Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000) 317
Solid Propellant Extinction by Laser Pulse
B. V. Novozhilov
Russian Academy of Science - Institute of Chemical Physics, 4, Kosygina St. - Moscow 117977 (Russia)
C. Zanotti* and P. Giuliani
Istituto per la Tecnologia dei Materiali e dei Processi Energetici TeMPE - C.N.R., via R. Cozzi 53, 20125 Milano (Italy)
�
Ausloschung von Festtreibstoffen durch Laserpulse Extinction de propergols solides par des impulsions lasers
� � � � �
Mit einem CO2-Laser wurde die Moglichkeit demonstriert, einen Avec un laser CO2, on a montre qu'il etait possible de generer un
�
�
Ubergangszustand der Verbrennung eines Komposit-Treibstoffes etat transitoire de combustion d'un propergol composite
� �
AP.HTPB=86.14 zu erzeugen, der unter stationaren Bedingungen AP.HTPB=86.14, qui brule dans des conditions stationnaires. Les
� �
brannte. Die experimentellen Ergebnisse zeigen auf, dass das resultats experimentaux montrent que le comportement de combustion
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Verbrennungsverhalten durch die Kurven bestimmt werden kann, die peut etre determine par les courbes qui separent, pour chaque pression
fur jeden Arbeitsdruck die kontinuierlichen Verbrennungslosungen de travail, les solutions de combustion continue des solutions d'ex-
� �
� � � � � �
von den Ausloschlosungen trennen. In dieser Arbeit liefert ein ver- tinction. Dans la presente etude, une approche theorique simpli�ee
� � �r � �
einfachter theoretischer Ansatz eine phanomenologische Erklarung fu donne une explication phenomenologique aux effets des impulsions
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die Effekte von Energiepulsen auf den Verbrennungsprozess und die d'energie sur le processus de combustion et a la reaction instationnaire
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nachfolgende, instationare Reaktion auf das Strahlungsende. Im qui suit la �n de la radiation. Dans le cadre de cette etude, la condition
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Rahmen dieser Studie wird die Ausloschbedingung in Abhangigkeit d'extinction est formulee en fonction de la temperature minimale qui
der Minimaltemperatur formuliert, die die Unterdruckung der Treib- entra�ne l'extinction de la combustion du propergol a la pression limite
� � �
� �
stoffverbrennung am Grenzdruck der Verbrennung verursacht. Die de combustion. Le travail theorique propose a surtout pour objectif de
� � � �
vorliegende theoretische Arbeit hat vor allem das Ziel, den kritischen determiner la valeur critique du Żux energetique pour differentes
Wert des StrahlungsŻusses fur verschiedene Drucke zu bestimmen, pressions sous lesquelles une extinction n'a jamais lieu, meme lorsque
� � �
� �
unter dem nie eine Ausloschung erfolgt, auch wenn die Pulsdauer la duree d'impulsion tend vers l'in�ni. Les limites d'extinction sont
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gegen unendlich geht. Dann sind die Ausloschgrenzen durch zwei ensuite de�nies par deux approches differentes, qui tiennent compte du
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verschiedene Ansatze de�niert, die das Verhaltnis der Relaxationszeit rapport entre le temps de relaxation en phase condensee et la duree
� �
in der kondensierten Phase und die Pulsdauer der Strahlungsenergie d'impulsion de l'energie de rayonnement. Deux cas limites donnes par
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mit einbeziehen. Zwei Grenzfalle, die durch die langsame bzw. l'interaction lente ou rapide de l'energie de rayonnement avec le
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schnelle Wechselwirkung der Strahlungsenergie mit dem Ver- processus de combustion peuvent etre utilises pour decrire le phe-
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brennungsprozess gegeben sind, konnen zur Beschreibung der nomene d'extinction et les resultats de ce travail montrent que la
� � � � � �
Ausloschphanomene genutzt werden, und die Ergebnisse dieser Arbeit tendance generale des valeurs limites calculees reproduit les resultats
�
zeigen, dass der generelle Trend der berechneten Grenzwerte die experimentaux.
experimentellen Ergebnisse wiedergibt.
the burning propellant extinction phenomena, and the results of this
Summary
work indicate that the general trend of the computed boundary limits
reproduces the experimental data.
The possibility to generate a combustion transient of a composite
AP.HTPB=86.14 propellant, burning under steady state conditions,
was experimentally demonstrated by using a CO2 laser energy pulse.
The experimental results point out that the burning propellant behavior
can be de�ned by the curves separating, for every operating pressure,
1. Introduction
the continuous burning from the extinction solutions. In this paper, a
simpli�ed theoretical approach gives a phenomenological explanation
of the energy pulse effect on the combustion process and the con- In the past, many papers devoted to the study of propellant-
sequent burning propellant response after the deradiation transient. In
laser-Żux-interaction have been published, and one of the
the framework of this study the extinction condition is formulated in
pioneer work in this �eld regards the ignition and gasi�cation
terms of the minimum temperature that causes the burning propellant
of double-base propellant induced by CO2 laser irradiation at
to quench at the Pressure DeŻagration Limit. The proposed theoretical
work is aimed, �rst of all, to determine the critical radiant Żux values, normal and high pressure as reported in Ref. 1. However, the
for different operating pressures, below which the burning propellant
capability to extinguish composite propellants, burning
extinction is never achieved even if the laser pulse duration tends to
under steady state conditions by a radiant energy pulse, has
in�nity. Then, the extinction boundaries are de�ned choosing two
been experimentally demonstrated only in the recent past(2).
different approximate approaches that take into account the ratio
between the condensed phase relaxation time and the radiant energy
This procedure offers the advantage to perturb the combus-
pulse duration. Two limit cases, de�ned as slow=fast interaction of the
tion process without modifying the Żuid dynamic �eld
radiant energy with the combustion process, can be used to describe
around the burning propellant sample and for that reason
the experimental tests can be carried out at constant pressure.
In this case, the burning propellant is removed from its steady
state combustion regime only by the radiant energy pulse,
* Corresponding author; e-mail: [e-mail] and that permits to analyze the extinction phenomena, if any,
# WILEY-VCH Verlag GmbH, D-69451 Weinheim, 2000 0721-3115/00/0612ą0317 $17.50�:50=0
318 B. V. Novozhilov, C. Zanotti, and P. Giuliani Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000)
only considering the laser pulse duration and makes the deradiation transient, decreases to reach values lower than
comparison among the obtained results more immediate. the critical surface temperature and extinction occurs. Here,
The experimental data, used in this theoretical approach, the latent heat per unit propellant mass needed to transform
are obtained using a CO2 laser system, working in continuous the condensed phase into gas can be considered as the main
wave mode to guarantee the best laser power stability and parameter that inŻuences the burning propellant behavior.
beam intensity distribution (Gaussian) during the tests. Thus,
the laser energy pulse has been generated activating a fast
2. Theoretical Model
electro-mechanical shutter, driven by personal computer,
properly designed for this scope.
A steady state combustion regime of any burning propel-
The burning propellant response to the radiant energy
lant is de�ned by the steady-state laws giving the dependence
pulse has been detected by means of a nonintrusive diag-
of the linear burning rate, u0, and of the surface temperature
nostic technique (Laser Doppler Velocimetry) that is able to
Ts0, on the external parameters(3) as in the following:
work at high frequency. Information, describing the phenom-
ena occurring during the combustion transient, has been
u0 � Fp; Taą Ts0 � Fp; Taą1ą
obtained and parameters concerning the extinction condi-
The simplest propellant model is used in this paper and the
tions have been de�ned. More information on the experi-
Eqs. (1) are taken in the form
mental apparatus, diagnostic techniques and results can be
v
found in Ref. 2 where, in order to simplify the data presenta- a
u0 � Ap ebT u0 � Be E=2RTs0 2ą
tion, all the collected results were referred to the incident
that, when operating at constant initial temperature, it is
laser Żux value corresponding to the maximum of the
useful to write the �rst expression of Eq. (1) in the following
Gaussian power distribution. Finally, no corrections were
way:
introduced to take into account the radiation absorption in the
v
gas phase or laser radiant energy reŻection from the propel- a
u0 � Dp D � AebT
lant burning surface and other possible losses of the radiant
As the steady burning rate at the PDL value(4) is different
energy along the path from the laser head to the propellant
from zero, we can use Eq. (2) to determine its value:
burning surface.
0
v
The most straightforward and natural method to elaborate a
u0 � Ap`ebT u0 � Be E=2RTs` 3ą
` `
a theory of solid propellant extinction by deradiation is to
Moreover, it has been shown(5) that the steady burning rate
treat nonsteady burning problems, formulated in the form of
dependence on the radiant heat Żux, I, is well described by
heat conduction equation in the condensed phase with a
the following relationship:
radiant heat source, supplemented with proper boundary
and initial conditions. v
a
I
u0 � Ap ebT �I=rcu0ąą 4ą
I
The aim of this work is to give a simple physical
while the second expression of Eq. (2) permits to write the
explanation of the observed phenomena and to develop a
correlation between the steady burning rate and the burning
theory of solid propellant extinction by a radiant energy
surface temperature in the presence of the radiant heat Żux.
pulse. This approach is far from the scope to describe the full
0
burning rate-time history but it allows to �nd the extinction
u0 � Be E=2RTsI 5ą
I
criterion connecting the most important parameters at the
extinction boundary.
First of all, the extinction criterion has been formulated 2.1 Extinction Condition
considering the existence of the Pressure DeŻagration Limit
(PDL) and then taking into account the ratio between the On the basis of the previous considerations it is necessary
condensed phase relaxation time and the laser pulse duration.
to formulate an extinction condition that implies that the
Two different approximate approaches to calculate the
combustion process can be quenched if the propellant
extinction boundary are examined. If the laser pulse duration
burning surface temperature reaches the value that corre-
is longer than the characteristic time of the solid phase, the sponds to the temperature at the PDL. Thus, the propellant
burning propellant, before the deradiation transient, combustion process is not activated when:
approaches the steady state regime and for this reason the
0
Ts Ts` 6ą
burning rate depends only on the laser radiant heat Żux and
operating pressure. In this case, the extinction criteria are or in terms of burning rate we can write:
determined from the analysis of the burning propellant
u u` 7ą
response features only depending on the steady state para-
meters.
The second case is characterized by the opposite condi- 2.2 Critical Radiant Heat Flux
tion; the propellant relaxation time is larger than the laser
pulse duration, thus, the radiant energy is deposited in a thin The experimental data show that a minimum radiant Żux
surface propellant layer that is quickly converted into gas. value I* can be found at every �xed pressure, below which
For that reason, the burning surface temperature, during the the burning propellant extinction is never achieved even if
Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000) Solid Propellant Extinction by Laser Pulse 319
eą ną
the laser pulse duration tends to in�nity, tp !1ą. The �rst Table 1. Radiant Fluxes I0 ą, I0 ą for tp )1
step of this work is addressed to the estimation of these ną eą
p I0 I0
critical radiant Żux values and Table 1 gives the experimental
kPa W=cm2 W=cm2
data de�ning the gaps inside which the extinction boundary
12 - 15
lies when the pulse duration tp !1. The critical radiant
15 15 21
heat Żux, I 0, is between Ieą and Iną Ieą4I 04Inąą.
20 42 63
0 0 0 0
If tp !1the propellant burns under steady-state com- 25 84 -
30 105 126
bustion regime ( p � const, I � const) the burning rate and
surface temperature are given by Eqs. (4, 5) and every
deradiation process begins with the burning rate changing
from u0 to u05u0. The �nal result of this combustion
I I
transient depends on the relations between the values of u0,
I
u0 and u0 and the experimental results indicate that three
`
different possible types of the combustion transients, induced
by the laser cut off, must be analyzed as depicted in Figure 1.
Figure 1a depicts the case where the laser Żux intensity is
not suitable to decrease, during the deradiation transient, the
instantaneous burning rate value below the one that is
reached at the PDL, so the combustion process can not be
quenched. An other solution is shown in Figure 1b where the
deradiation transient starts from a larger value of the laser
Żux and, in this case, the burning rate decrease is such to
obtain its value below the one corresponding at the PDL.
Therefore, a critical radiant Żux exists that should de�ne the
boundary limit between extinction=no-extinction burning
propellant behavior as shown in Figure 1c.
Every burning propellant can be considered an oscillatory
system having a natural frequency and damping features(6)
able to generate an oscillatory regime, damped in time (see
Figure 7(2)), and well de�ned if the combustion transient
yields the burning surface temperature close to Ts`. If no
damping effects are involved, the burning propellant extinc-
tion occurs when u0 u0 � u0 u0 but in the real case the
I `
experimental evidence shows that the difference u0 u0
I
should be larger in comparison with u0 u0.
`
The approximate estimation of the critical value of I must
consider this peculiar aspect involving a phenomenological
coef�cient a that takes into account the damping effect. So,
the new condition for the burning propellant extinction can
be written in the form:
u0 u04au0 u0ą8ą
I `
and the critical value uI can be obtained from the equality
u0 u0 � au0 u0ą9ą
I `
or
u0
`
u0 � u0 1 � a a 10ą
I
u0
Relationship between u0 and I has been obtained using
Eqs. (2, 4) which give
Figure 1. (a) Burning propellant response after the laser cut off
u0 I
I
ln � b 11ą
tp )1). I5I no extinction u05u0 . (b) Burning propellant
I I
u0 rcu0
I
response after the laser cut off tp )1). I4I extinction u04u0 .
I I
(c) Burning propellant response after the laser cut off tp )1).
and now by the Eqs. (10, 11) it is possible to calculate the
I � I critical condition u0 � u0 boundary between extinction and no
I I
critical value of the radiant Żux extinction.
320 B. V. Novozhilov, C. Zanotti, and P. Giuliani Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000)
rc u0 u0
` `
I � u0 1 � a a ln 1 � a a 12ą
b u0 u0
If the Zel'dovich parameter k � bTs0 Taą is introduced
we obtain:
rcu0Ts0 Taą
I 13ą
k
indicating that the critical radiant heat Żux is of the order of
the convective heat Żux inside the propellant condensed
phase rcu0Ts0 Taą.
2.3 Slow and Fast Interaction Between Combustion and
Figure 2. Burning rate behavior induced by laser pulses.
Irradiation
The parameters, for a given p, that characterize the laser
Appendix. On the basis of these considerations the solution
pulse able to generate a combustion transient that yields the
of Eq. (14) is:
burning propellant extinction are: the radiant energy Żux I
uI � u01 Ce t=tcIą15ą
and the minimum exposure time tp at which extinction
I
occurs. On the other hand, the burning propellant response
and C is a constant of the order of unity. The critical
depends on the condensed phase relaxation time tc and thus,
extinction condition can be written using Eq. (15) as
two limit cases of the burning propellant extinction can be
follows:
distinguished considering the relation between tp and tc.
u0 � u01 Ce tp=tcIą16ą
I I
1ą tp tc slow interaction.
which gives an explicit form of tp
If the laser pulse duration is larger or of the order of tc the
quasi steady state combustion regime can be reached before u0
I
tp � tcI ln C ln 1 17ą
the laser cut off.
u0
I
2ą tp tc fast interaction.
From the other side we can also write using Eqs. (4) and (11)
If the laser pulse duration is much shorter than tc the
rcu0 u0 rcu0 u0
I I I I
I � ln I � ln 18ą
instantaneous burning rate value, when the radiant Żux is cut
b u0 b u0
off, is far from the steady state regime.
Eqs. (17) and (18) allow to obtain a parametric representation
Of course, different theories, describing these two opposite
of the extinction function tpIą introducing the following
cases, must be used as reported in the following subsections.
values
u0 u0 rc
I I
z � z � q � 19ą
2.4 Slow Interaction
u0 u0 b
Using these values and Eq. (A8) for the relaxation time
If tp tc the propellant burning rate, before the laser cut
from the Appendix, Eqs. (17, 18) become:
off, is close to the u0 value and the possible types of burning
I
propellant response after the laser cut off are illustrated in k 1 z
tp � ln C ln 1
Figure 2.
z
u0ą2 z2
2
(1) t15tp, uIt1ą5u0 (curve 1) no extinction,
lu0
I
1 � z ln z
(2) t24tp, uIt2ą4u0 (curve 2) extinction, 20ą
I kbTs0 Taą
(3) t � tp, uItpą4u0 (curve 3) critical condition.
I
I � qu0z ln z
The asymptotic time behavior of the burning rate at t � tp
can be described by the phenomenological differential equa- I � qu0z ln z
tion
So, in the case of slow interaction the extinction curve tpIą
duI u0 uI can be obtained using the following procedure:
I
� 14ą
dt tcI
(a) for a given pressure u0, q and I are known, thus z is
that points out that the approaching rate of uI, to the steady calculated by the relation z ln z � I =qu0.
state value u0, is proportional to the difference between (b) for a set of parameters z4z it is possible to calculate
I
them. Here the characteristic time of the irradiated con- tpzą and Izą by using Eq. (20): The dependency tp on
densed phase tcI has been introduced as de�ned in the the radiant energy can be easily plotted.
Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000) Solid Propellant Extinction by Laser Pulse 321
(c) It should be noticed that when z � z we have I � I off the propellant does not burn because its surface tempera-
and as a result tp )1(critical conditions). ture is lower thanTs`.
The critical condition is shtpą �sgtpą or:
p������
�
It
a ktp � 23ą
rQ
2.5 Fast Interaction
As it is rather dif�cult to de�ne strictly the Q parameter,
ad hoc experiments should be made to estimate this energy
When the laser pulse duration is smaller in comparison
value. In this situation, we may put a � 1 and consider Q as an
with the propellant characteristic time two main physical
effective energy necessary to gasify the propellant. Thus,
processes, due to irradiation effect, must be considered.
Eqs. (22, 23) give the simple expression for the extinction
Firstly, the absorption of the high value of the radiant
conditions in the case of the fast interaction between
energy induces an important change of the thermal wave
combustion and irradiation.
structure inside the burning propellant and then the thin
heated layer at the burning surface may be converted into
kr2Q2
tp � : 24ą
gas if the laser heat Żux is rather large.
I2
The critical extinction condition imposes that after laser
Two remarks should be made at this point. Firstly, Eq. (24)
cut off the propellant burning surface temperature must be
does not contain any characteristics of the combustion wave
equal to the value reached at the PDL limit, Ts` and this
because the laser radiant Żux is larger in comparison with the
condition can be obtained by a simple dimensional con-
convective one, inside the propellant, at the burning surface.
sideration.
This condition implies that the convective effect during the
The thickness� of the heated layer by radiation grows with
p����
combustion transient is negligible. Of course, a more detailed
the time as kt while the position of the point where the
theory may reveal the inŻuence of combustion characteristics
condensed phase has the critical temperature, Ts`, obeys to
on the extinction conditions, however, as shown in the next
the same law:
section, experimental data, both for 12 kPa and 15 kPa, are
p�����
sh � a kt 21ą described by Eq. (24) rather well.
Secondly, we can expect that the order of the value of the
where a is a constant. From the other side, the thickness of
effective energy Q could be few hundred calories per gram
the propellant layer that is converted into gas, requiring Q
and such energy is released in the condensed reaction layer
energy per unit mass, is proportional to the radiant energy
during the combustion(7).
as described here:
It
sg � 22ą
rQ
3. Theory and Experimental Comparison
The curves reported in Figure 3 illustrate that different
The steady state burning characteristics of the
possible results could be obtained depending on the laser
AP.HTPB=86.14 composite propellant tested in Ref. 2 are
pulse duration. If the laser is switched off when (t � t1) the
reported in Table 2 and the numerical values of the propellant
thermal wave structure in the solid is such that the heated
thermophysical properties used in comparison of the theory
layer is larger than the gasi�ed one, thus, at that while the
with the experimental data are given in Table 3. They have
burning surface temperature is higher than Ts` and for that
been taken from Refs. 8, 9.
reason no extinction occurs. In the opposite case, t � t2 it
The whole set of the experimental results is reported in
happens that sgt2ą4sht2ą and during the solid gasi�cation
Ref. 2 and the useful data summarized in Table 4. Here the
the point with Ts` is inside the gasi�ed zone and after laser cut
minimum pulse duration at which the extinction exists is
denoted by teą, while the symbol tną corresponds to the
p p
maximum pulse duration at which no extinction is observed
in the experiments.
3.1 Critical Radiant Heat Flux
In this section a comparison between the experimental
radiant Żux I0 and the theoretical value I, was made. The
incident radiant Żux values associated to experimental
results, as mentioned in the introduction, used to validate
the theory are overestimated with respect to the effective
radiant energy deposited into the burning propellant. Deter-
mination of the energy losses, for every operating condition,
Figure 3. Critical condition for the fast interaction. is not an easy task due to the lack of information on the
322 B. V. Novozhilov, C. Zanotti, and P. Giuliani Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000)
Table 2. Propellant Combustion Characteristics
0
pu 102 Ts tc
kPa cm=sK s
5.8 1.12 680 8.12
12 2.20 711 2.06
15 2.71 721 1.36
20 3.55 733 0.79
30 5.18 755 0.36
Table 3. Propellant Thermophysical Properties
Symbol Quantity Unit Value
r density g=cm3 1.6
Figure 4. Critical radiant Żux. Theory and experiment comparison.
c heat capacity J=g K 1.7
k thermal diffusivity cm2=s 10 3
b temperature sensitivity K 1 3 10 3
n pressure exponent 0.932
3.2 Extinction Boundary
E activation energy J=mol 176 103
Ta ambient temperature K 300
The theoretical extinction boundaries, for different values
D factor in the burning law cm=s (kPa)n 2.17 10 3
of the operating pressure, are presented in Figs. 5ą7 and the
p` pressure deflagration limit kPa 5.8
u0 burning rate at PDL cm=s 1.12 10 2
experimental extinction=no-extinction data are reported as
`
0
Ts` surface temperature at PDL K 680
well.
l absorption length cm 3.6 10 3
One can see that at low pressure ( p � 12 kPa and 15 kPa)
both branches exist (s curves and f curves). The available
experimental data at p � 20 kPa are not suf�cient to obtain
radiant energy absorption by the gas and the burning
comparisons for the �rst branch.
propellant surface reŻectivity.
The only consideration on this matter is that the laser
radiant Żux before reaching the burning surface crosses two
ZnSe lenses and one ZnSe window having a radiant energy
loss of about 2% each. Thus, taking also into account that the
average energy value deposited on the burning propellant
surface is 3% less than the maximum of the Gaussian
distribution and that an error of 5% must be considered
in reading the total energy Żux, at least a reduction between
5% to 13% of the incident Żux is reasonable.
The theoretical values of the critical radiant Żux are
calculated by Eq. (12) where the only parameter that can
not be evaluated in the framework of this approximated
theory is the damping parameter a. Therefore, theoretical
curves for different a are plotted in Figure 4 showing that the
theory predicts the order of I rather well if the a parameter
grows with pressure from 1 to 2. Figure 5. Extinction boundary for p � 12 kPa.
Table 4. Laser Pulse Durations teąą, tnąą for Different Operating Pressures
p p
I0 p � 12 kPa p � 15 kPa p � 20 kPa p � 30 kPa
tną teą tną teą tną teą tną teą
p p p p p p p p
W=cm2 sssssss s
15 - - 5.6 - - - - -
21 0.9 1.0 2.6 2.8 - - - -
31 0.3 0.35 0.64 0.72 5.6 - - -
42 0.205 0.235 0.3 0.35 5.6 - - -
63 0.17 0.205 0.252 0.288 0.8 0.9 - -
84 0.132 0.17 0.14 0.168 0.358 0.404 - -
105 0.072 0.164 0.9 0.116 0.242 0.272 - -
126 0.048 0.06 - - 0.25 0.27 1.1 1.2
147 0.045 0.056 0.056 0.094 0.136 0.152 0.5 0.55
Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000) Solid Propellant Extinction by Laser Pulse 323
detailed data on some thermophysical propellant properties
and to widen the operating pressure range.
Last but not least a more powerful laser will be used to be
able to perform tests at higher radiant energy Żuxes in order
to enlarge the operating conditions.
From the theoretical point of view, a more strict and
detailed theory should be developed using nonsteady pro-
pellant combustion theory with appropriate steady state
propellant burning laws. It may be made in the framework
of the Zel'dovich-Novozhilov theory(3,6).
5. References
Figure 6. Extinction boundary for p � 15 kPa.
(1) B. N. Kondrikov, T. Ohlemiller, M. Summer�eld, ``Ignition and
Gasi�cation of Double-Base Propellant Induced by CO2-Laser
Irradiation'', Problems of Theory of High Explosive 83, 67ą78
(1974).
(2) C. Zanotti and P. Giuliani, ``Composite Propellant Ignition and
Extinction by CO2 Laser at Subatmospheric Pressure'', Pro-
pellants, Explosives, Pyrotechnics 23, 254ą259 (1998).
(3) B. V. Novozhilov, ``Nonsteady Burning and Combustion Stability
of Solid Propellants'', AIAA Progress in Astronautics and Aero-
nautics, Vol. 143, Washington, 1992, pp. 601ą641.
(4) C. Zanotti and P. Giuliani, ``Pressure DeŻagration Limit of Solid
Rocket Propellants: Experimental Results'', Combustion and
Flame 98, 35ą45 (1994).
(5) I. G. Assovskii and A. G. Istratov, ``Propellant Burning under
Light Irradiation'', Journal of Applied Mechanics and Technical
Physics 5, 70ą77 (1971).
(6) B. V. Novozhilov, ``Nonstationary Combustion of Solid Rocket
Fuels'', Nauka, Moscow, 1973 (Translation AFSC FTD-MD-24-
317-74).
(7) A. Zenin, et al., ``Nonsteady Burning and Combustion Stability of
Solid Propellants'', AIAA Progress in Astronautics and Aero-
Figure 7. Extinction boundary for p � 20 kPa.
nautics, Vol. 143, Washington, 1992, pp. 197ą231.
(8) C. Zanotti, et al., ``Nonsteady Burning and Combustion Stability of
Solid Propellants'', AIAA Progress in Astronautics and Aero-
The entire �tting parameters (I , C and Q) are of reason-
nautics, Vol. 143, Washington, 1992, pp. 399ą439.
able values and the regions where exist the slow and fast
(9) C. Zanotti, et al., ``Nonsteady Burning and Combustion Stability of
Solid Propellants'', AIAA Progress in Astronautics and Aero-
branches correspond to the theoretical predictions.
nautics, Vol. 143, Washington, 1992, pp. 145ą196.
4. Conclusion and Future Work Nomenclature
The experimental study reported in Ref. 2 and the theore- a dimensionless factor, see Eq. (21)
tical consideration of this work is only the �rst step investi- A burning law constant
gating such an interesting aspect of nonsteady propellant B burning law constant
combustion as the propellant extinction by a laser pulse. C integration constant
The results obtained by this approach, based on experi- D burning law constant
mental results collected in a limited range of pressure and E activation energy
using low values of the incident laser Żux, shows that it is F burning law function
possible to explain the peculiar form of the curves represent- h propellant heated layer thickness
ing the extinction-non extinction boundary limits. I radiant heat Żux
Even if many propellant parameters, used in this work, k Zel'dovich parameter
have been chosen from literature and they should be different l absorption length
from the ones characterizing the tested propellant we can say p pressure
that the theoretical approach is however consistent with the q given by Eq. (19)
experimental trend of the results. Q effective propellant latent heat for gasi�cation
For those reasons, the future work will be addressed to get R universal gas constant
more experimental information on the effective radiant Żux s heating or gasi�cation length
impinging on the burning propellant surface, to have more t time
324 B. V. Novozhilov, C. Zanotti, and P. Giuliani Propellants, Explosives, Pyrotechnics 25, 317ą324 (2000)
2 3
T temperature
6 7
I 1 0
u propellant linear burning rate
0
6TsI 7eu x=k
I
TI0xą �Ta � Ta
4 5
k
x space coordinate rcu0
I
1
z parameter given by Eq. (19)
lu0
I
a oscillation damping parameter
I
� ex=l A3ą
b propellant burning rate temperature sensitivity
k
1
F burning law function
lu0
I
k propellant thermal diffusivity
n pressure exponent Estimation of tc requires the evaluation of the value of the
heated layer characteristic length h that can be obtained from
Eq. (A3).
0
1
Superscripts
hc � TI0xą Taądx A4ą
0
TsI Ta 1
* critical value
and by simple integration we get
(n) non extinction
k Il
(e) extinction
hc � 1 � A5ą
o
o steady state u0 rckTsI Taą
I
The relationship between the characteristic relaxation time
and the characteristic length is:
Subscripts
h2
c
tc A6ą
k
o experimental value of radiant heat Żux
from which we have
a ambient
2
I radiant heat Żux
k Il
tcI � 1 � A7ą
h heating o
u0ą2 rckTsI Taą
I
g gasi�cation
0
where instead of TsI one can write Ts0.
` limit value
An alternative expression of the condensed phase relaxa-
s surface
tion time can be obtained from Eqs. (A7, 4) which has the
form:
k lu0 u0 2
Appendix I I
tcI � 1 � ln A8ą
u0ą2 kbTso Taą u0
I
Relaxation time of the condensed phase heated layer with
Without the irradiation this expression gives the usual form
the presence of the radiant heat source is calculated in the
for the characteristic time of the condensed phase
appendix. The steady state heat conduction equation in the
tc � k=u0ą2.
burning propellant condensed phase has the form:
d2T dT I
Acknowledgements
k u0 � ex=l � 0 A1ą
Prof. B. Novozhilov wishes to express his gratitude to the Italian
dx2 I dx rcl
Ministry of Foreign Affairs for the �nancial support during the four
months fellowship organized by the Landau Network - Centro Volta. It
at 15x 0 with the boundary conditions
is also his pleasure to thank Dr. E. Olzi as the director of the hosting
x ) 1 T � Ta Institute TEMPE-CNR. Dr. C. Zanotti wish to express his thank to the
A2ą Director-General A. Nishida and Prof. H. Kohno of the Institute of
x ) 0 T � Ts0 Space and Astronautical Science (ISAS) where he could write the �nal
version of this paper during his stay as Invited Foreign Researcher
In Eq. (A1) l is the radiation absorption length and k is the
Fellow.
thermal diffusivity of the condensed phase. The solution of
Eqs. (A1, A2) is: (Received December 21, 1999; Ms 2000=034)