JOURNAL OF PROPULSION AND POWER
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Vol. 17, No. 4, July August 2001
Random Packs and Their Use in Modeling
Heterogeneous Solid Propellant Combustion
S. Kochevets, J. Buckmaster,† T. L. Jackson,‡ and A. Hegab§
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University of Illinois at Urbana Champaign, Urbana, Illinois 61801
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It is shown that random packs of spheres of varioussizes can be constructed that modelammonium perchlorate-
in-binder propellants in the sense that both the size distributions and the packing fractions of industrial propellant
packs can be matched. Strategies for dealing with fractional numbers of large particles are addressed, as are
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strategies for dealing with a large number of very ne particles ne powder . Fine powder is necessary in a three-
dimensional pack to achieve the required stoichiometric ratio of ammonium perchlorate to fuel binder, but is not
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necessary in a two-dimensional disk pack. Some preliminary calculations of the two-dimensional combustion
eld supported by a disk pack are presented, in which full coupling between the gas phase, the condensed phase,
and the retreating nonplanar propellant surface is accounted for.
Nomenclature
reaction rate constants
mean diameter of particles
ture of a true propellant.For this, a random packing algorithmmust
be used, and a suitable one is identied and discussed in Ref. 2.
This algorithm is dynamic in nature and can closely pack spheres
of arbitrary size.
D1;2
dj
E1;2
L
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
activation energies
The calculations of Ref. 2 are restricted to bimodal packs, two
sphere sizes, and no serious attempt is made to simulate true pro-
pellants. A good portion of the present paper is concerned with an
examinationof experimentalAP size data, and the use of these data
length of a pack edge
total number of particles
number of particles in the th class
N
Nj
n j
n1;2
Ru
R1;2
rb
j
j
number fraction of particles in the th diameter classs
pressure exponents
universal gas constant
reaction rates
(
to de ne realistic packs with packing fractions volume percentage
)
of AP that closely approximate those of true propellants.
The remainder of the paper describes preliminary ame calcula-
tions for which the propellant data are de ned by a random pack.
These calculations incorporate in as simple fashion as possible
the following ingredients:AP decomposition;reaction between the
products of the AP decomposition and the binder gases; unsteady
heat conduction within the solid, allowing for the different me-
chanical and thermal properties of the AP and binder; an unsteady
nonplanar regressing surface; temperature-dependent transport in
the gas phase; and an Oseen approximationin the gas phase, which
bypasses the momentum equation. Our code can incorporate the
momentum equation when desired,but we have not chosen to do so
in the preliminaryresults,which we presenthere.Furtherdiscussion
of this issue, together with ame calculations for a sandwich pro-
surface regression rate
temperature
volume of particles in the th class
T
Vj
vj
X
Y
Z
´
Á
Ã
j
j
volume fraction of particles in the th class
mass fraction of ammonium perchlorate AP
mass fraction of fuel binder
mass fraction of AP decompositionproducts
surface function
surface location
level set function de ning the pack
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Subscripts
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pellant in which a full accountingof the Navier Stokes equationsis
AP
B
c
=
=
=
AP
binder
condensate
incorporated,may be found in Ref. 3.
II. Packing a Propellant
The industry constructs propellants by mixing a selection of AP
cuts in suitable proportions. Each cut is characterizedby a nominal
size, 200, 50, 20 ¹m, etc., but there is a wide range of sizes within
each cut. It is instructive to examine some true cuts, and here we
show dataprovidedby ThiokolCorporation,courtesyof R. Bennett.
The rst three columns of Table 1 de ne experimentalhistogram
data for a 200 ¹m cut. These data are acquired by passing the AP
I. Introduction
T has long been recognized that the burning rate of a heteroge-
neous propellantis in uenced by the propellantmorphology,by
I
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the size and size distribution of the ammonium perchlorate AP
particles. It follows that any serious attempt to simulate propellant
burning numerically must incorporatea packing algorithm, a strat-
(
througha sequenceof sievesof ever-decreasingmesh diameter rst
(
egy for de ning and constructing a model propellant. Lattice or
1
)
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column . The second column shows the percentage by volume of
AP that passes through each sieve. Thus, 100% passes through the
592.0-¹m sieve, that is, there are no particleslarger than 592:0 ¹m,
but only 99:96% passes through the 542.9-¹m sieve, so that 0:04%
)
crystal packs are easily de ned, but do not re ect the random na-
Received 17 November 2000;revision received 18 March 2001;accepted
c
for publication20 March 2001.Copyright 2001 by the American Institute
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by volume third column of the particles have diameter between
542:9 and 592:0 ¹m. At the other end of Table 1, we see that there
are no particles smaller than 52:33 ¹m, and 0:15% by volume have
diameters between 52:33 and 57:06 ¹m. It is convenientto replace
each diameter interval by its mean so that, for example, 0:04% of
of Aeronautics and Astronautics, Inc. All rights reserved.
Graduate Student, Department of Mechanical and Industrial Engineer-
ing, 1206 W. Green Street; kochevet@uiuc.edu.
†Professor, Department of Aeronautical and Astronautical Engineering,
104 S. Wright Street; limey@uiuc.edu. Associate Fellow AIAA.
‡Senior Research Scientist, Center for the Simulation of Advanced Rock-
ets, 1304 W. Springeld Avenue; tlj@csar.uiuc.edu.
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the AP has diameter 567:45 ¹m fourth column , the mean of 592:0
and 542:9.
§Postdoctoral Associate, Department of Aeronautical and Astronautical
Engineering, 104 S. Wright Street; hegab@uiuc.edu.
The volume fractionscan be convertedto number fractionsin the
N
following fashion. Given a portion of AP, suppose that
of the
j
883