I
INTRODUCTION
1
CFD-assisted particle size distribution measurements in the Keldysh Research Centre experimental set-up at ONERA ∗
†
Jouke Hijlkema and Pierre Prevot
ONERA - CFM, Mauzac, 31410, France
This article completes the work on the numerical simulations supporting the experimental determination of the particle size distribution of liquid aluminium oxide droplets resulting from the combustion processes inside solid rocket motors. Direct measurements under realistic conditions are extremely di cult and one of the rare experimental set-ups capable of doing just that is the KeRC installation present in our laboratory. However, the fact that the sampling takes place at a certain distance from the burning surface of the fuel grain allows the large vortices present in the ow to divert, mainly smaller droplets away from the sampling probe. This leads to a systematic bias in favour of the larger particles. By tweaking the size distribution of the injected particles in the numerical simulations, we have been able to reproduce the experiments numerically and thus, to determine the size distribution of the droplets directly at the surface of the propellant. Part of the results of these simulations have been presented at the 5th European Conference for Aeronautics and Space Sciences (EUCASS), this article gives complete and detailed account of this work.
Nomenclature Aluminised Butalite A composit propellant doped with less than 5 % of aluminium Butalamine
A composit propellant doped with aluminium oxide particles
Butalane
A composit propellant doped with aluminium particles
Butalite
A composit propellant without metalic particles
I. Introduction In order to increase performances, several launchers now ying use aluminium-doped propellant grains in their Solid Rocket Motors (SRM). This results in the creation of liquid aluminium oxide droplets amongst the combustion products, yielding an internal two-phase ow eld.
1, 2 Studies of the P230 SRM
(230-ton solid rocket boosters used on Ariane 5) as used on Ariane 5 revealed a possible, amplifying,
35 Besides the ampli cation of pressure
in uence of these particles on pressure and thrust oscillations.
oscillations, part of these particles get trapped and agglomerate in the aft-end cavity, resulting in 2 to
6 The slag formation leads to performance losses and can contribute to thrust oscillations
3 tons of slag.
when parts of it transit through the nozzle. To determine the in uence of the aluminium oxide particles on the overall ow and to try to understand and solve at least part of the above-cited problems, we need to determine the size distribution of the particles throughout their life inside the motor. In order to do so, several experimental set-ups
712
3, 1319 have been developed all over the world. The experimental set-ups all as well as numerical studies burn aluminised solid propellants and try to measure particle distributions but, besides the installation developed at the Keldysh Research Centre (KeRC), none that we know of gives satisfactory results. In 2005, ONERA (The French aerospace lab) obtained an installation equivalent to the KeRC experimental set-up and has been conducting experiments with it since. This paper describes a CFD coupled approach used to overcome the inherent bias to a speci c class size of particles that the Keldysh set-up suers from. ∗ Researcher, ONERA DMAE/LP,
[email protected] † Researcher, ONERA DMAE/LP,
[email protected]
I
INTRODUCTION
2
1 Description of the KeRC installation The Keldysh installation burns an aluminium-doped propellant in a cylindrical combustion chamber ( g. 1) designed to assure realistic operating conditions for pressure and temperature.
Figure 1: KeRC
After ignition, when the initial transient phase has passed, an electro-valve (5 in 1) permits a probe to collects a small sample of the two-phase ow. This sample moves through a cooling pipe (2 in g.1) to reduce its temperature. Then the larger particles get separated from the ow in a cyclone (3 in g.1) and the smaller particles get trapped in a splash chamber (4 in g.1).
A calibrated ori ce controls
the mass ow through the probing and collection mechanism and hence assures a pressure equilibrium between the probe and the chamber allowing for iso-kinetic sampling. Before a run, all of the collection mechanism (cyclone and splash chamber with the calibrated ori ce closed) is pressurized at slightly more than the expected chamber pressure. During the ignition and transient phase, the electro-valve assures the physical separation of the combustion chamber and the collection mechanism. Once the transient phase is over, the valve opens and there is a blow down from the collection mechanism through the cooling pipe into the combustion chamber.
This is meant to clean the probing system from eventual
slag or residues. After the pressure evens out, the collection phase starts. About 2 seconds later, the electro-valve shuts again and the sampling is over. An example of a typical pressure evolution for both the combustion chamber and the splash chamber is given in gure 2. The graph clearly shows the opening of the electro-valve at about 1 second, followed by a blow-down phase of around 0.2 seconds after which the pressures of the combustion chamber and the splash chamber are perfectly identical. We collect the residues from the cyclone and the splash chamber by rinsing both with alcohol and ltering the run-o through an ash-less paper lter. The lter and residues are then heated to 600 for
≈ 12
°C
hours to eliminate all carbon resulting from the decomposition of the Plexiglas cooling pipe
and all combustion residues. Finally a Coulter LS230 (with the small module) granulometer is used to analyse the left-over aluminium oxide particles suspended in acetone and exited by ultrasound to avoid clutter. Figure 3 gives the evolution of a typical particle distribution during these steps and shows the importance of the calcination step.
INTRODUCTION
3
5 Combustion chamber pressure Splash chamber pressure
Pressure (MPa)
4
3
2
1 Opening and closing of the electro-valve 0 0
1
2
3 Time (s)
4
5
6
Figure 2: Pressure dierence evolution
6
5
raw sample calcinated sample calcinated sample + US
4 volume ratio (%)
I
3
2
1
0 0.01
0.1
1
10
100
D (μm) Figure 3: Size distribution evolution for dierent steps in the process
1000
II
QUALIFICATION AND FIRST TEST RUNS
4
II. Quali cation and rst test runs The rst three experiments performed in Russia from 2002 to 2005 were satisfactory in the sense that a blind test on a cold butalamine grain seeded with calibrated inert
Al2 O3
particles was very reproducible
and yielded size distributions centred around the original particle sizes. The problem that emerged from these experiments is that all three results have a strong bias towards the larger particles. Figure 4 shows
50 %vol of small particles with a diameter 70 µm, we gather 10 %vol of small and 90 %vol
that for a block seeded with duralox 9 with a size distribution of of
3 µm
and
50 %vol
of large particles with a diameter of
of large particles. After reception of the installation, in 2005, a series of initial tests (on the same propellants tested in Russia) were carried out at ONERA. These tests con rmed the results obtained at the KeRC.
10
volume ratio (%)
8
Duralox 9 qualification 1 qualification 2 qualification 3
6
4
2
0 0.1
1
10 Diameter (μm)
100
Figure 4: Size distribution for KeRC experiments
As presented during the 47th Joint Propulsion Conference, we managed to con rm these ndings numerically. Large re-circulation zones were shown to be responsible for the diversion of the smaller particles, away from the probe, leading to a bias towards the larger particles (cf gure 7). These results, however satisfactory, were valid for inert particles only. Regular propellants contain aluminium that burns into aluminium oxide droplets.
Given the temperature eld, these droplets stay in a liquid form all along
their transit through the combustion chamber. In order to predict their size distribution in the sampling zone, complex phenomena like coalescence and secondary break-up need to be taken into account. This article describes CFD calculations eectuated to assist the analysis of a KeRC experiment we carried out in our laboratory.
III. The experiment The experiment we chose to simulate is the fth KeRC ring carried out in our laboratory (the main reason for this choice is that the experiment has been carried out using internal funds). The fuel grain used is a JD 9404AL aluminised Butalite and we sampled (after the ltering and calcination procedures described in I) 1.07 g of aluminium oxide particles. Figure 5 shows the particle size distribution of this sample. The mass ow at the surface of the fuel grain is estimated at 0.653
kg.m−2 .s−1
of
Al2 O3 .
IV
NUMERICAL SET-UP
5
5 100 measurments cumulative measurments 80
3
60
2
40
1
cumulative volume ratio (%)
volume ratio (%)
4
20
0 0
25
50
75
100
125
0
150
Figure 5: Measured particle size distribution
IV. Numerical set-up All the calculations presented in this article have been carried out using the ONERA in-house code
a
CEDRE . CEDRE is a multi-solver platform that allows for multi-phase and multi-physics numerical simulations. The solvers used here are called CHARME, the Navier-Stokes solver needed to calculate the gaseous phase, and SPARTE, the Lagrangian solver used for the dispersed liquid phase. The gaseous phase is tackled using classical second-order discretisations for both time and space. The dispersed liquid
13 sections IV.A.1 and IV.A.2 give an
phase is solved with a Monte-Carlo method described in detail in, outline of the principal particle interaction models used.
All the calculations presented here are 2D
axisymmetrical.
IV.A.
The kinetic particle model
The principle of a kinetic model of a dense liquid spray consists in describing the latter through the drop
f (1) (t, x, r, v) de ned such that f (1) (t, x, r, v)dx dr dv denotes the average number a radius in [r, r + dr] a velocity in [v, v + dv] situated in the volume [x, x + dx] at time t. (1) (1) the function f is de ned as an average meaning that the value of f , in a physical
distribution function of drops with Note that
sense, is the average over a large number of initial data of which we know just global parameters (mean, variance, . . . ). If we take all physical phenomena into account and use an in continuum mechanics classical approach,
f (1) can be written asb ∂f (1) + divx vf (1) + divv af (1) + Dturb f (1) ∂t = Qcoll f (2) + Qbup f (1)
it can be proved that the governing equation for
(1)
where the dierent terms of equation (1) have the following signi cation :
a http://cedre.onera.fr b Note that we have refrained
from adding an evaporation term because of the fact that we will deal with drops of liquid
aluminium oxide having a negligible evaporation rate under the circumstances we will encounter. Including evaporation eects is equivalent to adding the term
ω(t, x, r, v)
is the evaporation rate of a drop with velocity
v
∂(ωf (1) ) to the left hand side of equation 1, where ∂r
and radius
r
located at
x
at time
t
IV.A
The kinetic particle model
a(t, x, r, v) Dturb f
6
is the acceleration of an isolated drop with velocity
(1)
v,
radius
r,
located at
x
at time
t.
models the drops dispersion induced by the turbulent gas ow.
Qcoll
models the in uence of collision phenomena.
Qbup
models the in uence of breakup phenomena.
f (2) (t, x, r, v, r∗ , v∗ ) is the density of a pair of drops and is de ned such that f (2) (t, x, r, v, r∗ , v∗ )dxdrdvdr∗ dv ∗ ∗ ∗ ∗ is the average number of pairs of drops with radii (r, r ) in [r, r + dr] × [r , r + dr ] a velocity (v, v∗ ) in [v, v + dv] × [v∗ , v∗ + dv∗ ] located in the volume [x, x + dx] at time t. IV.A.1.
Breakup modelling
The general form for the breakup operator
Qbup (f (1) )(t, x, r, v)
Qbup
is given by
= −
number of drops
(t, x, r, v)
lost during breakup
+
number of drops
(t, x, r, v)
created during breakup
= −νbup (t, x, r, v)f (1) (t, x, r, v) ˆ + [νbup (t, x, r∗ , v∗ ) hbup (t, x, r, v, r∗ , v∗ )
(2)
R+ ×Rn
i f (1) (t, x, r∗ , v∗ ) dr∗ dv∗ where
n
νbup (t, x, r, v) is the breakup ( 0 if We ≤ Wcrit νbup = 1 if We > Wcrit τbup
where
τbup
is the dimension of space and
hbup (t, x, r, v, r∗ , v∗ )
dkvg − vp k √ = We
r
ρp ρg
6 1 + 1.2Oh0.74 p 4 (log W e)
is the breakup density function (cf
hbup (t, x, r, v, r∗ , v∗ ) = IV.A.2.
(t, x, r, v)
given by
is the average duration of a breakup given by
τbup and
rate of drops
27r∗3 exp 4 2rsmd
!
13 for all the details)
−3r rsmd
δv=vbup (r,r∗ ,v∗ )
Collision modelling
The collision operator
Qcoll ,
responsible for all drop-drop interactions.
Qcoll
can be split as follows
Qcoll = −Q− + Q+ Q− (t, x, r, v)dt dx dr dv represents the average number of drops that can be found at time t in the interval [x, x + ∆x] × [v, v + ∆v] × [r, r + ∆r] that will be removed from this phase space interval due + to a collision during the time period [t, t + ∆t]. A similar de nition is given for Q (t, x, r, v)dt dx dr dv where
as the number of drops appearing in the interval due to a collision during the same period. Under certain hypotheses (cf
13 )
Q− ,
as a function of
f (2) ,
c as for the
has almost the same expression
case of a rare ed gas of hard spheres of dierent radii and can thus be written as
Q− (t, x, r, v) = ˆ ˆ n f (2) (t, x, r, v, r∗ , v∗ )Es (t, x, r, v, r∗ , v∗ ) R+ ×Rn S −
k (v − v∗ ) · nk (r + r∗ ) c Using
2
o
dndr∗ dv∗
similar arguments as for the derivation of Boltzmann's equation for rare ed gases (see for instance20 )
(3)
IV.A
where
The kinetic particle model
x∗ = x + (r + r∗ ) n, dn
7
denotes the solid angle dierential element and
radius one for which the normal in all points is such that vector
x∗ − x
(v − v∗ ) · n < 0
while
S − is the half sphere of n is the direction of the
d as de ned in gure 6. We refer to the article of Cercignani20
at the moment of collision
for a detailed derivation of equation (3).
ϕ Op1
rp1 Vp2-Vp1
ϑ n
rp2
Op2
I
Figure 6: Collision parameters
Es
is a bounded positive function that models the hydrodynamic eects that might force small drops
to circumvent larger drops rather than to collide with them. By de nition we have
f (2) = gf (1) where
g
is the conditional drop density function (cf
drop density
f
(1)
13 ). Since we only have access to the average single
, it is necessary to de ne a relation between the conditional drop density
single drop density
f (1) .
g
and the
The simplest solution is the assumption of total chaos where we assume the
presence of drops to be completely decorrelated and we can write
g(t, x, r∗ , v∗ |r, v) = f (1) (t, x, r∗ , v∗ ) But because of the fact that drop velocities depend on the local gas velocity it is obvious that drops that are close in space will have more or less correlated velocities. This is especially true for small drops since they adapt more rapidly to the local gas conditions. We thus, have to introduce a correction function that links the conditional drop density function to the average single drop density function
g(t, x, r∗ , v∗ |r, v)
= Ev (t, x, r, v, r∗ , v∗ )f (1) (t, x, r∗ , v∗ )
(4)
21 have proposed a rigorous approach that permits, under certain hypotheses,
Simonin and Laviéville
the derivation of an expression of the function
Ev
but, for the sake of simplicity and given the fact that
the expression proposed by Simonin and Laviéville is valid only for drops of identical size, we will use a rougher estimation. The velocity correlation depends merely on the drops ability to adapt to the local gas conditions so if one of the two drops in a given pair has a characteristic relaxation time that is large compared to the local turbulence time scale then we may assume their velocities to be decorrelated and we pose
Ev = 1. d This
If, however, both drops have a characteristic relaxation time that is small compared to the local means that drops have to have a head on trajectory in order to collide
IV.A
The kinetic particle model
8
turbulence time scale then it is obvious that their velocities are correlated and we will need
Ev 1.
This reasoning leads to the following, crude, velocity correlation correction function
( ∗
∗
Ev (t, x, r, v, r , v ) =
0 1
if
r1 ≤ rcrit and r∗ ≤ rcrit ∗ if r1 > rcrit or r > rcrit
e is de ned as
where the critical drop size
r rcrit =
τturb
9µg 2ρl
So we now have an expression for the probability that a drop of radius a drop of radius
r∗ ∈ [r∗ , r∗ + ∆r]
and velocity
v∗ ∈ [v∗ , v∗ + ∆v]
r
and velocity
v
collides with
during a small time interval
∆t
that
can be written as
dP = π(r + r∗ )2 kv − v∗ kEs Ev f (1) dr∗ dv∗ ∆t If we parametrise the unit sphere by means of the angles than by using the normal
n,
and
Θ
(cf. gure 6 on page 7), rather
π
ˆ2 ˆ2πn f (1) (t, x, r, v)f (1) (t, x, r∗ , v∗ )
ˆ Q− (t, x, r, v)
Φ
equation (3) can be written as
= R+ ×Rn
0
(5)
0
Ev (t, x, r, v, r∗ , v∗ ) B (Θ, t, x, r, v, r∗ , v∗ )} dΘdΦdv∗ dr∗ where
B (Θ, t, x, r, v, r∗ , v∗ ) = 2
Es (t, x, r, v, r∗ , v∗ ) kv − v∗ k (r + r∗ ) cos Θ sin Θ The next step is the splitting of
Q+
in two separate collision operators, one representing the coales-
cence eects and the other representing the rebound eects.
+ Q+ = Q+ coal + Qreb If we use the relations for the post-coalescence velocity and radius as detailed in as the critical impact angle separating the coalescence and the rebound regimes and
13 and de ne
n
Θcrit
as the dimension
of space, then we can write the coalescence operator as
Q+ coal (t, x, r, v) = Θ ˆ ˆcritˆ2πn 1 f (1) (t, x, r∗ , v∗ )f (1) (t, x, r , v )Ev (t, x, r∗ , v∗ , r , v ) 2 0 R+ ×Rn 0 r 3n+2 B (Θ, t, x, r∗ , v∗ , r , v ) dΘdΦdv∗ dr∗ r where
(r , v )
is de ned as
(
r v
= =
√ 3
r3 − r∗3 r3 r 3 −r ∗3 v −
r ∗3 ∗ r 3 −r ∗3 v
and represents the radius and velocity of a drop denoted a drop of radius
r∗
and velocity
v∗
yields a drop of radius
is related to a critical impact parameter
Icrit
(6)
(detailed in
Θcrit (kv − v∗ k, r, r∗ ) = arcsin
r
such that a coalescence of this drop with
and velocity
13 ) through
v.
Icrit (kv − v∗ k, r, r∗ ) r + r∗
The critical collision angle
D(r ,v ) r 3n+2 in equation (6) is nothing but the Jacobian k r D(r,v) k resulting from the mapping 1 ∗ ∗ (r , v ) for xed r and v . The factor 2 appearing in equation (6) results from the fact that
The term
(r, v) → e Drops
with a radius smaller than the critical drop size are believed to adapt instantaneous to the local gas velocity
IV.B
Simulation details
9
each collision is counted twice during the integration while each coalescence of two drops results in the generation of only one new drop. In a similar manner, the rebound operator is given by
Q+ reb (t, x, r, v) = π 2
ˆ
ˆ
(7)
ˆ2πn 0 0 0 0 0 0 0 0 f (1) (t, x, r , v )f (1) (t, x, r∗ , v∗ )Ev t, x, r , v , r∗ , v∗
R+ ×Rn Θcrit 0
o 0 0 B Θ, t, x, kv − v∗ k, r, r∗ dΘdΦdv∗ dr∗ r , r∗ , v and v∗ are detailed in.13 = = R+ × Rn and group equations 3, 6 and
where the post-collisional radii and velocities So nally, if we introduce the space
0
0
0
0
7 together, we have
Q− (t, x, r, v) = π
ˆ ˆ2 ˆ2πn f (1) (t, x, r, v)f (1) (t, x, r∗ , v∗ )Ev (t, x, r, v, r∗ , v∗ ) = 0
0
B (Θ, t, x, r, v, r∗ , v∗ )} dΘdΦdv∗ dr∗ Q+ coal (t, x, r, v) = ˆ Θˆcritˆ2πn 1 f (1) (t, x, r∗ , v∗ )f (1) (t, x, r , v )Ev (t, x, r∗ , v∗ , r , v ) 2 0 0 = r 3n+2 B (Θ, t, x, , r∗ , v∗ , r , v ) dΘdΦdv∗ dr∗ r
(8)
(9)
Q+ reb (t, x, r, v) = ˆ
π
ˆ2 ˆ2πn 0 0 0 0 0 0 0 f (1) (t, x, r , v )f (1) (t, x, r∗ , v∗ )Ev (t, x, r, v , r∗ , v∗ )
= Θcrit 0
o 0 0 0 0 B Θ, t, x, r , v , r∗ , v∗ dΘdΦdv∗ dr∗ IV.B.
(10)
Simulation details
For the simulations presented here after, the following boundary conditions have been used. The interior walls of the engine have an adiabatic wall condition for CHARME and an elastic rebound condition for SPARTE. The nozzle has a classic supersonic out ow condition for CHARME and is transparent for SPARTE. On the sample probe, we impose a xed mass ow for CHARME and transparency for SPARTE. And, nally, we impose a xed mass ow on the surface of the fuel grain for CHARME and a given mass ow on the inner surface and an elastic rebound condition on the rest for SPARTE. We use the particle size distribution of the mass ow condition for SPARTE as an adjustable parameter, by tuning its values we try to reconstruct the experimental results. This approach can be seen as an attempt to resolve a sort of inverse problem. Figure 7 gives a typical ow eld computed with CEDRE showing the vorticity of the gaseous phase as a continuous eld with the dispersed phase superimposed as discrete particles. It is clear from this gure that the vortices diverge particles away from the probe and in doing so they concentrate the drops on the outskirts of the largest structures. This concentration results in high collision rates and hence more coalescence and therefore bigger particles. Due to the regression of the surface of the propellant grain, the internal geometry of the engine changes over time. This has an in uence on the internal ow eld, however, given the relatively short sampling period we considered a stationary solution on a xed grid. The chosen geometry corresponds to the situation about half way the sampling interval. A series of gas-only computations on ever denser meshes assured spatial convergence for a globally quadrangular grid of about 41000 points.
IV.C
Distribution 1
10
Figure 7: Typical ow eld structure
For all the calculations presented in the following sections we used
Al2 O3
as the dispersed phase. The
droplets are injected from the inner surface of the fuel grain at 1 m/s with a temperature of 2900 K and in injection frequency of 10 kHz.
IV.C.
Distribution 1
As a rst attempt of reconstructing the size distribution given in gure 5, we imagined that the bulk of the the third lobe in the measured trimodal distribution was the result of droplets coalescing as they transit the engine and that the size distribution of droplets as they are ejected from the fuel grain was strictly bi-modal, with particles concentrated in two classes. drops of about
15 µm.
Small drops of around
1 µm
and larger
We further xed an initial mass ratio of 50 % smaller drops which yields the
following characteristics for the injected droplets. Table 1: Distribution 1: injection classes mass ow (kg.m
−2
.s−1 )
class
diameter (µm)
1
1
0.3265
50
2
15
0.3265
50
mass fraction (%)
In order to determine the particle size distribution in a given zone at a given instance, it is relatively
f over all the numerical particles present in the designated area. Figure 8
easy to do a spatial average
shows the results of such an approach for two instants: 0.15 s and 0.19 s.
The chaotic nature of the
ow eld and the fact that for any given time the number of numerical particles
g present in the sample
probe is limited make that the deducted size distributions dier signi cantly for both instances. To nd a more realistic size distribution, both a space and time average is needed. CEDRE has the ability to store information about particles that interact with a boundary condition, and post-processing this data for the out ow condition of the sample probe yields the result given in gure 9. This approach has been shown to become invariant for calculation intervals bigger than
≈ 0.1
s. For what follows, we will only
present the size distributions at the exit of the sample probe, rst of all because this is data that interests us since it allows for a comparison with the experimental data, a second reason is that a modi cation of the CEDRE code to store particle data for an arbitrary zone is not yet completed.
f The
excellent and open-source visualisation software Paraview (http://www.paraview.org/) has programmable lters
that make this very easy
gA
numerical particle in the Monte-Carlo method represents a large number of identical physical particles
IV.C
Distribution 1
11
100
100
100
Volume (density)
Fraction (%)
Volume (density)
Volume (density)
Volume (flux)
90
Volume (flux)
90 80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 0
50
100
150
Volume (flux)
90
80
0 0
20
40
60
80
100
120
140
0
160
20
40
60
80
100
120
140
160
Diameter (µm)
(a) 0.15 s
100
100
100 Volume (density)
Volume (density) Volume (flux)
Fraction (%)
90
Volume (density)
Volume (flux)
90 80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 0
50
100
150
Volume (flux)
90
80
0 0
20
40
60
80
100
120
140
160
0
20
40
60
80
100
120
140
160
Diameter (µm)
(b) 0.19 s
Figure 8: Size distributions in dierent positions for dierent instances
Already for this rather coarse approach (particles emerging from the surface of the fuel grain are certainly not strictly bi-modal) gure 9 shows a very good comparison between the experimental data and the calculations for those particles resulting from the coalescence of members of the initial particle classes. In gure 9 this corresponds to particles bigger than
≈ 5 µm
between
and
15 µm
15 µm.
The population of particles with a diameter
cannot be the result of coalescence of the smallest particles (at least not
numerically), since they adapt instantly to the local gas velocity. For this reason, the relative velocity of a pair of collision candidates tends to zero as they approach so that the chance of an actual collision vanishes.
The only possibility to create smaller particles from bigger ones is by means of secondary
breakup. The calculations have shown that this is not happening because, besides from the nozzle exit, local Weber numbers are too low to allow for any form of droplet breakup. This hints that the initial size distribution is not strictly bi-modal (as we already expected) but contains particles with diameters between
≈ 5 µm
and
15 µm.
The next section covers an attempt to simulate this.
IV.D
Distribution 2
12
15 Experiment Calculation
12.5
Volume ratio (%)
10
7.5
5
2.5
0 0
25
50
75 100 Diameter (μm)
125
150
Figure 9: Distribution 1: comparison between experimental and numerical results
IV.D.
Distribution 2
In an attempt to better capture the particles with sizes that cannot result from coalescence, we added two more classes with intermediate diameters and we redistributed the masses (more or less) proportionally to the slope of the experimental values. Table 2 gives the numbers for the classes used while gure 10 clearly shows a better t for the smaller particles (beware, for more details, the y-axis scale has been adapted between gures 9 and 10) We note that the integral of the experimental curve in both gures 9 and 10 seems slightly higher than the sum of the surface of the computational bars. This would be a problem since both should add up to 100 %. The explanation for this is that, due to a recirculation zone close to the symmetry axis, particles get trapped and coalesce without limit, this numerical artefact results in one or two super massive particles that tilt the scale for the numerical results. have a diameter inferior to 160
µm.
µm
(100 % of the particles sampled experimentally
while 0.75 % of the numerical particles detected are bigger than 160
We assume that the 0.75 % of missing mass is negligible.) Table 2: Distribution 2: injection classes mass ow (kg.m
−2
.s−1 )
class
diameter (µm)
1
1
0.16325
25
2
5
0.081625
12.5
3
10
0.16325
25
4
15
0.244875
37.5
mass fraction (%)
IV.E
Distribution 3
13
8
Experiment Calculation
Volume ratio (%)
6
4
2
0 0
25
50
75 100 Diameter (μm)
125
150
Figure 10: Distribution 2: comparison between experimental and numerical results
IV.E.
Distribution 3
In order to re ne the results even more, two more classes of particles have been added (cf. table 3) the mass distribution was modi ed also based on the results presented in gure 10.
Figure 11 shows the
results of these computations (beware, for more details, the y-axis scale has again been adapted between gures 10 and 11). We see an overall good comparison between the experimental data and the numerical results, especially the apparition of the third mode of particles with a diameter of around 40
µm.
Table 3: Distribution 3: injection classes
2
class
diameter (µm)
mass ow (kg/m )
mass fraction (%)
1
1
0.09795
15
2
5
0.0653
10
3
7.5
0.09795
15
4
10
0.119499
18.3
5
12.5
0.1306
20
6
15
0.141701
21.7
V
CONCLUSIONS
14
4
Experiment Calculation
Volume ratio (%)
3
2
1
0 0
25
50
75 100 Diameter (μm)
125
150
Figure 11: Distribution 3: comparison between experimental and numerical results
We could continue re ning the classes of smaller particles but we stop here since we expect only minor improvements. A better approach would be to replace the discrete classes between 5
µm
µm
and 15
with a continuous , bell-shaped distribution. This is part of the future plans.
V. Conclusions The experimental determination of the particle size distribution of
Al2 O3
droplets emerging from the
surface of a burning solid propellant grain is extremely complicated. To our knowledge, there exists no experimental installation capable of doing just that under realistic temperature and pressure conditions. Since the installation of the KeRC set-up in our laboratory, many dierent modi cations and strategies have been tested, each with it's own particular advantages and drawbacks. The method presented in this article uses a dierent approach in trying to simulate the entire experiment in order to be able, as some sort of inverse problem method, to determine the boundary conditions that lead to the experimental observations. This allowed us to reconstruct, with a fair degree of con dence, the particle size distribution originating from a Butalane fuel grain.
Several questions are still open and need to be addressed in more
detail:
What is the in uence of the change in internal geometry during the sampling period?
The fact that the turbulent structures of the ow diverge a signi cant amount of smaller particles away from the sampling probe must have a negative in uence on the sensitivity of the method. What is the error margin due to this separation eect?
Once these questions answered, this numerical approach can be used to both support our KeRC experimental work and inspire and explore technical modi cations to reduce the inherent bias present in the actual set-up. The bottom line is that the development of ever richer and detailed models is important in order to understand the complex physical phenomena at play. This is especially true in rocket propulsion where the extreme functioning conditions make direct observations near to impossible. We feel that the future of the KeRC set-up has become just a little brighter.
REFERENCES
15
References 1 Duterque, J. and Hommel, J., Etude de l'agglomération et de la combustion des particules d'aluminium dans les propergols solides,
La Recherche aérospatiale, , No. 4, 1993, pp. 124.
2 Beckstead, M. W., A Model for Solid Propellant Combustion, in Proceedings of the 14th Jannaf Combustion Meeting, , 1977.
3 Cai, W., Ma, F., and Yang, V., Two-phase vorticoacoustic ow interactions in solid-propellant rocket motors,
Journal of Propulsion and Power, Vol. 19, No. 3, 2003, pp. 385396.
4 Cesco, N., Dumas, L., Hulin, A., Pevergne, T., and Fabignon, Y., Stochastic Models to the Investigation of Slag
Accumulation in a Large Solid Rocket Motor, AIAA Paper 19973118, 1997.
5 Guery, J., Godfroy, F., Ballereau, S., Gallier, S., Pieta, P. D., Orlandi, O., Robert, E., and Cesco, N., Thrust Oscillations in Solid Rocket Motors,
Exhibit, Hartford, CT, Vol. 2008-4979.
AIAA Paper 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and
6 Salita, M., De ciencies and Requirements in Modeling of Slag Generation in Solid Rocket Motors,
Propulsion and Power, Vol. 11, No. 1.
Journal of
7 Traineau, J., Kuenzmann, P., Prévost, M., Tarrin, P., and Delfour, A., Particle size distribution measurements in a 28th .
subscale motor for the Ariane 5 solid rocket boosters,
AIAA Joint Propulsion
8 Gosse, S., Hespel, L., Gossart, P., and Delfour, A., Morphological characterization and particle sizing of alumina particles in solid rocket motor,
Journal of Propulsion and Power, Vol. 22, No. 1, 2006, pp. 127135.
9 jin, L. P., hua, B. J., ming, Y. X., and bo, Z. Z., Collection and analysis on the condensed-phase particles in the chamber of SRM,
Journal of Solid Rocket Technology, Vol. 31, No. 5, 2008, pp. 461.
10 Salita, M., Survey of recent
Al2 O3
droplet size data in solid rocket chambers, nozzles and plumes, Vol. 1, 1994,
pp. 117.
11 Salita, M., Use of water and mercury droplets to simulate Al O collision/coalescence in rocket motors, Journal 2 3 of Propulsion and Power, Vol. 4, 1991, pp. 505512. 12 Friemel, R., Etude expérimentale et numérique d'une sonde de prélèvement isocinétique en écoulement diphasique dense. Application à la granulométrie des produits de combustion des propulseurs à propergol solide., Ph.D. thesis, ENSAE, 2005.
13 Hijlkema, J., Modélisation cinétique et simulation numérique d'un brouillard de gouttelettes. Application aux propulseurs à poudres, Ph.D. thesis, ENSAE, 1999.
14 Achim, P. and Berlemont, A., Comportement dispersif des gouttelettes d'alumine dans les MPS; prise en compte 3me .
des interactions particulaires,
Ecoulements internes en propulsion solide,
colloque R&T
15 Dumas, L., Chauvot, J., and Schmeisser, K., Modeling of slag deposition in solid rocket motors,
Porpulsion conference, pp. 952729.
31th
AIAA Joint
16 Wahab, S., Kan, X., and Yu, L., Numerical simulation of multi-phase combustion ow in solid rocket motors with
metalized propellant,
Journal of Aerospace Power, Vol. 24, No. 7, 2009, pp. 16541660.
17 Villedieu, P. and Hylkema, J., Kinetic modelling and numerical simulation of dense liquid sprays,
uid dynamics'98, pp. 325330.
18 Dupays, J., Two-phase unsteady ow in solid rocket motors,
Computational
Aerospace Science and Technology,
Vol. 6, No. 6,
2002, pp. 413 422, doi:DOI:10.1016/S1270-9638(02)01182-3.
19 Cesco, N.,
Étude et modélisation de l'écoulement diphasique à l'intérieur des propulseurs à poudre,
Ph.D. thesis,
École Nationale Supérieure de l'Aéronautique et de l'Espace, Toulouse, France, 1997.
The Boltzmann equation
20 Cercignani, C., , Springer, 1988. 21 Laviéville, J., Deutsch, E., and Simonin, O., Large eddy simulation of interactions between colliding particles and a homogeneous isotropic turbulence eld,
Gas-Solid Flows ASME, Vol. 228.
