Combustion, Explosion, and Shock Waves, Vol. 38, No. 3, pp. 313 321, 2002
Nonstationary Regimes of Transformation
of Multilayered Heterogeneous Systems
P. M. Krishenik,1 A. G. Merzhanov,1 and K. G. Shkadinskii2 UDC 536.46
Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 3, pp. 70 79, May June, 2002.
Original article submitted June 7, 2001.
A nonstationary mathematical model of thermal propagation of flame in a layered het-
erogeneous system is proposed. The structure and dynamics of the frontal exothermal
transformation in quasihomogeneous, transitional, and relay-race regimes are studied.
Averaged characteristics of the front and dynamics of transformation of individual
elements of a  discrete combustion wave are analyzed using the model proposed.
A correlation is established between the combustion of a model medium and real
heterogeneous compositions. It is shown that the maximum combustion velocity is
reached at an intermediate level of medium dispersion in a transitional parametric
region.
Key words: combustion waves, heterogeneous systems, multilayered nonstationary
regimes, modeling.
The classical theory of flame propagation deals a distance ´ from each other. Heat transfer between
with combustion regimes under the assumption that the the reactive plates was performed through a gap of
initial combustible mixture is homogeneous [1, 2]. In thickness ´ filled by a gaseous oxidizer or an inert sub-
the case of combustion of heterogeneous media with a stance. Depending on the scale of heterogeneity, two
nonuniform initial structure of the mixture, one can ex- limiting regimes of combustion (relay-race and homo-
pect that the flame front is also heterogeneous. This geneous) were found and described in an experimental
is the principal difference between the front of isother- analysis of propagation of a combustion wave in such
mal transformation of heterogeneous systems and the a system. Merzhanov [7] proposed a stationary mathe-
combustion wave in homogeneously mixed media. matical model that describes propagation of transverse
To study the laws of combustion of heterogeneous solid flame in a system of homogeneous layers of charge
systems, a number of models have been proposed, in- with allowance for thermal resistance of the contacts
cluding the classical models of the  sandwich type [3], under the assumptions of the absence of thermal iner-
cellular models [4], a wide class of models that describe tia of the latter and gradientless heating of layers in
combustion of porous compositions, etc. One of the pa- the wave. Such assumptions allowed obtaining approx-
rameters determining the structural characteristics of imate expressions for the flame velocity in two limiting
the medium is the macroscale of heterogeneity, which regimes: homogeneous and relay-race. Filimonov [8]
can be responsible for special features of combustion studied the influence of radiative heat transfer on sta-
of heterogeneous systems. Vadchenko and Merzhanov tionary combustion regimes of a heterogeneous system
[5] proposed a model system of a heterogeneous medium within the framework of the stationary model [7]. A nu-
for experimental investigation of solid-flame combustion merical analysis of the stationary model can be found
processes [6]. This system was a multilayered struc- in [9], where the main attention is paid to the analy-
ture consisting of thin plates of thickness d located at sis of the laws of transition from one limiting regime of
combustion to the other. In [10], which summarizes the
1
Institute of Structural Macrokinetics and Material
results of [5 9], the waves in multilayered systems un-
Science, Russian Academy of Sciences,
der consideration were called  discrete waves, which
Chernogolovka 142432; [e-mail].
2
reflects the special features of combustion of hetero-
Institute of Problems of Chemical Physics,
Russian Academy of Sciences, Chernogolovka 142432.
0010-5082/02/3803-0313 $27.00 © 2002 Plenum Publishing Corporation 313
314 Krishenik, Merzhanov, and Shkadinskii
geneous systems. To support this idea, Rogachev et "Ä ¸ = "¾¾¸ + Å‚-1"Ä ·, (1)
al. [11] analyzed the experimental results on the struc-
where
ture of a heterogeneous combustion wave and concluded
"Ä · = (1 - ·) exp(¸/(1 + ²¸)) = Å‚F (¸, ·), (2)
that all wave regimes examined can be described on the
basis of two shapes of reactive waves: quasistationary
Å‚-1(1 - ·) exp(¸/(1 + ²¸)), 0 d" · d" 1,
and  discrete flickering ones. Problems of direct ex-
F (¸, ·)=
0, · e" 1;
periments stimulate the development of new, more ade-
quate physical models of  discrete combustion waves.
 equation of heat conduction in the ith inert
Combustion of heterogeneous systems is accompanied
layer [id + (i - 1)´ < ¾ < i(d + ´)]
by specific features of chemical transformation under
ÃcÁ"Ä ¸ = Ã"¾¾¸. (3)
conditions of nonlinear thermal interaction between re-
Conditions of thermal coupling of the layers are
active cells, which depends on structural characteristics.
In the present paper, we propose a nonstationary
¸ R = ¸ I, "¾¸ R = Ã"¾¸ I; (4)
mathematical model of thermal propagation of flame in
conditions of initiation (Ä > 0 and ¾ = 0) are
a layered heterogeneous system. The flame-front struc-
ture is analyzed, and the dynamics of propagation of
¸ = ¸in; (5)
solid flame in a multilayered medium is examined. Aver-
initial conditions (Ä = 0 and ¾ > 0) are
aged characteristics of the front and dynamics of trans-
formation of individual elements of a  discrete com-
¸ = ¸0, · = 0. (6)
bustion wave are determined. A correlation of combus-
Scaled quantities and dimensionless variables: t" =
tion laws for the model medium and real heterogeneous
2
RT" k-1 exp (E/RT")cR/QE is the time, x" =
mixtures is found.
[(R/cRÁR)t"]0.5 is the length, T" is the characteris-
tic temperature of the process, Ä = t/t", ¾ = x/x",
2
¸ = (T - T")E/RT" , and · is the depth of transfor-
MATHEMATICAL MODEL
mation of the reactive layer. To pass to dimensionless
Å» Å»
sizes of heterogeneous cells d and ´, we used the char-
Å» Å»
To study the features of combustion of a heteroge-
acteristic scaled length x": d = d/x" and ´ = ´/x". For
neous mixture, we use a simplified (model) system con-
simplicity, the bar over these quantities is omitted.
sisting of alternating combustible layers of thickness d
Note, this problem has two adiabatic temperatures,
and inert or gaseous interlayers of thickness ´. We use
which can be chosen as the characteristic tempera-
the following physical assumptions.
ture T":
1. The layers react in a gasless regime and do not
(1)
Tad = T0 + Q/cR,
change their thermophysical characteristics and sizes.
The thickness of the heterogeneous cell d + ´ is much which is the temperature of adiabatic combustion of the
smaller than the diameter of the whole structure. combustible (reactive) layer and
2. An adiabatic combustion regime is considered;
(2)
Tad = T0 + QÁRd/(cRÁRd + cIÁI´),
heat losses to the ambient medium may be ignored.
3. Heat transfer from one reactive layer to another which is the mean adiabatic temperature established af-
is performed through an immobile inert medium whose ter equalization of temperatures in the system of com-
thermophysical properties are also constant. bustible and inert layers.
4. The activation energy of exothermal transfor- Here R is the universal constant, E is the activa-
mation is assumed to be rather high for the frontal tion energy, k is the preexponent, cR and cI, ÁR and
propagation regime to exist. The size of the hetero- ÁI, and R and I are the heat capacities, densities,
geneous system considered allows the establishment of and thermal conductivities of reactive and inert lay-
the frontal regime of flame propagation. Initiation of ers, respectively, and Q is the thermal effect of the
combustion is performed from the butt-end face of the reaction. The following dimensionless parameters are
2
structure by a high-temperature source. used: Å‚ = RT" cR/EQ is the ratio of the reaction-
We describe the process of exothermal transforma- zone width to the heating-zone width in the combus-
tion of a heterogeneous system by the following dimen- tion wave, ² = RT"/E is the temperature sensitivity of
sionless equations: the reaction rate, ÃcÁ = cIÁI/cRÁR is the ratio of heat
 equation of heat conduction and macrokinetics capacities of unit volumes of inert and reactive layers,
in the ith combustible layer [(i - 1)(d + ´) < ¾ < id + and à = I/R is the ratio of thermal conductivities
(i - 1)´] of inert and reactive layers.
Nonstationary Regimes of Transformation of Multilayered Heterogeneous Systems 315
Fig. 1. Space time distribution of temperature (a) and heat-release function F (¸, ·) (b) at the time Ä = Ä" in a
quasihomogeneous regime of combustion of a multilayered system for d/´ = 0.5, ÃcÁ = 0.25, Ã = 0.1, Å‚ = 0.117,
² = 0.153, ¸0 = -5.67, ¸" = ¸in = 0, and d = 0.3125.
In a heterogeneous system, the process of flame reaches a periodic stationary regime. In a stationary
propagation was described by one temperature ¸(Ä) [7], frontal regime, the spatial dependence of the temper-
and the temperatures of the neighboring plates were re- ature field can be taken into account using the time
lated as ¸i-1(Ä - Ä1) = ¸i(Ä) = ¸i+1(Ä + Ä1), where shift Ä1. Because of heterogeneity and discrete char-
Ä1 is the shift in time. Depending on the value of Ä1, acter of the chemical heat release rate, in solving the
the proposed equations described two limiting regimes above system, it is reasonable to determine the front
of combustion: quasihomogeneous and relay-race. It velocity on the basis of the change in temperature [for
"
was shown that the model of [7] reduces to the classical example, the shift of the isotherm (T + T0)/2].
equations of the flame-propagation theory [1, 2] in the The mathematical model presented refers to the
first case and to equations of thermal explosion with a class of  stiff systems of differential equations. An
stage of heating in the second case. analysis of its solutions is complicated by the necessity
The nonstationary model (1) (6) offers a possi- of long-time tracing of nonlinear dynamics of front prop-
bility of studying the process of combustion and ini- agation until the establishment of a periodic stationary
tiation by a heated wall until the moment of flame regime.
stabilization. It is intuitively obvious that, in a
multilayered system where the scale of heterogene-
ity l = d + ´ tends to zero, one should expect ho-
QUASIHOMOGENEOUS REGIME
mogenization of the medium with the averaged char-
OF COMBUSTION
acteristics " = (1 + ´/d)IR/(R´/d + I) and
(cÁ)" = [(cRÁR + cIÁI´/d]/(1 + ´/d), thermal effect
The problem considered is multiparametric. The
Q" = QcRÁRd/(cRÁRd + cIÁI´), and thermal diffu-
quasihomogeneous regime of combustion is illustrated
sivity a = "/(cÁ)". The characteristic width of the
in Fig. 1. The sufficient condition for the quasihomo-
front of exothermal transformation of such a medium
geneous regime of combustion similar to that shown in
is Lf = a/V , and the combustion velocity itself can be
Fig. 1 is the small thermal thickness of the layers of
estimated according to [12]:
combustible and inert substances, which is equivalent
2 2
V = ak exp (-E/RT")RT" /E(T" - T0), (7)
to the condition (d + ´)/Lf Å‚.
The layered structure of the heterogeneous system
(2)
(T" = Tad ).
is illustrated by the vertical lines. The characteristic
Nevertheless, because of the heterogeneity of the com- scale of heterogeneity d + ´ is significantly smaller than
position, the front structure differs from the wave struc- the front width Lf. The overall structure of the front
ture in homogeneous systems. The periodic nature is similar to the classical structure of transformation
of the heterogeneous composition does not allow the of homogeneous systems. The combustion front con-
frontal regime in the form of a mathematically rigor- sists of a zone of heating of the initial composition and
ous travelling wave. Nevertheless, as in usual combus- a comparatively narrow zone of the exothermal reac-
tion, the front  forgets about the initiation process and tion, which transforms the heated initial composition to
316 Krishenik, Merzhanov, and Shkadinskii
medium (ÃcÁ = 0.25 and à = 0.1). Hence, the instan-
taneous velocity of the front V¸, which is determined by
"
the displacement of the isotherm (T + T0)/2, is a peri-
odic function. In analyzing combustion of such systems,
it is convenient to use the period-averaged value as the
velocity of the quasihomogeneous front V . In studying
combustion of heterogeneous systems, it is important
to determine the characteristic size of cells at which the
homogeneous regime of transformation is still observed.
It was found by a numerical analysis that it is sufficient
to have approximately three effectively reacting layers
in the reaction zone for the front to have a quasihomo-
geneous structure. Hence, using the classical analysis of
stationary characteristics of a homogeneous combustion
wave, one can estimate the limiting size of the elements
of the heterogeneous system at which the quasihomoge-
neous combustion regime is still observed.
Thus, we can conclude that the front can be charac-
terized by a travelling wave (similar to [12]) if the com-
bustion front is analyzed with a comparatively rough
 resolution (which usually occurs in an experiment).
In a detailed analysis of combustion of a heterogeneous
medium (similar to [11]), however, the motion of the
visible boundary experiences fluctuations, which reflects
the  discrete structure of the heterogeneous front. Fig-
ure 2c shows the instantaneous velocity V¸ of the quasi-
homogeneous front (d = 0.3125). Forced fluctuations
Fig. 2. Dependence of the instantaneous combustion ve-
of velocity V¸ are related to the difference in thermo-
locity on time during propagation of a transverse com-
physical properties of the reactive and rarefied reactive
bustion wave in a multilayered system for d/´ = 0.5,
layers. Obviously, for ÃcÁ = Ã H" 1, the amplitude
ÃcÁ = 0.25, Ã = 0.1, Å‚ = 0.117, ² = 0.117, ¸0 = -5.67,
and d = 5 (a), 1.25 (b), and 0.3125 (c). of fluctuations decreases, and we have V¸ H" V . For
the case presented in Fig. 1, despite the quasihomoge-
neous structure of the front moving with a mean veloc-
high-temperature products. The heat flux from the nar- ity V = 0.062, its characteristics are periodic functions;
hence, the instantaneous velocity V¸ is also nonstation-
row high-temperature zone heats several reaction layers
ary.
whose individual temperature field is close to spatially
homogeneous. The temperature of combustion prod-
(2)
ucts is close to the equilibrium temperature Tad (here
(2)
T" = Tad ). The characteristics of the front structure
TRANSITIONAL REGIME
and its propagation velocity can be approximately de- OF COMBUSTION
termined via effective thermophysical parameters and
averaged macrokinetics. Figure 1b shows the heat- In the case of a correlated increase in the size of
release function F (¸, ·) at a fixed time Ä", which corre- reactive and inert particles, which retains a constant
sponds to a stationary space time distribution of tem- mean-mass concentration of the mixture, the quasi-
perature (marked by the asterisk in Fig. 1a). The source homogeneous regime transforms to another limiting
of heat release has a columnar structure, since the heat regime: relay-race regime. Figure 3 shows the charac-
release is concentrated in reacting plates. The structure teristics of exothermal transformation of a multilayered
of the envelope F (¸, ·) corresponds to heat release in a system at the transitional stage. This regime is formed
homogeneous medium. Its spatial displacement parallel if the scale of heterogeneity is comparable to the scale
to itself is caused by a correlated change in F (¸, ·) in of the reaction zone of the homogeneous combustion
each reacting layer (marked by arrows). The tempera- wave, i.e., (d + ´)/Lf H" Å‚. The space time distribution
ture field retains a stepwise character because of the dif- of temperature is shown in Fig. 3a. The quasihomoge-
ference in thermophysical characteristics in the layered neous character of front propagation is lost here, and
Nonstationary Regimes of Transformation of Multilayered Heterogeneous Systems 317
Fig. 3. Nonstationary characteristics of the transitional regime of combustion in a multilayered system for d/´ =
0.5, ÃcÁ = 0.25, Ã = 0.1, Å‚ = 0.117, ² = 0.153, ¸0 = -5.67, ¸" = ¸in = 0, d = 5, and Ä = 2565.39 (1),
2601.39 (2), 2612.92 (3), 2615.59 (4), 2617.46 (5), 2618.53 (6), 2619.30 (7), 2620.93 (8), 2621.64 (9), 2622.95 (10),
2705.32 (11), 2929.74 (12), 3029.62 (13), 3065.62 (14), 3074.54 (15), 3078.82 (16), 3080.69 (17), 3081.15 (18),
3082.27 (19), 3082.77 (20), 3083.48 (21), and 3084.97 (22); (a) space time distribution of temperature in the
combustion process; (b) detailed temperature dynamics of transformation and ignition of the neighboring reactive
layers; (c) heat-release function F (¸, Ä); (d) heat fluxes in the reactive layer.
the reactive layers burn one by one in the regime of ignition of the reactive layer corresponds to an increase
the relay-race mechanism of initiation. The remaining in the instantaneous combustion velocity. In the process
layers either have already burned or have a low tem- of the short-time period of transformation of the layer,
perature, and their transformation may be neglected. significant temperature gradients appear at the bound-
Propagation of the combustion wave is characterized ary of reactive and inert layers because of the difference
by relaxation fluctuations of all characteristics of the in thermophysical properties of the layers (Ã = 0.1).
front, including the velocity V¸ (Fig. 2a and b). The This leads to an increase in the velocity V¸ and forma-
character of fluctuations is determined by two factors: tion of a new extremum. Since the  detachment of
superadiabatic temperatures of transformation of reac- the maximum temperature from the equilibrium value
tive layers with increasing scale of heterogeneity and is not large in this case, the maximum of the velocity
heat transfer in a multilayered system. As the transfor- V¸ is lower than the extremum associated with another
mation quasihomogeneity is violated (see Fig. 2b), the factor. With further increase in the scale of heterogene-
temperature in the reaction zone (T") starts to  detach ity, the temperature in the reaction zone is significantly
(2)
higher than the equilibrium temperature (see Fig. 2a).
from the equilibrium temperature Tad , which leads to
The maximum value of the velocity V¸ caused by in-
an increase in the instantaneous velocity V¸ and the for-
tense exothermal transformation exceeds the maximum
mation of its local maximum. In Fig. 2b, the moment of
318 Krishenik, Merzhanov, and Shkadinskii
value of the velocity V¸ due to heat transfer. After reactive layer is determined as
a short time of exothermal transformation, there be-
¾i+1
gins a long stage of heat transfer between the reacted
¸
and the nearest nonreacted layers. This corresponds
Qw(¾, Ä) = (1 - ·) exp d¾,
1 + ²¸
to rather low values of V¸. As the cell size increases,
¾i
the maximum amplitude of fluctuations of the velocity
V¸ first increases (´ = 5 and d = 2.5), but then it de-
where i + 1 is the boundary between the combustible
creases with further increase in the cell size (´ = 10
and inert layers. Figure 3d shows the heat fluxes Qn
and d = 5). The character of fluctuations remains un-
and Qw versus time (the initial moment is the time
changed, and the maximum value of V¸ is reached at
of complete combustion of the previous reactive layer).
the moment of ignition of the reactive layer. The pe-
Though this regime is transitional, chemical heat re-
riod of oscillations reflects the periodicity of the multi-
lease at the stage of heating may be ignored. At the
layered system. The mean combustion velocity of the
time Ä" H" 377, the equality Qn = Qw holds, and the
heterogeneous medium V in passing to the relay-race
stage of inert heating may be assumed to be finished.
combustion regime is mainly determined by the time of
A heated reactive substance is formed by an external
inert heating of the next reactive layer and by the pe-
source at this stage. During inert heating, the transfor-
riod of induction of its self-heating. The dynamics of
mation depth of the substance changes insignificantly.
the stabilized process of heat transfer and heat release
At Ä > Ä", the main contribution to substance heating
F (¸, ·) in the front is illustrated by two periods of the
is made by chemical heat release, which starts to play
multilayered system (see Fig. 3b and c) corresponding
the determining role in the heat balance. The stage
to the reactive layers marked by one and two asterisks in
of ignition ends when the zero temperature gradient is
Fig. 3a. If the reference point Ä = 2120.0 is assumed to
reached at the  combustible inert layer boundary at
be the combustion time of the reactive layer preceding
the time Äz = 441. At later times, heat release due to
the layer marked by one asterisk (max · > 0.99), then
the chemical reaction exceeds heat removal to compar-
the curves F (¸, ·, ¾) and ¸(·, ¾) correspond to the rela-
atively cold layers of the substance. This is the reason
tive times Ä = 2565.39 3084.97. The stationary period
for a self-accelerating increase in the maximum temper-
of fluctuations is characterized by a comparatively long
ature. Since the layer is heated nonuniformly already
stage of heating of the medium. The next reactive plate
at the stage of ignition, the wave character of trans-
behaves in the regime of dynamic thermal explosion,
formation of the substance in the reactive layer is ab-
and the characteristic time of self-ignition is one order
sent. The maximum temperature in the reaction zone
smaller than the time of thermal relaxation. The tem-
is much higher than the equilibrium combustion tem-
perature of combustion products in the plate reaches
perature (¸" H" 0).
superadiabatic values. On the one hand, the energy lib-
Merzhanov [7] evaluated the delay of ignition of the
erated during combustion is spent on heating of the ini-
combustible layer in the relay-race regime of transfor-
tial heterogeneous medium. On the other hand, heat ex-
mation. It was assumed that exothermal heat release at
change between the layers occurs behind the front, and
the stage of heating may be neglected. This assumption
the equilibrium temperature of combustion products is
is also valid for the characteristic transitional regime. In
established. Superadiabatic heating significantly affects
heat-flux estimations, the temperatures of the previous
the instantaneous velocity V¸ and the mean combustion
and next plates were assumed to be constant in time and
velocity V . According to the data in Fig. 3b and c, three
equal to the temperatures of the burned (¸ad) and initial
stages of combustion of a layer can be identified: heat-
(¸0) plates, respectively. Results of a numerical analysis
ing through a layer of an inert interlayer, ignition, and
show that the heat flux through the inert layer during
rapid exothermal transformation. After an induction
the induction period is a rather complicated function of
period (dotted curves in Fig. 3a or curves 11 and 12
time. At the first (short-time) stage of heating, this is
in Fig. 3b), when the substance can be considered as
an increasing function, which is caused by the increase
an inert body, there starts a self-accelerating process of
in temperature in the reactive layer. Then it becomes a
exothermal transformation of the next reactive plate.
decreasing function qualitatively similar to [7]. Hence,
We analyze the changes in the heat flux toward
approaches used in analyzing flame propagation in the
the reactive layer as a function of time Qn = ¸¾ ¾=¾ ,
i
relay-race regime may be used to determine the charac-
where ¾i is the coordinate of the ith boundary between
teristics of the transitional regime.
the inert and combustible layers. The amount of heat
liberated as a result of chemical transformation in the
Nonstationary Regimes of Transformation of Multilayered Heterogeneous Systems 319
RELAY-RACE COMBUSTION REGIME bustible layer has burned. In this sense, the example
shown in Fig. 4 reflects the relay-race character of het-
In the relay-race combustion regime, the thermo-
erogeneous combustion [6, 7]. The time of front dis-
physical characteristics and sizes of the reactive and in-
placement through one period of the structure (d + ´)
ert layers are close to each other and (d + ´)/Lf > 1.
actually coincides with the time of initiation of the next
Results of a numerical analysis confirm the pres-
reactive layer, since the stage of combustion is negligibly
ence of (experimentally observed) roughly quasihomo-
small. In this connection, we note one special feature.
geneous and relay-race combustion regimes.
The presence of a  flash at the end of combustion of
Figure 4 shows the nonstationary space time pro-
the plate, which is caused by the excess of enthalpy in
files of temperature in a heterogeneous layered system
the gasless front of combustion and weak thermal con-
modeling a coarsely dispersed medium. In the sta-
ductivity of the inert interlayer may significantly affect
tionary combustion regime, the pattern of combustion
the processes of heat transfer between the layers. Thus,
for each  combustible inert pair (periods 1 4) is re-
in the case of an optically transparent inert interlayer
peated with a certain shift in time Ä1 [7] (in our ex-
(for instance, a gas medium), radiant heat transfer may
ample, Ä1 H" 5262). In calculating this combustion
turn to be a dominating factor in the course of heat
regime, we used the value of the characteristic temper-
transfer between the layers, which will change the char-
(1)
ature T" = Tad = T0 + Q/cR. acter of transformation of the heterogeneous medium.
Analyzing the profiles, we can conclude the follow- An analysis of this regime shows once again that
ing: the process of heat transfer ( token passing ) becomes
" Each combustible layer burns out in the frontal a determining factor of the propagation velocity, which
(1)
depends on the structure of the heterogeneous system
regime, and the temperature is higher than Tad .
and the thermophysical properties of the medium.
" Heat from the burned layer (in the nonstationary
regime) is spent on heating of the  fresh inert layer and
the already heated (previous) layer; the latter occurs MEAN VELOCITY
because of the  superadiabatic heating. OF THE COMBUSTION FRONT
" Heat transfer through the inert layer causes heat-
The presence of superadiabatic heating signifi-
ing of the next combustible layer and leads to ignition
cantly affects the instantaneous velocity V¸ and the
of the latter.
mean combustion velocity V . The reconstruction of
" When the initiated combustion front approaches
combustion regimes and the parametric region of their
the opposite surface of the combustible layer, a sud-
existence depend not only on the scale of heterogeneity
den increase in temperature may occur near the surface,
of the media but also on the physicochemical properties
which is the manifestation of the  enthalpy effect [13].
of the layers. It was found from a detailed numerical
We emphasize that the reaction in the combustible
analysis that, as the front propagates in the transitional
layer leads to its ignition only after the previous com-
regime, the nonreacted reactive layers experience the
thermal effect of not only the previous layer but also,
partly, the  far-away reactive layers. For Å‚ 1, the
ignition delay (Äin) in the relay-race regime (and in the
transitional regime close to the latter) is determined by
the value of the averaged integral heat flux from the
nearest reacted plate (p). We choose the time Ä = 0
Ü
as the end of transformation of the ith reactive layer;
¾" = (i - 1)(d + ´) + d. We introduce a dimensionless
mean flux
Äin ¾"+´
|p|d¾ dÄ
0 ¾"
p = ,
Ü
´Äin
where p = ¸ . If the inert interlayer in the heteroge-
neous system is a gas, we can use the approximation
" "
|p(¾)| H" (¸¾ - ¸¾ +´)/(¾i - ¾i-1). Since the tempera-
Fig. 4. Space time distribution of temperature in
ture of the reacted layer decreases and the temperature
a multilayered system in the relay-race combustion
regime for d = ´ = 100, ÃcÁ = 0.1, Ã = 1.0, Å‚ = 0.1534, of the next layer increases, this function decreases dur-
² = 0.118, ¸0 = -6.508, and ¸" = ¸in = -0.65.
ing initiation. Estimation of p(¾) is complicated by the
320 Krishenik, Merzhanov, and Shkadinskii
CONCLUSIONS
A detailed analysis of the structure of the combus-
tion front and dynamics of wave propagation in hetero-
geneous systems shows that the characteristics are much
more complicated than it followed from the macroscopic
 averaged wave analysis. The mechanism of front
transformation in the systems considered has a nonsta-
tionary discrete character and depends on both kinetic
and structural properties of the media.
In mixtures of identical composition, depending
on the scale of heterogeneity, either a  quasistation-
ary regime of flame propagation close to the classical
Fig. 5. Mean combustion velocity versus the
regime and typical of homogeneous media or a relay-
scale of heterogeneity of the medium for d/´ =
race ( nonstationary ) regime is observed. The dynam-
0.5, ÃcÁ = 1.0, Ã = 0.1, ¸0 = -5.67, Å‚ = 0.117,
ics of the latter depends on the characteristic scale of
² = 0.153, and ¸" = ¸in = 0.
heterogeneity. At an intermediate level of heterogeneity,
a maximum is found on the mean combustion velocity,
"
fact that the temperature ¸¾ at the boundary between
which is associated with the complicated character of
the combustible and inert layers at the time of com-
heat transfer between reactive particles. Obviously, de-
plete transformation of the layer depends not only on
spite the  model orderliness of the initial composition,
the chemicophysical parameters of the reactive layer it-
these properties are also inherent in real systems, since
self but also on the rate of heat exchange between the
the transition from one combustion regime to another is
layers. Therefore, to determine this value, one has to
caused by the mechanisms of heat transfer and ignition
solve a full nonstationary problem. Nevertheless, to es-
of reactive cells rather than with the orderliness of the
timate from above the characteristics of combustion in
structure.
the relay-race regime, one can use the approaches de-
In subsequent studies, it is necessary to consider
scribed in [7], which implied that the temperature of
the influence of radiant heat transfer in the process of
the reactive layer at the end of its transformation is
heat exchange between the layers and the stability of
¸ad, and the heat flux during the ignition-delay period
the flame front in the transition to limiting regimes of
is constant and equal to (¸ad - ¸0)/´.
transformation. These issues are of special interest in
We consider the velocity of the flame front V as a
analysis of multilayered systems.
function of the combustible layer width d (´ = 0.5d) in
This work was supported by the Russian Foun-
the course of transition from the homogeneous regime of
dation for Fundamental Research (Grant No. 99 03
transformation to the relay-race regime (Fig. 5). In the
32392a).
case of  narrow layers, the transformation of the sub-
stance occurs in the homogeneous regime. The velocity
in this parametric domain depends weakly on the scale
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