor of 100-500) occurred within the rapidly accelerating, compressive

19

heating stage. At this point, (denoted by singleheaded arrows in Fig. 3), the combustion wave speed ranged from 2.5 to 3.0 mm/izs. At low densities, the combustion wave continued to accelerate following this rapid change in radiant output, which indicated that the described phenomenon was a predetonation event. The run distance to detonation was taken from the streak records at the point at which flame spread first reached a steady-state value (indicated by a double-headed arrow in Fig. 3) and this distance for various initial densities of CP is shown in Fig. 4. This figure clearly shows that there was a minimum run length to detonation in CP near an initial density of 1.4 g/cm 3 [solid volume fraction (4>~*) of 0.71. Confirmation of the distance at which detonation began was checked by postmortem examination of the steel housing for location of maximum enlargement of the bore. It was often difficult to judge with high precision the maximum bore, particularly for those columns loaded at low CP initial densities. However, at densities greater than 1.4 g/cm 3, maximum stress (as evidenced by the location of longitudinal deformation in the housing) and maximum bore diameter correlated very closely with the run distance to detonation, determined from the distance versus time streak records. In a separate experiment, tests were conducted to study the extent of the early stage of conductive burning. This mode of flame spread

8.0 i 7.S 7.0 6.5

6.0

I

I

I

I

~e J

I

l

I

I

I

I

i v

I

CALCULATIONS • STREAK-. ,~'

5.5 5.0 I-

U)

3

4.5 4.0

•

1.10

t

I 1.20

I

I 1.30

Q

i

I 1.40

i

i 1.50

i

I 1.60

INITIALDENSITY(O/Crn3)

Fig. 4. The run distance to detonation at various initial densities of CP.

20 following ignition was observed through the rear of a polished sapphire header. These observations indicated that the low speed (conduction-driven) deflagration was limited to a 0.5mm region surrounding the bridgewire. Past this region, the deflagration rate abruptly increased with the onset of convective burning. Additional tests were also conducted with shortened CP columns interfaced with a glassfiber rod and prism arrangements to allow the streak camera to record both the flame trajectory and the impingement of the flame front on a planar cross section of the column. In this fashion, the shape of the flame front during deflagration (parallel and perpendicular to the bridgewire orientation) was deduced based on the arrival time of the optical signal. Flame fronts were seen to be essentially planar. There was no evidence of a peripheral lag, suggesting that the Lexan window provided adequate confinement. These records also suggest that convective burning rapidly spreads laterally, annihilating possible early time wave front curvature which may have formed locally around the bridgewire in the ignition region.

MELVIN R. BAER ET AL. was the assumption that each phase occupied every point in the flow field. Associated with each phase was a material density, Pa, and a volume fraction, ~ba. In addition, we assigned a velocity, Va, pressure, Pa, and internal energy, ea, to each phase and conservation laws of mass, momentum, and energy relationships were formulated which linked transport processes in each phase with phase interactions. It is the phase interactions which brought the discrete nature of the mixture into the formulation. These balance laws (in conservative form) are Mass: 0

0

~-~ (t~aPa) q'- ~

(1)

Momentum: 0

0

(t~aPa Va) "1--2-'- (q~aPa+~aPaVa2)=ma+; dX at

(2)

Energy: 0

0

0t (q~aPaEa)+~X

((OaPafa+dPaPa)Va+qa)=ea+; (3)

3. T H E O R Y

In this study, we applied a theory of reactive two-phase flow developed in a previous investigations [5, 8]. In these works, a continuum mixture theory was proposed to describe onedimensional flame spread for a two-phase (solid/gas) system that treated each phase as compressible and in thermodynamic nonequilibrium; as such, each phase possessed its own pressure, density, temperature, entropy, etc. Rather than repeating formal derivations and the development of the mixture model, we shall state only the final forms of the conservation laws and focus on the constitutive aspects of the model. The constitutive assumptions used in this study were strictly tailored to the explosive CP and were motivated by independent experimental observations. Forming the basis of a continuum description

(t~aPaOa) ----C a + ;

where the total energy of each phase, Ea, is given b y 0a 2 Ea -- e a + - -

2

(4)

and the conductive energy flux is denoted by qa. The + superscript denotes an interaction between phases that was constrained so that conservation of mass, momentum, and energy of the total mixture was maintained. In this nonequilibrium treatment of fully compressible multiphase flow separate equations of state for each phase were used. Several parts of the model were linked to pressure differences that require accurate equation of state descriptions. For the solid phase we used a well-established thermodynamic description for the Helmholtz free energy in terms of the

DEFLAGRATION-TO-DETONATION TRA NSITION

21

density 0s and the temperature Ts [17]:

TABLE l

Equation of State and Transport Properties

¢~(p~, Ts)=Cv ~ [ ( L - T ~ ' ) × (l+£sPs(l:

Variable (cgs units)

~s))

CP

O,~ cv~ KT

"

F~p,

T, I J

N Dcj AHd~ ~j

The reference state was taken as T*s = 300K and the unstressed crystal density was Or = 1.97 [g/ cm3]. The values of the bulk modulus Kr, the specific heat Cv~, the Grfineisen ratio Fs, and the parameter N (see Table l) were determined using experimental shock loading data [ 10]. The pressure, Ps, and the internal energy, es, could be then determined from the thermodynamic identities as follows [5]: Os 2 Ol~s

cgp, '

0~s

e,, = ,t/,,,- T~ O--Tss

(6)

For the combustion gas equation of state, we used the Jones-Wilkins-Lee (JWL) equation of state [18], which takes the following form:

Pg=A

(

O),Og )

1 Rl¢a~p o

exp

(Rl{/}sPs)

s

+B 1

w~

Pg

_

Rzdp~O~,/ + Oopgeg.

exp

6.9 107

pgret

+o(N(N-^ 1) L\P~ /

P~ =

1.97 (g/cm ~) 6.47 × 106 (erg/gm K) 4 × 10 tt (dyne/cm 2) 1.14 (g/cm ~) 2.1 (1.12 + 6.670 1 ) x l0 s (cm/s) (5.36 - 2 . 3 2 ¢ , ~) x 10 I°(erg/gm) X (dyne/cm:) 0.01 33.0 6.1 x 109(dyne/cm -~) 143.8 0.73 2.115 2.32 X 107 (erg/gm K) 8 x 10 4 ( g / c m s ) 1.24 X 103 (erg/cm s K) 4.2 x 104 (erg/cm s K) 4.43 1.95 2.39 x 10 ~2(dyne/cm 2) 0.119 x 1012 (dyne/cm 2) 0.23 4.5 x 105 (cm/s)

s

Pg (7)

It is noted that at low gas densities the exponential terms are small and the limiting form of this equation of state is the ideal gas law. The constants A, B, R~, R 2 , 60 were experimentally determined and varied with the initial density of

a2 P* ct c2 c3 C,v /z~.

kg k~ Rt R2 A B o: as

I

the porous CP. Two densities have been examined to match hydrodynamic calculations of experimental cylinder expansion tests [ 19]. Data for ~b~ -- 0.61 and 0.83 were given in [10]; to determine the JWL constants at other densities an extrapolation method [20] was used. Linear fits of R~, R2, and co were established using experimental data of detonation velocity, and the heat of detonation was estimated from chemical equilibrium calculations using the TIGER code [21]. At Chapman-Jouget detonation conditions, the following relationships [22] were used: pgCJ( 1 1 ) A H d e t = eg CJ --

(8)

pgCJ __ q~ s P s ° ocJ2 ,

(9)

( r g + 1)

22

l°g

MELVIN R. BAER ET AL.

= p..EgOPg

(10)

Pg Opg

The values of Dcj and AHdet are given in Table I. Gas-phase temperatures were computed from

Tg=--~-g(eg+AHdet

M q~s+ u~ ~xx ~bs=

C0pg

lOPs Ps

VPs Ps

Pg

R B R °------~ exp 2q)s Ps

~o

0\\

Pg

,

(11)

//

O4~s_ f + Cs+ Vs' ms+ = - r a g + = P g -~x

(13)

Pg-~x-/

, (17)

Ps

(18)

,

( P s - Pg-/3s)

Ts - Tg) +EsCs+,

- h(

+

and #c is the compaction viscosity. In describing the combustion of the granular explosive we employed a grain burn model given by Eq. (12), where a depends on specific surface area. Strand burn tests have been conducted by Stanton [231 for CP (see Fig. 5) and a linear increase of burn rate with increasing gas pressure was found. In comparison, CP strand burn velocities were typically an order of magnitude faster than those observed for HMX [24]. Unfortunately, this type of burn rate determination was conducted at pressures which are much lower from conditions relevant to detonation and we were forced to accept an extrapolated estimate of the burn rate at detonation conditions. To complete the description of the combustion we specified the surface area associated with the linear grain regression rate. An estimate of the 100

/.to

Pg-/3s)

~)g= 1 --(~s,

where Cvg is the constant specific heat of the combustion gas mixture. As developed in [5, 7] the Second Law of Thermodynamics suggests admissible forms of the phase interaction terms, heat flux, and a dynamic equation for volume fraction. On this basis, the phase interaction terms for mass, momentum, and energy exchange (denoted by the superscript) were proposed, of the following form [51: Cs + = --Cg + =otPg n, (12)

- (Ps-/~s)

(Ps-

#c

which was constained by the saturation condition,

A R -°-~-; exp

es÷ = - e g ÷ =

In addition, the entropy inequality suggests a model for changes in the solid volume fraction which is consistent with the treatment of pressure nonequilibrium [5],

l

j

i

i

i

llll

I

I

I

1 ~

I

I

II,,_

er,, o ,,,:

LcP S T R A N D

DATA-

.'"e

F (Stanton) "~.V'""

(14)

'l

where the drag force f is given by v

f= Cd+

(Vs-- Ug),

10

"

STRAND

I

i

DATA

/

(15)

om o~ is the shape factor, Cd is the drag coefficient, and h is the heat transfer coefficient. The heat flux for each phase, qa, is given by Fourier's law of heat conduction:

1

[

i

i

i

10 3

aTa -/ca --

0x

(16)

i

i

i

i

i

i

10 4 GAS

qa =

i i i iI

PRESSURE

i , ai 10 5

(10 4 dyne/cm

2 )

Fig. 5. Strand burn rate data for CP with comparative HMX data [24].

DEFLAGRATION-TO-DETONATION TRANSITION specific surface area of the granular reactant was made by considering packed spheres of a mean diameter. In a preliminary examination of the burn rate law, this surface area produced deflagrations which were faster than observed experimentally for porous charges. Thus, a multiplicative correction to the specific surface area was included in the burn rate which decreases the rate of combustion at high initial densities. 2 With the strand data and the modified specific area of the granulated reactant, the burn rate for CP was formulated as -6~sOs

o //

Pg ~n=l

c~+---dr F(4~)~)_~_

,

(19)

where F(~b;) =min{12.5, 0.5 exp(4.1~bg)} [cm/s]. (20) Burning at this rate was allowed only when the two-phase mixture reached an " i g n i t i o n " temperature defined by a volume fraction-weighted temperature. This ignition criterion was based on a temperature at which rapid decomposition of the solid reactant was expected [25]. Thus, burning of the solid reactant was when (Os Ts +

~g

Tg) >

Tig n -

450K.

(21)

Admittedly, this criterion is a simplified one and much remains to be clarified concerning various thermal/mechanical/chemical mechanisms that initiate combustion at very high pressures. In this study, we have found that the ignition temperature was not a sensitive model parameter and, thus, accepted this criterion. Our description of interphase exchange of momentum, ms + , included three effects. The momentum exchange due to volume changes was given as the gas pressure multiplied by the gradient of solid phase volume fraction. The remaining terms described interphase drag and

23

the momentum exchange resulting from mass exchange. In this study, the interphase drag relationship was determined from recent experimental data. Using a shock-tube technique, Shepherd and Begeal [26] have experimentally determined the permeability of granular explosive beds. Flow characteristics were measured over a wide range of overpressures ( < 20,000 psi) in columns which were precompacted to a known density. The initial density dependence on permeability for granular CP is shown in Fig. 6. Note that the Ergun drag law [27], commonly used in parallel studies, overpredicts the permeability, and the data more closely follow one posed by Rumpf-Gupte [28]--particularly at low porosities. To treat the high Reynolds number flow, we followed an approach suggested in Dullien [29] and scaled the interphasial drag using Darcy numbers similar to those predicted by the Ergun correlation. Thus, the drag coefficient was defined as C

d

___( Ol R e ) #g 1 + , K

(22)

(~)s

where #g is the gas viscosity, and the permeability, K, and the Reynolds number Re were given by do%g 4.5 x= - ,

(23)

Ot2 '

i

,

i

i

i

,

I

i

I

,

POROU$ IIHOCK TUBE D A T A

lO-t

(MEAN

PARTICLE

SIZE"

10-18/~m DIA.)

10"1° >. 10-11 Ikl I

10"12 10"13 10-14

J 0,2=~33.01

|

| ,

10-11 -" in the limit of total pore closure the specific area available to surface burning should be substantially reduced. Unfortunately, the packed sphere analogy does not correctly yield this limit since the specific surface is given by S / V = 6 4 ~ J d o.

1.4

1.5

1.11

1.7

1.8

1.9

2.0

INITIALDENSITY (glc m3) Fig. 6. The variation of permeability of granular CP at various initial densities.

24

MELVIN R. BAER ET AL.

Re = Cgogdp l v~ - v, I #g

(24)

The data fit constants oq, c~z are given in Table 1. The interphase exchange of energy, es +, occurs due to the work done by the drag forces, work done in compacting the granular bed, local convective heat transfer, and exchange of energy by mass transfer. For the convective heat transfer we used a heat transfer coefficient, h, given as follows [30]: h = 12kg~bg2/3 dp 2

(1 + 0.2 Re2/3).

~s= P,(¢s)- PJ(¢ ~ ),

Pi p- -*=

1 + c ] l ¢ s - c21c3, (27)

(25)

A sensitivity study using modified heat transfer coefficients produced minor variations of the flame spread as viewed through the transient growth of gas pressure. We have additionally incorporated variable transport properties #g and kg in the constitutive models implementing the dense gas correlations outlined in Reid et al. [31]. Representative C-J state gas mixtures, calculated using the TIGER chemical equilibrium code [21], were used in these estimates and low-pressure properties were calculated using Chapman-Enskog theory [32]. The volume fraction rate equation is a simplification of one posed by Carroll and Holt [33] whereby one neglects inertial effects. In this model of dynamic compaction, material viscosity effects are balanced to local pressure differences. The compaction viscosity, ~c, was estimated using a simplified transport model similar to that posed in the kinetic theory of gases using a mean particle diameter, dp (as the mean free path), the solid phase sound speed, as, and the crystal density, p~: I.zc -- dpasPs .

pressure associated with interparticle stress which resisted the changes of the granular structure. The configuration pressure /3s was estimated using experimental data of pressure versus partial density of statically-loaded granular columns. Shown in Fig. 7 is the reported data for CP as given in Ref. [25]. The function/3s was constructed so that no resistive stresses existed at the initial pressed density state. Thus

(26)

Changes of volume fraction take place due to the opposing influences of pressure nonequilibrium, and the combustion which causes volume change by mass exchange. Imbedded in this model was a pressurelike quantity -/3~. We interpreted this quantity as the configuration

where ¢* is the initial solid volume fraction of the unreacted material and P * , ca, c2, c3 are data fit constants given Table 1. 4. NUMERICAL METHOD Solution to the partial differential equations describing the reactive multiphase flow was numerically obtained by a method-of-lines (MOL) technique in which spatial gradients are discretized by central differencing. The resulting set of ordinary differential equations (ODEs) was then integrated using previously developed 10.0 E o

i

,

i

,

5.0

c

% v ul E

u~

1.0

o

o.5

U) tu n. el Z Q <[

o,

0.1

*

0.7

I

0.8

I

I

0.9

1.0

(~)s (Solid Volume Fraction) Fig. 7. Quasi-static loading pressure at various solid volume fractions used in defining the configurational stress of granular CP.

DEFLAGRATION-TO-DETONATION TRANSITION software [34]. Details of the difference methods, incorporation of boundary conditions, and artificial viscosity are described elsewhere [5, 35]. The initial conditions used to start flame spread involved permitting a few nodes to burn at a low pressure. Consequently, convective burning was immediately induced and thus the very early stage of conductively-driven flame spread was not examined in this work (experimental evidence discussed earlier indicated that the conductive heating zone was small, - 0 . 5 mm in CP). The number of preignited zones was varied in preliminary calculations and, in each case, the combustion wave was seen to evolve rapidly into a self-sustaining deflagration driven by convective effects. Further, it was noted that variations of initial conditions had a minor affect on later accelerated combustion wave structure. The incorporation of fast burn rates and low permeability in the model caused the set of equations to be numerically stiff [36]. This problem was resolved with appropriate ODE software. MOL is truly a generalized method, although not always recognized as such--many finite difference, multistepping methods are subsets of this approach. The robustness of the method is due to the use of ODE solvers which have been extensively developed to handle efficiently the time integration. With the availability of the vectorized, large storage, computer systems, such as the CRAY-1, several thousand coupled ODEs can easily be integrated using a variety of solvers. These external routines monitor the solution, internally control the time stepping, and indicate to the user when a method is being used inefficiently. The apparent stiffness of the coupled ODEs suggests that finite difference methods such as McCormick or L a x - W e n d r o f f [37] may not be appropriate for the solution of these types of problems. Typically, when stiffness is encountered in an explicit solution, a drastic reduction of the time stepping is required to maintain stability and accuracy. When this reduction occurs the method becomes computationally expensive or even impractical. To bypass these problems, an implicit ODE solver (DEBDF)

25

[38], using backward differencing formulas, was necessary for modeling flame spread in granular CP beds. Although the incorporation of an implicit solver typically uses more overhead of computer storage and time, computer run .ime of a DDT case was typically several hundred seconds on the C R A Y - I . 5. N U M E R I C A L C A L C U L A T I O N S A N D C O M P A R I S O N TO E X P E R I M E N T In this section we compare numerical results from the reactive two-phase model and the experimental observations described earlier. In the one-dimensional calculations to follow, a porous 1 cm long column was represented using a computational domain of 101 evenly spaced nodes. (Additional calculations using two- and fourfold increases in computational nodes were conducted and adequate model numerical resolution was assured.) As an idealization, we considered this porous column to be of uniform density and composed of granulated reactant with a uniform particle diameter (dp) of 15/zm. This mean particle diameter was suggested from experimental photographic scans of the deconsolidated prepressed material [26]. ~ Shown in Figs. 8-10 are numerical-experimental comparisons of the trajectory of the burn front with time within columns of initial densities 1.2, 1.4, and 1.6 g/cm 3. The trajectory of the luminous burn front was taken from imageenhanced photographic streak records and compared to the nodal location of the burn front predicted by the model. As seen in these figures, the flame spread from deflagration into detonation was predicted reasonably well. During deflagration, the calculations indicated two distinct modes of flame spread (convective and compressive)--each region characterized with a rapid transition to the next mode of burning. Interestingly. experimental observations of the deflagration by streak photography confirmed

Recall that the CP crystals fragment upon pressing in a colunm and the mean particle diameter may vary with initial volume fraction.

26

MELVIN R. BAER ET AL. 15

[ -

I

I

CP

1S

I

I

1.2 g/c,'r7 3

I

CP

10 -0IU

--

I

I

I

-

1.6

9/c.~ 3

~o

Dc d =

0Z

Z ,< I,-

Dcj

= 6.2 mm

I,,,-

Q

a

Z :)

,,,

Z

5

-

m

CALCULATION. ~

--

o

",'--T

• J

* • STREAK DATA I 2

"

1

i 3 TIME (ju~)

i 4

I

5

6

Fig. 8. A comparison of calculations and experimental data of the temporal variation of the combustion front for CP with initial density of 1.2 g/cm 3 (4~ = 0.6).

15

i CP

I -

i

I

I

1.4 9/cm 3

A E

10

= 5.6 m m / p s /

Dcj

W

U Z I-

/

Q

5 S

m

STREA

m

y e e

J

i

I

•

o 1

T

R

~

"OALOLATION

'

2

S

3 TIME ( F I )

4

i 6

Fig. 9. A comparison of calculations and experimental data of the temporal variation of the combustion front for CP with initial density of 1.4 g/cm 3 (4)* = 0.7).

the existence of these two linear growth regions prior to the transition to detonation. These regions occurred within a time frame of 2 /~s relative to the onset of detonation, and flame spread was typically 1-2 mm/#s. (In contrast, many propellants and explosives exhibit lower rates of deflagration and this stage of convective/compressive burning may span times of 10100 ms.) For all densities of interest, rapid flame spread was observed to undergo the transition to detonation in a smooth fashion

0

~

I

1

CALCULATIONS I

2

3 TIME (~s)

I

[

4

5

6

Fig. 10. A comparison of calculations and experimental data of the temporal variation of the combustion front for CP with initial density of 1.6 g/cm 3 (~b~ = 0.8).

rather than by an abrupt change suggested from earlier observations. Consistent with measured values, detonation wave velocities were predicted to increase from 5.2 to 6.5 mm/#s as the initial density increased from 1.2 to 1.6 g/cm 3. Shown in Figs. 11-13 are the temporal and spatial variations of pressure and temperature of the solid and gas phases and the solid volume fraction as predicted by calculations for an initial density of 1.4 g/cm 3. In the gas phase, early time pressure wave development was dominated by the effects of gas permeation. Following ignition, the gas evolved by combustion quickly established a pressure gradient to induce flow into the unreacted porous material. Only a small rise in pressure (less than a few kilobar) was typical of this burning regime. As burning proceeded, the gas generated by combustion exceeded that which was permeated ahead of the combustion wave and the pressure within the reaction zone increased. With the pressure dependence of the burn rate, the gas generation rate became faster and a large pressure gradient was established at the burn front. At this point, inertial effects became important and the hydrodynamics formed a compressive shock wave. An overlay of solid and gas pressures shown in Fig. 11 clearly indicated pressure differences of tens of kilobars, particularly near the burn front.

27

DEFLAGRATION-TO-DETONATION TRANSITION SOLIDPHASEPRESSURE

SOLID PHASE TEMPERATURE

1Oo

~'o.o GAS PHASE TEMPERATURE

GAS PHASE PRESSURE

to. o #o.o Fig. 1 1. The t e m p o r a l and spatial v a r i a t i o n of solid- and g a s - p h a s e p r e s s u r e s for CP with initial density of 1.4 g / c m 3

co," = 0.7).

Fig. 12. The t e m p o r a l and spatial v a r i a t i o n of solid- and g a s - p h a s e t e m p e r a t u r e s for CP with initial density of 1.4 g/ cm 3 (~b~ = 0.7).

Figure 12 displays the bulk temperatures of each phase. During convective burning, the hot combustion gases were forced into unreacted pores and the gas-phase temperature profile exhibited a thermal boundary layer ahead of the flame front. This thickness was roughly 1 mm long. As the burning accelerated, the gas within the reaction zone was trapped and gas compression occurred as the bed compacted. As the deflagration changed to compressive burning, the combustion gases within the primary flame zone approached 10,000K. Although not di-

rectly measured, high gas temperatures have been observed in the streak photographic records where, prior to detonation, the radiance output of the combustion gases was observed to increase quickly by two orders of magnitude. The calculations did not include the effects of ionization or dissociation and thus predicted gas temperatures may be too high. In the solid phase, the early rise of bulk temperature was due to the convective heat exchange. However, the compressive heating through the effects of solid-phase compression were clearly evident as

28

MELVIN R. BAER ET AL. SOLID PHASE VOLUME FRACTION

,e o

tO. o

Fig. 13. The evolution of solid volume fraction during the combustion of CP with initial density of 1.4 g/cm 3 (~h,° =

0.7). the flame spreads from compressive burning into detonation. To show the development of compaction during burning, the variations of solid volume fraction are shown in Fig. 13. During convective burning significant compaction occurred, driven by the pressure nonequilibrium that existed between the phases. During deflagration the extent of the compacted region extends beyond the burn front at a maximum distance of 2 mm and maximum solid volume fraction of ~s = 0.9 was predicted. As a result of this flow restriction, the gas permeation layer collapsed and pressure built up rapidly. The increased gas pressure accelerated the flame and eroded the compaction zone until it reaches the steady state associated with detonation. Upon reaching detonation all of the compaction and compressibility effects were within the reaction zone as opposed to leading the combustion as observed during the deflagration stages. Predictions of the run distance to detonation at various initial densities of CP are shown with experimental data in Fig. 4. This distance was inferred as the point where a steady detonation (constant velocity) combustion wave had formed. Although the experimental data exhibited some data scatter, it was evident that a

minimum run distance to detonation occurred near a density of 1.4 g/cm 3. Calculations produced a similar trend and an understanding of this minimum was suggested from the modeling. At the higher initial densities, the increase in run length can be attributed to two effects. With increased density, early time combustion behavior will cause significant compaction and this void closure will reduce flame spread prior to development of significant pressure buildup because of reduced convective heat transfer. In addition, at the high densities, less surface area participated in the combustion. At the other extreme, (low density), high gas penetration into the porous reactants prevents the development of the permeable " p i s t o n " due to lack of strong interphasial momentum exchange. 6. S U M M A R Y A N D C O N C L U S I O N S Experimental optical measurements of the flame spread within porous CP charges were taken through a Lexan-filled window in a stainlesssteel tube. The image-enhanced, streak photographs of the luminous burn front during flame spread provided a diagnostic tool for studying the various modes of combustion from deflagration to detonation. We compared numerical calculations of the flame spread to these experimental flame front trajectories. In this one-dimensional model of reactive multiphase flow, we have incorporated experimentally determined grain recession rate, permeability, and configurational pressure for granular CP. A multiplicative correction to the granular specific surface area was included in the burn rate model which decreased the rate of combustion at low porosities. An important feature of this reactive two-phase theory was the incorporation of pressure nonequilibrium. We allowed each phase to obey separate equation of state descriptions and enforced a dynamic model for changes in volume fraction which was consistent with the thermodynamics of multiphase flow. Therefore, each phase was treated to be compressible and the mathematical difficulties, known to occur for multiphase flow with pressure equilibrium [39], were avoided. Nu-

DEFLAGRATION-TO-DETONATION TRANSITION merical solutions required the use of an implicit method which accurately followed the disparate length and time scales induced by the coupled effects of multiphase hydrodynamics and phase interaction. Details of numerical calculations showed that the effect of gas permeation has a dominating influence on the wave growth during early accelerated combustion provided that sufficient confinement, abundant surface area, and rapid chemistry support the burning. The configuration and transport characteristics of the granular reactant (i.e., pressed particle size, porosity, permeability) are critical model parameters in determining this mode of combustion. However, as combustion induces rapidly growing pressure, mechanical compaction and the multiphase hydrodynamics combine to produce the transition from deflagration to detonation. The important model inputs critically linked to this latter stage of burning depend are those parameters which govern the rate of dynamic compaction and the equations of state describing highly compressed gas and solid states. To substantiate the choice of model inputs, we have relied on a wealth of independent experimental measurements and have demonstrated that the incorporation of this data in a multiphase flow model predicts reasonably well the burn front trajectories. Although many of these model inputs were strictly applied to the explosive, CP, the framework of the multiphase reactive flow formulation is sufficiently general to be applicable to other granular energetic materials, provided that the constitutive and equations of state inputs can be determined. This model is far from being complete and there is a need to refine the combustion descriptions to include the effects of solid phase decomposition/ pyrolysis/gas-phase kinetics [40] with the variety of " h o t - s p o t " mechanisms which enhance combustion wave growth. Our future work will be directed to addressing these important issues.

tional Laboratory, Livermore) and the encouragement and support of M. Lieberman. We wish to thank R. J. Haushalter for his experimental designs, for implementing new photographic techniques, and for carefully conducting the experiments in this work. Our thanks are additionally extended to F. M. Tamashiro f o r automating the data reduction and generation o f the image processing. REFERENCES 1.

2.

3.

4. 5. 6.

7.

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10.

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12. 13.

We gratefully acknowledge the many fruitful discussions with D. B. Hayes, J. Kennedy, P. Stanton, D. A. Benson, H. Krier (University o f Illinois), and E. Lee (Lawrence Na-

29

14. 15.

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Received 10 October 1985; revised 30 January 1986