Role of the Structure of Heterogeneous Condensed Mixtures in the Formation of Agglomerates, CHEMIA I PIROTECHNIKA, ...

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Combustion, Explosion, and Shock Waves, Vol. 38, No. 4, pp. 435{445, 2002
Role of the Structure of Heterogeneous Condensed Mixtures
in the Formation of Agglomerates
S. A. Rashkovskii
1
UDC 536.46
Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 4, pp. 65{76, July{August, 2002.
Original article submitted August 3, 2001.
Mathematical simulation of the structure of metallized heterogeneous condensed mix-
tures is performed. Evolution of a system of aluminum particles is studied in the
case a heat wave passes over the mixture. It is shown that rapid heating of a het-
erogeneous condensed mixture forms a system of \clusters" of contacting aluminum
particles, which may sinter to form a porous system that melts and disperses into
individual droplets with further heating under the action of surface-tension forces.
After coalescence, these droplets form agglomerates. The structure of \clusters" of
contacting particles is studied, and the mean-mass size of metal particles is deter-
mined as a function of dispersion of the components and their concentration in the
heterogeneous condensed mixture. It is shown that contacting aluminum particles in
the heterogeneous condensed mixture form fractal-like structures, which may play a
signicant role in the course of combustion of the mixture.
Key words: agglomeration of aluminum, combustion of composite rocket propel-
lants, propellant structure, ammonium perchlorate, binder.
INTRODUCTION
reaction with oxygen-containing products of decompo-
sition of other components.
The process of metal agglomeration in HCM com-
bustion has many stages [1{8] and starts in the depth
of the condensed phase (c-phase) from inert heating of
components in a heat wave. Heating of the contact-
ing metal particles in HCM leads to their sintering [9],
melting, and coalescence due to surface-tension forces
into larger structures | subagglomerates, which pass to
the surface and join other subagglomerates to form ag-
glomerates. Enlargement of agglomerates on the burn-
ing surface is associated with capturing of new subag-
glomerates or with joining other subagglomerates on the
burning surface. The nal stages of the agglomeration
process are ignition, combustion, and separation of ag-
glomerates from the burning surface [1{8].
The \initial conditions" of agglomeration are deter-
mined by the HCM structure, in particular, by the pres-
ence of contacting metal particles in the initial HCM [6].
Naturally, the presence of contacting metal particles in
HCM is not a sucient condition for agglomerate for-
mation. Some other conditions related to sintering of
the contacting particles, their melting, and coalescence
Modern heterogeneous condensed mixtures (HCM)
may contain several types of disperse components with
dierent dispersion and properties. In the course of
combustion, some of them act as oxidizers (ammonium
perchlorate, HMX, RDX, etc.), and some may work as
combustibles (for instance, metals: aluminum, magne-
sium, and boron). A typical feature of HCM is the ran-
dom character of distribution of disperse components in
them. HCM combustion involves interaction (mechani-
cal, thermal, chemical, etc.) of various components and
a change in their phase state and chemical composition.
The character of this interaction aects the structure
of the reaction region, the burning rate of HCM as a
macroscopic system, and also the chemical composition
and structure of combustion products. This is most pro-
foundly manifested in combustion of metallized HCM,
which is accompanied by coalescence of metal particles
into agglomerates. In this case, the metal fuel in the liq-
uid and gaseous phase may actively enter the chemical
1
Moscow Institute of Thermal Engineering,
Moscow 127276; rash@rash.mccme.ru.
0010-5082/02/3804-0435 $27.00
c
2002
Plenum Publishing Corporation
435
436
Rashkovskii
into subagglomerates should be satised. These con-
ditions are fullled for a certain dispersion of the ini-
tial metal particles in HCM and certain parameters of
the heat wave, which are related to the HCM burning
rate [1]. Thus, the presence of contacting metal parti-
cles in HCM is a necessary but not a sucient condition
for the formation of agglomerates.
There are no doubts now that the HCM structure
exerts a determining eect on HCM combustion and, in
particular, on the formation of condensed combustion
products (agglomerates).
Indirectly, the HCM structure has been always
taken into account in simulation of the agglomeration
process. Examples can be the model of \pockets" [1,
4{6], which are some structural cells formed by ammo-
nium perchlorate (AP) particles within which agglom-
erates are formed, or models that use the coordination
number of metal particles in HCM composition [6, 10],
which is a factor determining the capability of metal
particles to join in the course of heating in the heat
wave.
It should be noted that the pocket model is phe-
nomenological, since the notion of a pocket itself is
rather conventional. An analysis of the structure of real
HCM and structures obtained by mathematical simula-
tion [11], HCM has no isolated (in the rigorous meaning)
regions bounded by particles of oxidizers, where inde-
pendent formation of agglomerates can occur. Using
the terminology of Babuk et al. [4], we can state that
the formation of agglomerates always includes an \in-
terpocket" mechanism, which implies that metal parti-
cles grouped near dierent particles of the oxidizer can
join into a single agglomerate. In addition, the pocket
model does not allow one to describe the dynamics of
agglomerate formation.
Obviously, the closed theory of agglomeration
should be statistical and should be based on a dynamic
model that describes all stages of agglomerate develop-
ment. Such a model was proposed in [7, 8] and employed
the spectrum of agglomerates determined by the HCM
structure as the initial data.
The objective of the present work is to study the
structure of metallized HCM and its inuence on the
initial process of agglomerate formation.
els allow one to hope for direct (non-phenomenological)
modeling of processes in the c-phase of HCM in the case
of transition of a heat wave, which lead to coarsening of
metal particles | agglomeration.
A general method for modeling the statistical struc-
ture of disperse systems, including HCM, was proposed
in [11]. This method oers an eective description of
both metal-free and metallized systems with an arbi-
trary size distribution of disperse components. The
disperse components of HCM are considered as hard
spheres. Simulation of the HCM structure implies ran-
dom arrangement of a system of particles with a given
size distribution inside a certain region of space.
The algorithm of random arrangement of an arbi-
trary system of hard spheres in a given region of space
is based on solving the system of equations [11]
d x
i
dt
=
X
x
i
x
j
jx
i
x
j
j
ij
;
(1)
j6=i
ij
=
1 if jx
i
x
j
j < r
i
+ r
j
;
0 if jx
i
x
j
j> r
i
+ r
j
;
(2)
where x
i
is the radius vector of the center of the ith
sphere, r
i
is the radius of the ith sphere, and t is the
\time."
System (1), (2) is solved under random initial con-
ditions: the initial coordinates of the centers of particles
are set by a generator of random numbers. The calcula-
tion is continued until no pairs of intersecting particles
remain in the system. Justication of the method and
its detailed description can be found in [11].
Heating of the c-phase of HCM in the heat wave
leads to sintering of the contacting metal particles [9].
Groups of sintered metal particles, which are isolated
from each other, form \clusters." The clusters may have
a complicated internal structure characterized by the
coordination number, which is the mean number of con-
tacts per one particle of the cluster [6]. Obviously, the
higher the coordination number of the cluster, other
conditions being equal, the higher its strength.
The size of the clusters of contacting metal parti-
cles in HCM and their connectivity are determined by
the mass content of metal in HCM, dispersion of its par-
ticles, and mass content of disperse oxidizers in HCM
and the spectrum of their particles. The clusters can
be rather strong spatial structures that perform certain
functions in the course of HCM combustion and metal
agglomeration. Thus, extended clusters can play the
role of thermal bridges intensifying the heat ux from
the burning surface inside the c-phase and, hence, in-
creasing the HCM burning rate. Rather large clusters
can serve as a basis of the frame layer near the HCM
burning surface and retain agglomerates on the burning
HCM STRUCTURE
AND ITS BASIC ELEMENTS
Some recent papers deal with modeling of the HCM
structure or its characteristic cell [11{13]. The nal ob-
jective of these works is to relate the statistical HCM
structure with the laws of its combustion. These mod-
Role of the Structure of Condensed Mixtures in the Formation of Agglomerates
437
surface, favoring their growth. It is possible to imag-
ine a situation where an agglomerate is retained on the
burning surface by a cluster extended into the depth of
the c-phase, which is a \donor" of this agglomerate. As
the binder burns out, the burning surface moves; due to
melting of the upper part of the cluster, part of its mass
passes to the agglomerate, and the agglomerate itself,
moving along the cluster under the action of capillary
forces, follows the burning surface until the whole clus-
ter passes to the agglomerate or the repulsive force from
gaseous combustion products becomes greater than the
cluster strength.
Melting of metal particles in the heat wave may
lead to the coalescence of the initial metal particles
in the cluster and to the formation of a large drop-
subagglomerate. In this case, a subagglomerate is a
cluster in a melted state. At the same time, rapid melt-
ing of rather large clusters can lead to the formation
of a liquid porous structure, which is unstable. In this
case, cluster melting can be accompanied by its split-
ting (dispersion) under the action of capillary forces and
gas-dynamic forces from the side of gaseous combustion
products with the formation of several smaller drops-
subagglomerates, the sum of their masses being equal
to the mass of the cluster. Note, the process in the
combustion wave is rather fast, and there may be not
enough time for the ultimate coalescence into a single
drop or splitting into several drops. In this case, the
sizes of subagglomerates arriving on the burning sur-
face are greater than their sizes in the form of a single
(monolithic) drop. This process is characterized by the
ratio of the time that passed from the moment of melt-
ing of the initial metal particles before their arrival on
the burning surface to the time of coalescence of the
cluster into a single drop due to capillary forces. We
give the quantitative criterion of this process.
The time t
melt
from the beginning of melting of the
drop to its arrival on the burning surface can be evalu-
ated using the model proposed in [1]. Simple transfor-
mations allow us to obtain the formula
i.e., only for those particles that have enough time to
reach the melting point in the heat wave within the c-
phase.
The characteristic time of coalescence of melted
particles under the action of surface tension has the or-
der D
cl
=, where and are the viscosity and surface
tension of the metal of the drop in the melted state and
D
cl
is the characteristic size of the cluster prior to melt-
ing.
For D
cl
=t
melt
1, the particles rapidly arrive
on the HCM burning surface, and the cluster has not
enough time for ultimate coalescence into a single drop
or for splitting into several small drops. Vice versa,
for D
cl
=t
melt
1, evolution of the cluster under the
action of capillary forces is completed in the HCM sur-
face layer, and one or several melted drops arrive on the
burning surface, their total mass being equal to the mass
of the initial cluster. Thus, large clusters may arrive on
the burning surface in the form of a system of sintered
particles, and their melting and integration into a drop-
agglomerate occur already on the burning surface or in
the gas ame. At the same time, comparatively small
clusters, due to capillary forces, may transform into one
or several drops-subagglomerates prior to reaching the
burning surface.
Within the HCM c-phase and frame layer, subag-
glomerates are not mobile; therefore, coalescence of in-
dividual subagglomerates before reaching the burning
surface is little probable and is possible only on the
burning surface or in the gas ame of HCM.
Thus, the process of formation of agglomerates
from initial metal particles can be conventionally di-
vided into the following stages: (i) sintering of initial
metal particles into clusters; (ii) melting of clusters;
(iii) capillary compression of clusters and their splitting
into smaller ones with the formation of subagglomer-
ates; (iv) arrival of subagglomerates on the burning sur-
face and collision with agglomerates already located on
the burning surface; (v) their coalescence into larger ag-
glomerates; (vi) entrainment of agglomerates from the
surface under the action of gaseous products of HCM
decomposition.
The rst three stages of the process are examined
in the present work.
In what follows, we consider HCM containing two
disperse components, which are collectively called AP
(coarse particles) and aluminum (ne particles). We as-
sume that all particles of one kind are identical. This is
true for real HCM containing narrow fractions of pow-
ders of disperse components.
The volume concentration of particles of pow-
dered aluminum in HCM is determined by the formula
Al
= &
p
=
Al
, where & is the mass fraction of aluminum
u
2
h
1
1 + c
Al
Al
u
2
D
Al
=12
T
s
T
0
T
melt
T
0
i
; (3)
t
melt
=
ln
where and are the thermal conductivity and ther-
mal diusivity of HCM, c
Al
and
Al
are the specic heat
and density of aluminum particles, u is the HCM burn-
ing rate, D
Al
is the diameter of the initial aluminum
particles, T
0
and T
s
are the initial HCM temperature
and the temperature of the HCM burning surface, and
T
melt
is the melting point of aluminum. Formula (3) is
valid only for aluminum particles of diameter [1]
D
Al
<
12
c
Al
Al
u
2
T
s
T
melt
T
melt
T
0
1=2
;
438
Rashkovskii
in HCM; the diameter of particles of disperse AP is D
AP
and their volume concentration in HCM is
AP
. To
estimate the HCM density (
p
), we used the formula
p
= (1
AP
Al
)
b
+
AP
AP
+
Al
Al
, where
b
,
AP
, and
Al
are the densities of the binder, AP, and
aluminum,
b
= 900 kg/m
3
,
AP
= 1950 kg/m
3
, and
Al
= 2700 kg/m
3
.
From considerations of dimensionality, the mean-
mass diameter D
43
of subagglomerates formed after
melting of clusters of contacting aluminum particles can
be represented in the form
D
43
= D
Al
f(D
Al
=D
AP
;
Al
;
AP
); (4)
where f is a dimensionless function, which depends on
whether the dispersion of the melted clusters occurred
or not. Depending on the ratio of diameters D
Al
=D
AP
,
we conventionally speak about nely disperse aluminum
(D
Al
=D
AP
! 0) or about the relatively coarsely dis-
perse aluminum if D
Al
=D
AP
acquires a nite, though
small value.
Note that the notion of a \nely disperse particle"
in this work is purely geometric and is related to the
relative size of aluminum and AP particles only.
Fig. 1. Mean-mass diameter of subagglomerates ver-
sus the volume concentration of nely disperse alu-
minum in HCM: the open and lled points show the
data without and with allowance for dispersion of
clusters, respectively.
FINELY DISPERSE ALUMINUM
The dependence of D
43
=D
Al
on the eective con-
centration of aluminum in HCM
e
Al
for the case consid-
ered is plotted in Fig. 1 by open points. For
e
Al
<
cr
,
where
cr
0:15, all the calculation values (indepen-
dent of the \specimen" size) are grouped around one
linear dependence
D
43
= (1 + a
e
Al
)D
Al
;
We consider HCM containing nely disperse parti-
cles of aluminum. Formally, from Eq. (4), we obtain
D
43
= D
Al
f(0;
Al
;
AP
): (5)
From the viewpoint of modeling of the HCM structure,
this limiting transition is equivalent to independent ar-
rangement of aluminum and AP particles: AP particles
are located in the HCM volume without allowance for
aluminum particles, and aluminum particles are located
in the space between AP particles as in the empty space.
Thus, to model the structures formed by nely dis-
perse particles in HCM, it is sucient to place these
particles randomly, with an eective volume concen-
tration
e
Al
=
Al
=(1
AP
), in the empty space. In
the dimensionless form, where all sizes are normalized
to the diameter of aluminum particles, the problem re-
duces to arrangement of spheres of unit diameter in a
cube [11] and is characterized by only one parameter
e
Al
. The calculation was performed for \specimens"
of sizes 10 10 10, 20 20 20, and 30 30 30.
The method of identication of clusters of contacting
particles is described in [11].
We consider the mean-mass sizes of subagglomer-
ates formed as a result of melting of clusters without
dispersion, i.e., we assume that the total mass of the
cluster transforms into one subagglomerate | spherical
drop.
(6)
where a = 5{6. For
e
Al
>
cr
, the mean-mass size of the
clusters drastically increases: the calculation points de-
viate noticeably from the linear dependence (6), and a
signicant scatter of the calculation values for an identi-
cal concentration of aluminum is observed. The smaller
the size of the calculation \specimen," the greater the
scatter; the greater the \specimen" size, the slower the
numerical points deviate from dependence (6) with in-
creasing
e
Al
. These features are explained by the fact
that, for
e
Al
<
cr
(
cr
0:15), the system con-
tains only isolated clusters whose sizes are substantially
smaller than the size of the calculation \specimen."
With increasing aluminum concentration, the size of
clusters (on the average) increases; the appearance of
a cluster extended through the entire \specimen" (per-
colation cluster) and including a signicant mass of the
initial aluminum particles becomes more probable [11].
For
e
Al
>
cr
, a percolation cluster whose mass varies
from one calculation to another emerges in the system
with a probability close to unity, which increases the
Role of the Structure of Condensed Mixtures in the Formation of Agglomerates
439
Fig. 2. Number of initial aluminum particles in clus-
ters for dierent initial concentrations of aluminum
in HCM (the curve indicates the upper boundary).
Fig. 3. Number of particles in the cluster versus its
maximum size.
scatter in the values of D
43
. Thus, the concentration
cr
0:15 is the percolation limit for a system of iden-
tical spherical particles.
The number of initial aluminum particles in dif-
ferent clusters for dierent eective concentrations of
aluminum is plotted in Fig. 2 for specimens 202020
and 30 30 30.
For each value of
e
Al
, there is a maximum number
of particles united into one cluster. The curve
HCM \specimens." The exponential growth of the ag-
glomerate size with increasing aluminum concentration
in HCM was experimentally observed in [3]. In prac-
tice, this is manifested in the form of very large single
agglomerates found after burning of the HCM specimen.
In some recent works, the fractal character of en-
ergy release in HCM combustion is grounded [14]. Ob-
viously, this is possible only if the initial HCM structure
contains some fractal structures. An analysis of the re-
sults of mathematical simulation shows that there are
no fractal structures (in the rigorous meaning of this
notion) in the initial HCM structure, but it was found
that the clusters of the contacting metal particles in the
initial HCM form fractal-like structures.
An analysis shows that the clusters, on the average,
are extended in one of the directions. Figure 3 shows
the dependence of the number of particles in the clus-
ter on the maximum linear size of the cluster (L is the
greatest distance between the centers of its particles)
for various values of
e
Al
. The calculation points are
grouped around the power dependence
N = A(L=D
Al
)
D
; (8)
which is typical of fractal structures. Here D is a coe-
cient that may be considered as the \fractal dimension"
of the cluster and A is a constant factor. For identical
ne aluminum particles, D = 2 and A = 0:6.
A similar dependence was obtained for soot aggre-
gates formed in combustion of hydrocarbon fuels [15];
the fractal dimension of the aggregates is 1.74. This
is indicative of a single mechanism of formation of the
N
max
= 3 exp (32:4
e
Al
)
(7)
limits all the calculation values from above and may be
considered as the dependence (in the statistical mean-
ing) of the maximum number of initial particles in the
cluster on the eective concentration of aluminum in
HCM. The estimate shows that dependence (7) limits
the mass of the maximum cluster with a probability of
higher than 0.96.
Thus, the mass of the greatest cluster and, hence,
the maximum possible diameter of subagglomerates
(without dispersion) equal to D
max
=D
Al
= N
1=3
max
1:44 exp (10:8
e
Al
) increase exponentially with the con-
centration
e
Al
. Figure 1 shows the dependence of
D
max
=D
Al
on
e
Al
. The dependence is plotted only for
the values
e
Al
<
cr
, for which it makes sense. The-
oretically, for an innite specimen, without allowance
for dispersion, we have D
max
=D
Al
! 1 as
e
Al
!
cr
.
The nite value of the diameter of the maximum sub-
agglomerate for
e
Al
=
cr
, which was obtained in cal-
culations, is related to the nite size of the calculation
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