Numerical simulations of flows of reacting two-phase media in a two-velocity, two- temperature approximation are used to study the shock-wave initiation of ...
Combustion, Explosion, and Shock Waves, Vol. 35, No. 3, 1999
Numerical Simulation of Shock-Wave Initiation of H e t e r o g e n e o u s D e t o n a t i o n in A e r o s u s p e n s i o n s of A l u m i n u m Particles A . V . F e d o r o v 1 a n d T . A . K h m e l '1
UDC 532.529+541.126
Translated from Fizika Goreniya i Vzryva, Vol. 35, No. 3, pp. 81-88, May-June 1999. Original article submitted November 18, 1998. N u m e r i c a l s i m u l a t i o n s of flows o f r e a c t i n g two-phase m e d i a in a two-velocity, twot e m p e r a t u r e a p p r o x i m a t i o n are u s e d to s t u d y t h e shock-wave i n i t i a t i o n o f d e t o n a t i o n in a e r o s u s p e n s i o n s o f a l u m i n u m particles in oxygen. T h e c o n d i t i o n s in a h i g h p r e s s u r e c h a m b e r u n d e r w h i c h d e t o n a t i o n can develop a f t e r r u p t u r e of a d i a p h r a g m are det e r m i n e d . Two i n i t i a t i o n scenarios are established t h a t d e p e n d on t h e localization of t h e i n i t i a t i o n source. I t is s h o w n t h a t i n i t i a t i o n brings on a self-sustained d e t o n a t i o n r e g i m e ( C h a p m a n - J o u g u e t or i n c o m p l e t e l y compressed, d e p e n d i n g on t h e r e l a x a t i o n p a r a m e t e r s ) . T h e r e q u i r e d i n i t i a t i o n e n e r g y is e s t i m a t e d a n d i g n i t i o n c r i t e r i a are f o r m u l a t e d . T h e possibility o f d e t o n a t i o n initiation w h e n insufficiently s t r o n g shock waves are reflected f r o m a rigid wall is discussed.
INTRODUCTION Gaseous suspensions of aluminum particles belong to the class of media in which detonation processes are not ideal. In experiments, unburnt particles are observed among the detonation products, perhaps because the formation and growth of an oxide film on the surface of the particles inhibits complete combustion. Combustion and heat transfer between the gas and particles take place simultaneously, so it is possible to have a state in which the heat release by the chemical reaction is less than the heat expended in heating the particles, and the total heat release is nonmonotonic. Under these conditions, in free detonation, besides the Chapman:]ouguet regimes, there are undercompressed stationary regimes with a transition through a frozen sound speed and a completely supersonic final state or with an intersonic final state (the relative velocity is greater than the equilibrium value, but less than the frozen sound speed). Similar detonation regimes have been analyzed for relaxing gases [1-3]. For gaseous suspensions of aluminum in oxygen, the stationary detonation regimes, including undercompressed ones, have been studied [4, 5] on the basis of a mathematical model 1Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. 288
of heterogeneous detonation [6] that was consistent with experimental data [7] on the detonation velocity as a function of the initial concentration of the particles in the aerosuspension. A map of the solutions in space has been obtained [5] in a two-velocity, two-temperature formulation (it included the detonation velocity and relaxation parameters, with the regions and manifolds corresponding to each type of stationary solution indicated) and the features of the local flow structure in the various detonation regimes were studied. It has been confirmed numerically [8, 9] that the Chapman-Jouguet detonation regimes and the undercompressed detonation regimes with supersonic final states (with respect to the frozen sound speed in the one-velocity formulation [8] or with respect to the equilibrium sound speed in the twovelocity formulation [9]) are stable. In these regimes, the detonation front does not interact with the adjacent rarefaction wave and does not attenuate it.
In the present paper, we use a two-velocity, two-temperature approximation for the mechanics of reactive heterogeneous media to extend an earlier study [10] of shock-wave initiation of detonation in gaseous suspensions of aluminum particles for different values of the relaxation parameters which determine the development of one or another selfsustaining detonation regime [9].
0010-5082/99/3503-0288 $22.00 ~) 1999 KluwerAcademic/PlenumPublishers
Numerical Simulation of Shock-Wave Initiation of Heterogeneous Detonation FORMULATION
OF THE PROBLEM
We shall consider a half space bounded by a solid wall, part of which is a high pressure chamber isolated by a diaphragm. Let us assume that an initiator in the high pressure chamber has released energy, aluminum particles burnt, and the mixture entered a new state characterized by a high temperature and pressure. Outside the high pressure chamber, the parameters in a low pressure chamber correspond to the initial state. At time t = 0, the diaphragm is ruptured. The propagation of the plane detonation waves which then develop in the low pressure chamber in the two-phase reactive mixture can be described by the following system of equations: c9pl
(9(plul)
0-T +
o--S--
a(p u ) at
&
j,
a(p2u2)
Op2
-
j,
+
a--g-
-
f + Ju2,
-
a(p u +p) +
a(p2u2)
Oz +
a(p2u]) _ _ bx
_
f
_
c~z
=
(1)
Yu2,
a(mE1) a[(mul(E1+Win)] O'---'~- +
289
-q -
fu2
+
JE2,
cg(p:E2) O(p~u2E2) c9--'---~ + Oz - q + f u 2 - JE2.
To close this system of equations, we invoke an equation of state (assuming a low volume concentration of the particles) p = p1RT,
particles, respectively. In Eq. (3), ~ = p2/p is the relative mass concentration of the particles, p = Pl +P2, Ea is the activation energy, ~fin is the fraction of unburnt particles, T~sn is the ignition temperature, and v~ is the burning time. The characteristic times for the velocity (ru) and thermal (VT) relaxation processes are, in general, variable and depend on It has been shown [4] that including this dependence does not qualitatively affect the properties of the flow structure behind a detonation wave front and confirms, for example, the existence of a solution with an internal acoustic point (undercompressed detonation), so in this paper the times 7"u and VT and the relaxation parameters a = VT/7"u and/3 = TT/T ~ are assumed frozen, as in [9]. The initial value-boundary value problem for the system of Eqs. (1)-(4) is formulated as follows:
E1 = - ~ + cv,l T,
(2)
..2
E2 = 2
F i g . 1. H a r d i n i t i a t i o n of a C h a p m a n - J o u g u e t wave.
+ cv,2T2 + Q,
t = O, ~ =
a formula for the combustion of the aluminum particles that takes incomplete burnup into account, re Ea x exp ( - ~--~2) max(0, sign (T2 - T~zn)),
(3)
and formulas for the flow around the particles and heat exchange between the gas and particles, f = P2 (ul - u2), q -- p2c~,,2 ( T - 7"2). ru VT
(4)
Here p is the pressure, pi, ui, Ei, and ev,i are, respectively, the average density, velocity, total energy per unit mass, and specific heat of the ith phase (i = 1, 2), T and Tz are the temperatures of the gas and particles, and Q is the heat released in the chemical reaction. The subscript 1 and 2 refer to the gas and
~t, ~o,
0
