Combustion, Explosion, and Shock Waves, Vol. 39, No. 1, pp. 59 63, 2003
Quasi-Homogeneous and Pseudospin Modes
of Zirconium-Wire Combustion in Air
S. G. Vadchenko1 UDC 546
Translated from Fizika Goreniya i Vzryva, Vol. 39, No. 1, pp. 69 73, January February, 2003.
Original article submitted January 22, 2002.
A novel experimental technique is proposed for examining the transition mechanism
from quasi-homogeneous to heterogeneous combustion  burning of a variable-pitch
spring. Depending on the pitch of air-combustible zirconium springs, two combus-
tion modes are possible. Quasi-homogeneous (layer-by-layer) combustion is observed
in the case of small-pitch springs; as the spring pitch increases, quasi-homogeneous
combustion transforms into heterogeneous (pseudo-spin) combustion. Conditions for
the occurrence of various combustion modes, depending on the spring diameter and
pitch, are studied.
Key words: heterogeneous combustion, spin, zirconium oxidation.
In the present-day theory of heterogeneous com-
bustion, studying of burning features of specially con-
structed heterogeneous model systems is of great inter-
est. For instance, combustion in multilayered systems
composed by combustible and inert layers was studied in
[1, 2]; the theory of such processes was given in [3 6]. In
these works, concepts of primary importance concern-
ing quasi-homogeneous and relay modes of combustion
were developed. Another heterogeneous model struc-
ture composed by thin alternating fuel and oxidant lay-
ers (for instance, metal and nonmetal layers or layers
of two dissimilar materials) was proposed in [7, 8]. In
these works, the possibility of  high-velocity combus-
tion modes was demonstrated.
In the present article, a new heterogeneous model
(spring) is proposed and examined; this model also per-
mits various combustion modes  solitary waves of the
Fig. 1. Experimental scheme: 1) specimen holder;
2) initial wire; 3) combustion front; 4) combustion
spin type [9] or collective waves resulting from thermal
products; 5) video-camera; 6) igniting wire.
interaction of spring coils.
ten wire diameters. The spring is ignited by a 0.1-mm
EXPERIMENTAL PROCEDURE
diameter molybdenum filament heated by a short elec-
tric pulse. The limiting case of spring combustion (for
The scheme of the experiments is shown in Fig. 1.
h ") is combustion of a straight wire. Combustion
The specimens are springs of diameter D = 0.3 1.1 mm,
can be initiated in the upper or lower part of the spec-
wound from zirconium wire of diameter d = 0.09 mm
imen. The process is registered by a video-camera, and
with a fixed pitch h. The minimum spring length is
the combustion mode and front velocity Uf are found
from the video-recording.
1
Institute of Structural Macrokinetics and Material
The experiments were performed in air under at-
Science Problems, Russian Academy of Sciences,
Chernogolovka 142432; [e-mail]. mospheric pressure.
0010-5082/03/3901-0059 $25.00 © 2003 Plenum Publishing Corporation 59
60 Vadchenko
Fig. 2. Straight-wire burning rate versus the angle Õ:
negative inclination angles refer to combustion initi-
Fig. 3. Frontal (Uf) and predicted frontal (Uf ) burning
ation from below, Õ = 0 to the horizontal position of
rates versus the angle Õ.
the wire, and positive inclination angles to combus-
tion initiation from above.
lowed us to reveal the existence of two combustion
EXPERIMENTAL RESULTS
modes  quasi-homogeneous combustion and pseu-
AND DISCUSSION
dospin combustion. In the first case, the combustion
Influence of Gravitational Convection and front exerts a layer-by-layer motion, without any sub-
Edge Effects. Figure 2 shows the straight-wire burn- stantial delays in going over from one coil to the other.
ing rate U versus the angle of wire deflection from the This mode is typical of dense, small-pitch springs. In
horizontal line (Õ = 0). Engaging attention is the the second case, starting from a certain coil, combus-
burning-rate minimum for a position of the wire close tion propagates along the wire and is visually observed
to the horizontal one, caused by variation of the ef- as a heat-source motion during spin combustion. In the
fective cross section of the convective airflow affecting latter case, the main spring-combustion features (a de-
heat transfer from the combustion front. A study of the crease in the frontal and circumferential burning rates
burning-rate distribution over the length of a straight with increasing spring diameter) are in good qualitative
vertical wire as a function of combustion-wave propa- agreement with the features of ordinary spin combus-
gation direction showed that, if combustion is initiated tion [7].
from above, the steady-state combustion-front velocity Combustion-Front Velocities and Time
is established over a wire length approximately three Shift. Figure 3 shows the combustion-front velocity
times as short as that in the case of initiating combus- (Uf) as a function of the angle Õ between the spring-
tion from below. The observed burning rate differs from coil tangent line and the combustion-front velocity for
the steady-state value over the starting length of the various spring diameters. The angle Õ was calculated
wire because of the additional heating from the igniting by the formula
wire and at the end of the wire because of conductive
h
heat transfer into the specimen holder. Õ = arctan . (1)
Ä„(D - d)
The steady-state burning rate of the spring as a
function of the spring diameter and pitch was stabilized Here the angle Õ is chosen as an argument since it is
after 1 to 5 spring coils were burnt and remained con- a universal value for springs of various diameters and
stant almost up to the end of the process. pitches, permits a comparison between the burning rate
To exclude the influence of convection and edge of springs and the straight-wire burning rate (h "),
effects (additional heat flux from the igniting wire and and provides a convenient scale of values for spring
heat transfer to the specimen holder) on the process, pitches varying from zero to infinity.
we conducted tests with initiation of the reaction from The following specific features of the curves Uf(Õ)
above; measurements were performed over the length of are worth noting:
the spring with an established burning rate.  no substantial burning-rate discontinuity occurs
Modes of Spring Combustion. Video-recording as the quasi-homogeneous combustion mode gives way
of propagating combustion waves along the spring al- to the pseudospin one;
Quasi-Homogeneous and Pseudospin Modes of Zirconium-Wire Combustion in Air 61
Fig. 5. Unit-length burn-up time versus the angle Õ.
Fig. 4. Time shift versus the angle Õ.
 starting from a certain angle Õ, the spring burn-
ing rate becomes higher than the straight-wire burning
rate (the dashed curve in Fig. 3);
 the burning rate displays a maximum at a cer-
tain angle Õ.
By analogy with the combustion-wave propagation
process in a system of plates [1], one can also introduce
the time shift tsh, being the time required for one spring
coil to burn out completely:
tsh = h/Uf. (2)
The dependences tsh(Õ) are shown in Fig. 4.
Such representation of data reveals several character-
istic zones, most distinctly observed during combustion
of small-diameter springs. In the quasi-homogeneous
Fig. 6. Domains of existence of various combustion
combustion-mode region, as the angle Õ (of individual modes: the angle at which the transition from the quasi-
homogeneous to pseudospin combustion mode is ob-
coils) increases, the time tsh remains approximately con-
served (1), the angle at which the frontal burning rate is
stant. The transition to the pseudospin regime is first
higher than the straight-wire burning rate (2), the an-
accompanied by the growth of tsh; afterwards, a plateau
gle Õlim (3), and the angle below which the dependences
is observed in the tsh curve up to a certain value of
tsh(Õ) display the plateau (4).
the angle Õ; and, finally, the time tsh monotonically in-
creases. With increasing spring diameter, the length of
the plateau decreases.
small-diameter springs, these curves display an inflec-
Circumferential Burning Rate and Unit-
tion point. With increasing spring diameter, the max-
Length Burn-up Time. From the combustion-front
imum degenerates into a plateau (D = 0.58 mm), and
velocity, one can calculate the corresponding circumfer-
for the diameter D = 1.1 mm, the time ts monotonically
ential burning rate
increases with increasing angle Õ.
Parametric Domains of Existence of Vari-
(Ä„(D - d))2 Uf
Us = Uf + 1 = (3) ous Combustion Modes. A number of factors affect
h2 sin Õ
the combustion modes and burning rate; these factors
or the unit-length burn-up time ts = 1/Us (in the partially mask the observed effects. These factors in-
present study, we assume the unit length to equal clude heat transfer to the ambient medium, diffusion-
1 mm). It should be noted that the quantities Us and limited transport of oxygen to the wire surface, and non-
ts in the quasi-homogeneous combustion mode are fic- equal accessibility of different parts of the wire at spring
titious. The curves ts(Õ) are shown in Fig. 5. For sections with different spring diameters and pitches.
62 Vadchenko
Nonetheless, using the experimental data, we built a conductive heat transfer along the wire. The number of
parametric diagram of the domains of various combus- heated coils is given by the formula
tion modes in the  angle Õ (spring pitch) spring diam-
eter coordinates (Fig. 6). From the left and below, the
l cos Õ
n = (5)
diagram is bounded by the limiting values of the angle
Ä„(D - d)
Õ given by the formula
d
and ranges from n = 4.5 (D = 0.3 mm and Õ = 8.6ć%)
Õlim = arctan . (4)
Ä„(D - d)
to 0.47 (D = 1.1 mm and Õ = 60ć%). Thus, due to con-
ductive heat transfer along the wire, combustion always
The whole combustion domain can be divided into
two subdomains, diffusion and kinetic ones. The po- occurs at the initial temperature of spring coils higher
than the ambient temperature.
sition of the dividing curves between the subdomains
Owing to conductive and radiative heat transfer
can be roughly estimated from the intersection points
through the gas gap between the coils, the initial tem-
of the curve Uf(Õ) and the straight-wire burning-rate
perature of the wire additionally rises, causing an as-
curve. These points give the maximum values of the
sociated increase in the burning rate; as a consequence,
angle Õ at which the reaction is still limited by oxygen
the length of the heated zone of the wire and the frontal
transport to the reaction zone. The minimum values of
heating length of the spring also undergo changes.
the angle Õ can be found as angles corresponding to the
We measured the straight-wire burning-rate tem-
end of the plateau in the tsh(Õ) curves, at which the
perature coefficient in the temperature range from 293
time tsh starts increasing. Since the oxygen transport
to 533 K. The curve obtained can be fitted by the for-
to the wire surface is diffusion-controlled, one of the
mula
reasons for the occurrence of the plateau in the tsh(Õ)
curves or inflection points in the ts(Õ) curves may be
d ln U
the transition from the diffusion region of the reaction
kT = = 2.45 · 10-3 K-1.
0
dT0
to its kinetic region.
Thus, the quasi-homogeneous combustion is ob-
served in the diffusion region, while the pseudospin com- Comparing the dependences U(T0) and Uf(Õ), one
bustion both in the diffusion and kinetic regions. can conclude that, for the maximum observed excess
Effect of Initial Temperature. Obviously, the of the combustion-front velocity over the straight-wire
combustion-front velocity should tend to the straight- burning rate (D = 0.4 mm and Õ = 57ć%), it suffices to
wire burning rate with increasing spring pitch or in- heat the spring coils to the temperature T0 = 390 K,
creasing angle Õ; hence, this velocity should contin- which may well be the case due to heat transfer along
uously increase. For conditions without heat trans- the wire and across the gas phase.
fer between spring coils, the straight-wire burning rate As was noted above, the plateau in the tsh(Õ)
can be used to estimate the fictitious combustion-front curves or the inflection point in the ts(Õ) curves, indica-
velocity, which is independent of the spring diameter, tive of a relative growth in the circumferential burning
by the formula Uf = U sin Õ (the calculated curve is rate, may be caused by the transition from the diffu-
shown in Fig. 3). However, because of the interaction sion region of the reaction to its kinetic region. Another
between spring coils (heating of the next coil by the possible reason for the increase in the burning rate in a
previous, burning one), the experimentally measured certain range of angles Õ may be approaching of com-
combustion-front velocity exceeds the predicted burn- bustion conditions for fixed-pitch springs to conditions
ing rate in the quasi-homogeneous combustion mode by of the super-adiabatic mode of combustion theoretically
a factor of 2.5 5; as the angle Õ increases, the measured substantiated in [3 5]. The actual reason for the plateau
and predicted values come closer and closer together. can be established by studying zirconium-wire combus-
Apparently, the combustion-front velocity in the tion in oxygen or using, in the experimental procedure
pseudospin-mode domain depends on the circumferen- described above, wires capable of self-sustained com-
tial velocity. As was noted above, the study of the varia- bustion in inert media, such as nickel-coated aluminum
tion of the burning rate along a straight wire revealed a wires.
strong influence of heat transfer to the specimen holder; The author thanks Academician A. G. Merzhanov
this effect was observed at a final section of the wire of for his valuable advice concerning the experimental
length H"3 mm, or 30 wire diameters. This distance can scheme and for discussion of the results obtained.
be identified as the heating-zone length in the straight- This work was supported by the Russian Founda-
wire combustion wave. As the wire is coiled into a tion for Fundamental Research (Grants Nos. 99-03-
spring, there arise coils heated only at the expense of 32392 and 99-03-32020).
Quasi-Homogeneous and Pseudospin Modes of Zirconium-Wire Combustion in Air 63
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