# Inert plug formation in the DDT of granular energetic materials Page: 4 of 6

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where pc is the compaction viscosity, P is pres-

sure, and Q is volume fraction (7). The subscripts

"s" and "g" refer to solid and gas, respectively.

The configuration pressure, 0, is modeled using

min[-max((#3 - ,o), 0)rln(Og)/Pg, /3ma] where

T is a stress coefficient (7). The volumetric heat

transfer coefficient, N , is assumed to have the

form, N = (3kHO,)/(a2) where ap is an effec-

tive particle radius, ap = apomin(1, (8/Pso)i/3)

Vigorous reaction significantly affects interphase

heat transfer. To crudely account for this, kH

H(I - 0.5)ki + (1 - H(I - 0.5))k2 where I is

the induction time variable (0 < I < 1 ), H is

the Heaviside step function, and ki & k2 are

constants. The source term assumed for the en-

ergy equation is E = -PgF - R(Ti a - Tg).

The first term corresponds to depositing all com-

paction work with the solid (7), and the second is

Newton's law for heating the solid where Tint is

the interface temperature.

Heating of the Solid

We need an estimate for Tint to be used in the

heat transfer potential and as an ignition criterion.

The solid energy equation can be written as,

De P Dp+ (Ps - Pg)F N(Tg - Tint)

Dt p2 Dt + (ss) + (.ss)

(1)

The terms on the right hand side of Eq. (1) cor-

respond to heating (increase in internal energy)

via compression, compaction (frictional, plastic

work, etc.), and heat transfer processes, respec-

tively. This heating is not uniform; "hot spots"

form which act as reaction centers. As a first

approximation we assume compaction heating is

deposited at the particle interface (in addition to

convective heat transfer), and compressive heat-

ing is uniformly distributed. We employ inte-

gral methods (e.g., (9)) to obtain an expression

for the interface temperature. The temperature

profile in the particle is assumed to be of the

form (T(r) - T,)/(Tint - T,) = (r/apo)'m, where

T, is the centerline temperature and m is a

constant. We obtain the bulk temperature, Ts ,

in terms of T, and Tint by integrating the as-

sumed temperature profile to yield Ts = (mT, +3Tint)/(m + 3). Before significant reaction oc-

curs, the boundary condition at the interface of

a spherical particle is then k, (aT/r)rP

(ap/309) [N(Tg - int) + (P - Pg)Y]. Applying

this boundary condition to eliminate T, gives

(m + 3)T + 3BiTg apoap(P - Pg).F

nt m + 3Bi + 3 3 ,ks(m + 3Bi + 3)'

(2)

where Bi = hapo/3k, = kHapo/3apk3 . When

vigorous reaction begins, this expression is no

longer appropriate because reaction was neglected

in its derivation. Therefore, Tint is not allowed

to exceed 600 K.

Effect of Pore Size on Ignition

Ignition and deflagration of energetic materials

are greatly influenced by the gas-phase zone (pre-

heat and reaction zone), often called the stand-off

distance, Lg. The pore size, LP, in a damaged

explosive is estimated to vary from about 0.2 to

15 pm for porosities of 0.01 to 0.4, assuming an

effective particle size of 35 gm. The stand-off

distance of HMX is estimated to vary from about

0.5 to 50 m for pressures of 100 to 0.1 MPa,

respectively (e.g. (11)). Therefore, conditions can

exist where gas-phase ignition would be inhibited

by the pore size (Lp < Lg ); as well as, the case

where LP > Lg (see Fig. 2) (cf. Chapt. 3 of (2)).

To investigate the effects of pore size on ig-

nition and burning we consider an additional ig-

nition constraint for gas-phase driven combus-

tion to occur, Lp/Lg > const. Lg in the

simplest approximation scales with kg/(psrbCg)

(e.g. (11)) where kg , ps , and Cg are the

conductivity of the gas, density of the solid and

heat capacity of the gas, respectively. In steady

gas-phase driven combustion rb ~ kp(Pg/Dp)bn

where bn is a constant and DP is a scaling

constant, so Lg ~ kg/(pskp(Pg/D)bnCg) . A

measure of the pore size is Lp ~ Vr where

rs is the permeability. Assuming the correla-

tion for permeability adopted by Baer & Nun-

ziato (BN) (10), we find Lp ~ ap5225. Thus,

Lp/Lg ~ ap42S(psk(P/D)b Cg)/kg . Lump-

ing nearly constant parameters into a specified

threshold parameter, Rko , we require for gas-J

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Son, S.F.; Asay, B.W. & Bdzil, J.B. Inert plug formation in the DDT of granular energetic materials, article, September 1, 1995; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc624740/m1/4/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.