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)
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)'
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-
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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.