Combustion, Explosion, and Shock Waves, Vol. 37, No. 3, pp. 274 284, 2001
 Roughness of the Gasless Combustion Front
V. K. Smolyakov1 UDC 536.46
Translated from Fizika Goreniya i Vzryva, Vol. 37, No. 3, pp. 33 44, May June, 2001.
Original article submitted December 27, 1999; revision submitted July 4, 2000.
Concepts of local curvatures of the combustion front ( roughnesses ) caused by pen-
etration of the melt into the heating zone are developed. Based on the estimates of
characteristic times and scales of individual stages, different combustion modes and
their relation with the structural parameters of the initial mixture are revealed.
INTRODUCTION For RB > RA, we have
1/3
CA�B 1
R" = RB 1 + . (1)
CB�A 1 - m
Most existing theoretical concepts of combustion of
A unit volume of the powder medium contains N =
gasless heterogeneous systems are based on the postu-
3
(4ĄR"/3)-1 cells. One particle B in each cell corre-
late of thermal homogeneity of the powder medium [1].
sponds to nA particles A:
According to this postulate, each section of the combus-
CA�B RB 3
tion wave contains a statistically representative set of
nA = . (2)
CB�A RA
particles that have an identical temperature. The con-
The use of the cellular model within the framework of
cept of thermal homogeneity is based on the analysis of
the postulate of thermal homogeneity was beneficial for
the problem on combustion of a model layered system,
studying the kinetic [4] and structural [5] factors in com-
a  sandwich composed of layers of solid reagents that
bustion of systems of self-propagating high-temperature
form a solid product [2]. In accordance with [1, 2], the
synthesis (SHS). One of the consequences of the model
criterion of ignoring the temperature difference of the
of thermal homogeneity is the planar front of adiabatic
neighboring particles is the small ratio of the character-
combustion.
istic times of heat transfer (t1) and mass transfer (t2) in
Using the precision technique of studying combus-
the condensed phase, which is expressed by the Lewis
tion waves, Merzhanov et al. [6] and Rogachev et al. [7]
number Le = t1/t2 1. This circumstance allows one
found local curvatures of the combustion front,  rough-
to ignore mass transfer on scales comparable with the
nesses [6], which are observed in systems that form the
size of the characteristic zones of the combustion wave.
liquid phase. The local curvatures of the combustion
Based on the postulate of thermal homogeneity, the
front are most noticeable in combustion of highly porous
cellular model of the structure of the powder medium
specimens formed by large melting particles. Combus-
was developed and applied [3]. The essence of this
tion modes in which thermal heterogeneity of the fore
model is the representation of the mixture by a set of re-
part of the combustion front is considerably manifested
action cells. For a two-species mixture (A+B), each cell
were called  flickering [7]. In system reacting with-
consists of a particle of a larger reagent (for example, B)
out emergence of the liquid phase, the front  rough-
surrounded by small particles of the other reagent (A);
ness was not observed [7]. The existence of  flickering
the number of small particles is determined by a given
ratio of the reagents. The size of the initial cell R" regimes only in the presence of a liquid in the combus-
tion front indicates that the possible reason for the front
depends on the mass content of the reagents CA and
 roughness is related to the action of melts.
CB (CA + CB = 1), their densities �A and �B, particle
The objective of the present work is to determine
sizes RA and RB, and porosity m.
the effect of the liquid on the local curvatures of the
1
Tomsk Department of the Institute of Structural
combustion front and to evaluate the effect of this factor
Macrokinetics and Material Science Problems,
on combustion of heterogeneous mixtures reacting with
Russian Academy of Sciences, 634055 Tomsk;
[e-mail] participation of melts.
274 0010-5082/01/3703-0274 $25.00 � 2001 Plenum Publishing Corporation
 Roughness of the Gasless Combustion Front 275
PHYSICAL MODEL titanium particles over this length are H"15 and H"20,
OF THE PHENOMENON respectively. For a temperature difference of H"1600 K
in the zone considered, the temperature decreases by
more than 100 K in each cell. The temperature differ-
The rate of chemical interaction in a heteroge-
ence is even more significant in a cell formed by small
neous system is determined by the surface area of con-
fusible particles of Si and a more refractory particle
tacts of unlike particles. Its value in the initial mix-
of Ti. For the mixture 5Ti + 3Si with particle sizes
ture of reagents is small. For significantly different
of 2RA H" 2 � 10-4 m for titanium and 2RB H" 3 � 10-5 m
particle sizes, most particles of the finely dispersed
for silicon, the cell size is 2.8 � 10-4 m, and the length of
reagent A have no direct contacts with the particle of
the heated layer is H"10-3 m [12]. This length contains
the reagent B. Thus, e.g., for the mixture Ti + B for
H"4 cells, and the temperature difference in an individual
RB/RA = 10, the number of particles A (boron) in a
cell is more than 200 K.
stoichiometric reaction cell is H"500, as follows from (2),
Thus, in the heating zone, the neighboring reaction
and for a porosity m = 0.5, the relative size of the cell
cells and individual particles in one cell may have dif-
is R"/RA H" 14.5. For the mixture 5Ti + 3Si with the
ferent temperatures; the more so that the main thermal
ratio of titanium and silicon particles RA/RB = 7, the
resistance of the granular medium is concentrated at the
number of silicon particles in a cell is nA H" 170, and
points of particle contacts [13]. Melting of the fusible
the relative size of the cell is R"/RA H" 10.
reagent is followed by spreading of the melt and chemi-
The appearance of the liquid phase exerts an acti-
cal transformation. Since the liquid temperature differs
vating action on the chemical transformation rate. As a
from the temperature of finely dispersed refractory par-
result of wetting and spreading of the melt, the reaction
ticles, especially at the cell periphery, spreading of the
surface drastically increases, and the scale of hetero-
melt is accompanied by convective heat transfer. The
geneity decreases. Therefore, the temperature of emer-
part of the liquid spreading toward the initial mixture
gence of the liquid phase Tliq may be considered as the
penetrates into the less heated part of the cell and forms
natural  cut-off of the source of heat [8].
hot regions, where the distribution of reagents is more
In the case, which is most common in practice,
uniform than in the initial mixture. Local reaction sites
the particles of the fusible reagent B are greater than
may be formed in these regions. Penetration of the liq-
the particles of the refractory reagent A, and the melt
uid into the initial mixture leads to a local curvature of
spreads over capillaries formed by the substance A. Si-
the front boundary of melt spreading; the greater the
multaneously with spreading of the liquid reagent B,
ratio R"/x", the greater the curvature (Fig. 1). The
chemical interaction with particles A (capillary walls)
disorder in location of large fusible particles and cells
occurs in the cell. If the time of melt spreading tc in
in the mixture (stochasticity of the initial structure) is
the cell is small as compared to the time of the chemical
manifested in nonsimultaneous proceeding of these pro-
reaction at the melting temperature tr, then the liquid
cesses in the neighboring cells, which finally leads to
spreads faster than complete chemical interaction oc-
 flickering of the combustion wave.
curs. The amount of the product formed thereby (the
The pattern described refers to the case of melt-
solution of the reagent A in the melt of the reagent B or
ing of the larger reagent. If smaller particles melt first,
the solid product AB) may be insignificant despite the
as in the example given for the system 5Ti + 3Si, then
fact that exactly the chemical reaction over the wetting
spreading of the melt of the reagent A (silicon) over the
perimeter is one of the factors that determines spread-
particle B (titanium) occurs simultaneously with coag-
ing [9, 10]. Conditions of satisfaction of the inequality
ulation of the contacting liquid drops A. The curvature
tc < tr will be discussed below.
is formed due to the inflow of the silicon melt or the
The estimates of the heating zone width x" H" a/u
liquid solution of both reagents into the initial mixture
(a is the temperature diffusivity and u is the burning
(the melting points differ by H"300 K).
rate) and analysis of temperature profiles of the com-
Depending on the structural and thermophysical
bustion wave [11, 12] indicate that the representation of
characteristics of the mixture, the burning rate of in-
a heterogeneous mixture in the heating zone as a sta-
dividual cells and, hence, the macroscopic burning rate
tistical ensemble is very rough. Thus, according to [11],
may be controlled by the chemical reaction rate, the
in combustion of the mixture Ti + B with particle sizes
melt spreading rate (rate of increasing reaction surface),
of 2RB H" 4.5 � 10-5 m for titanium and 2RA H" 10-6 m
or the heating rate of the cell up to the temperature of
for boron, the length of the sector from the initial tem-
appearance of the melt. In the first case, where the slow
perature T0 H" 290 K to the temperature at which the
stage is chemical transformation, the combustion front
liquid appears Tliq H" 1900 K does not exceed 10-3 m.
propagates over a solid liquid suspension containing an
The numbers of reaction cells (2R" H" 7 � 10-5 m) and
276 Smolyakov
ESTIMATES AND HIERARCHY OF
CHARACTERISTIC TIMES AND SCALES
Depending on the parameters of melt inflow into
the cold mixture and heat-release and heat-transfer
rates at the interphase boundary, either heating of the
flowing liquid or cooling of the melt down to the crystal-
lization temperature is possible. The precise mathemat-
ical description of penetration of the melt into the heat-
ing zone should include consideration of the nonuniform
flow of the liquid over the sections of the medium com-
posed of refractory particles with simultaneous chemical
interaction at the  solid reagent melt interface. The
solution of this problem faces severe difficulties. Never-
theless, for the purpose of the present work (to propose
a model of the process and evaluate the action of this
factor on propagation of the combustion front), it is
sufficient to consider a simplified one-dimensional for-
mulation of the problem.
We determine the parameters of spreading of the
larger reagent. The structural characteristics of the
medium are assumed to be unchanged. We also as-
sume that the change in the capillary radius rc is small
as compared to its length, and the change in the mo-
mentum of the melt, caused by the mass exchange be-
tween the liquid and solid phases due to chemical in-
teraction, may be ignored as compared to the viscous
Fig. 1. Planar analog of the medium structure:
friction force. We also assume that, as the tempera-
1) boundary of melt spreading; 2) reaction cell;
ture increases, the decrease in viscosity of the liquid is
3) particles A; 4) particles B; 5) capillary; 6) pores;
compensated by the decrease in fluidity due to dissolu-
7) final product.
tion of the reagent A in the melt (if a liquid product is
formed) or by the narrowing of the capillary (if a solid
product is formed). The latter assumption allows us to
insignificant amount of the product. The curvature of
consider the viscosity of the melt � in the cell to be con-
the boundary separating the region of melt spreading
stant. Since the object of investigation is mixtures with
from the initial mixture leads to the curvature of the
essentially different components, the main mechanism
surface of the maximum heat-release rate whose dis-
of spreading in a cell is the volume spreading [14, 15].
placement toward the initial mixture is used to find the
In addition, we note that the motion parameters of a
burning rate. If the slow stage is spreading, then the
liquid with good wetting of the solid phase in granular
combustion front almost coincides with the spreading
media (usually simulated by beaded capillaries) differ
surface. In the case of large  roughness, the combus-
from the corresponding values in cylindrical capillaries
tion front may decompose into separate microfronts in
only by correction factors. Therefore, we confine our-
the cells. When the limiting factor is the stage of heat-
selves to consideration of the melt flow in a cylindrical
ing, propagation of the combustion front ensures a set
capillary.
of microexplosions in the cells. The mutual effect of
Under these assumptions, the motion of the liquid
the processes in individual cells is determined by their
along the capillary from an infinite source is described
thermal relationship.
by the equation [14]
In the present paper, we consider two cases: 1)
d dl "p 8�l dl
curvatures are small (R"/x" < 1) (propagation of the
l = - ,
2
dt dt � �rc dt
combustion front is controlled by the chemical reaction
or melt spreading); 2) curvatures are large (R"/x" 1) where t is the time, l is the length of the path covered by
(the slowest stage is the heating). In the first case, with the liquid, "p = 2�/rc is the capillary pressure, � is the
the limiting transition R"/x" 0, the results should be surface tension, and � is the density of the liquid (it is
in agreement with the model of thermal homogeneity. further assumed that the densities of all components of
 Roughness of the Gasless Combustion Front 277
the heterogeneous medium are identical). The solution The rate of heat removal to the solid phase (capil-
of this equation has the form lary walls) is
2
1/2 1/2
W1 H" 2ą(T1 - T2)/rc. (7)
rc "p t
l = t - t" 1 - exp - ,
4� t"
The heat-transfer coefficient between the liquid and
2 granular medium (ą) is characterized by the Nusselt
where t" = �rc /8� is the time of establishing of the
number Nu = 2ąRA/, where  is the thermal conduc-
quasi-stationary regime. For � H" 103 kg/m3, rc H"
tivity. The values of Nu are found from the formula [13]:
10-5 10-6 m, and � H" 10-3 Pa� sec, this time is
10-5 10-7 sec. The corresponding spreading length
Nu = 2 + 0.106RePr1/3.
is 3 � 10-6 1.4 � 10-4 m, i.e., for thin capillaries, the
Here Re = 2v�RA/� is the Reynolds number, Pr =
length of establishing of the quasi-stationary regime is
�c/ is the Prandtl number, and c is the heat capacity.
commensurable with their diameter [14]. Therefore, for
From analysis of possible mechanisms of volume spread-
rc H" RA H" 10-5 10-6 m and R"/RA > 10, we may use
ing of the liquid in pores, it is known that Re < 10 [15].
the quasi-stationary equation
Since for metal melts we have Pr1/3 H" 1, the Nusselt
2
dl rc "p
number may be assumed to be constant: Nu H" 2. Note
= v = (3)
dt 8� l
that the stabilized heat removal in a laminar flow in a
tube is characterized by the value Nu H" 3.66 [16].
and the corresponding time tc and path length l:
Because of the exponential dependence
2
4l2� rc "ptc
of the diffusion coefficient on temperature
tc = , l2 = . (4)
2
rc "p 4�
D(T ) = D0 exp(-E/RT ), where D0 is the preex-
ponent, E is the activation energy, and R is the gas
For large capillary diameters, the values of tc, l, and
constant, the decrease in temperature reduces the heat
flow velocity v should be estimated using the general
income to a greater extent than the heat removal.
solution or by calculating the averaged values, reducing
Therefore, assuming that rc H" RA, we write the
the unsteady flow to an effective steady flow [15]. Note
necessary condition for the formation of the liquid
that the time of the capillary flow tc is greater than the
product as
spreading times predicted by other mechanisms, i.e.,
the spreading velocity determined by Eq. (3) is mini- D1(Tliq)
Q� 2ą(Tliq - T2).
mum [14, 15].
rc
The necessary condition for melt flow toward the
This relation may be represented as the ratio of the
cold mixture is sustaining of the liquid temperature
time of chemical interaction at the melting point of the
equal to or higher than its melting point. The heat-
reagent B
release reaction rate at the  liquid solid substance in-
2
tr H" rc /D1(Tliq) (8)
terface may be represented as
and the characteristic time of heat exchange between
d�
W = Q� , (5)
the phases
dt
tą H" Q�rc/2ą(Tliq - T2).
where Q is the thermal effect and d�/dt = k(T )f(�)
is the reaction rate [k(T ), f(�), and � are, rate con- This ratio has the form of the inequality
stant, kinetic law, and depth of transformation, re-
tr tą. (9)
spectively]. In heterogeneous systems with a scale of
The physical meaning of condition (9) is that the heat
heterogeneity greater than the molecular-kinetic scale
income at the interface of the solid and liquid phases due
(H"10-8 10-9 m), the slowest stage of chemical transfor-
to chemical interaction is greater than the heat removal
mation is usually the transport of reagents. Therefore,
for heating of the solid phase. In this case, fluidity of
we assume the reaction rate to be equal to the diffusion
the melt is retained.
mass-transfer velocity
In the heating zone, prior to melting of the
d� D(T )
reagent B, the temperature distribution in a steady
H" (6)
2
dt rd
combustion wave moving with a macroscopic velocity u
is described by the expression
(rd is the characteristic scale of diffusion). In the case
u
of formation of liquid (rd H" rc) and solid (rd H" RA)
T2 = (Tliq - T0) exp x + T0, -" x 0. (10)
a
products, we understand T and D(T ) as the tempera-
ture Ti and the diffusion coefficient Di(Ti) in the melt In (10), the origin (x = 0) coincides with the surface
(i = 1) or in the solid product (i = 2). of the fusible particle (beginning of the capillary); T0 is
278 Smolyakov
the initial temperature of the mixture. Using Eq. (10), physical meaning: the smaller the burning rate, the
we represent the characteristic time of heat exchange as greater the depth of heating of the substance and the
greater the spreading length.
2
rc Q�
tą H" .
For spreading at a certain length l" in the cases
Nu(Tliq - T0)[1 - exp(x/x")]
of formation of liquid and solid products, the following
Its minimum value is
inequalities should be valid:
2
Q�rc
a t0 aTliq�
t0 H" . (11) ą
ą
u < ln 1 - . (14)
, u <
Nu(Tliq - T0)
l" tr l"(Tliq - T0)
If t0 > tr, then the inflow of the melt into the initial
ą
If conditions (14) are violated, the melt cannot spread
mixture is not limited by heat transfer.
at a depth greater than l" because of the liquid crystal-
In the case t0 tr, the spreading length lą is lim-
ą
lization in the capillary.
ited by heat exchange between the liquid and the capil-
The existence of three characteristic times [those
lary walls, and it may be either greater or smaller than
of heat transfer (tą), spreading (tc), and chemical reac-
the capillary length in the cell. We find the coordi-
tion (tr)] implies possible relations between them.
nate corresponding to the limit of melt spreading. The
For slow heat transfer, the spreading length in the
"
temperature of the solid phase (capillary walls) T2 , for
capillary is not limited by heat removal to the cold walls
which we have W = W1 and T1 = Tliq, is determined
[conditions (9) and (14) are satisfied] and equals the
from the equation
maximum value determined by the size of the stoichio-
"
d� 2ą(Tliq - T2 ) metric cell: l" = l0 = R" - RB. The time of spreading
Q� = ,
is found from Eq. (4). The size of  roughnesses is
dt rc
l0/x" < 1. In this case, two variants are possible:
and the coordinate is found from Eq. (10). In the case
of formation of a liquid product, we have
1) tą tr tc (the limiting stage is the chemical
reaction);
(Tliq - T0)t0
" ą
T2 H" Tliq 1 - ,
Tliqtr
2) tą tc tr (the limiting stage is the capillary
(12)
spreading).
a t0
ą
lą H" - ln 1 - ,
u tr Note that the last parts of the chain of these inequal-
ities were studied by Nekrasov et al. [17] within the
where tr and t0 are defined by Eqs. (8) and (11), re-
ą
framework of the model of thermal homogeneity. In
spectively. Since we used the minimum estimate of tą,
the present work, in contrast to [17], the assumption of
formulas (12) yield the upper estimates.
thermal homogeneity is not used. From the comparison
If a solid product is formed as a result of the reac-
" of the characteristic times tc and tr, it follows that, if a
tion, the equation for the temperature T2 has the form
liquid product is formed and
" "
D2(T2 ) 2ą(Tliq - T2 )
3
Q� = .
rc > 2(R" - RB)2�D(Tliq)/�, (15)
2
RA rc
the combustion is mainly determined by the chemical
Its solution in the linear approximation is
interaction rate, and spreading has practically no effect
t0 (Tliq - T0)/trTliq
" ą
on the process. In a granular medium, the capillary ra-
T2 H" Tliq 1 - H" Tliq(1 - �),
1 + t0 (Tliq - T0)/�Tliqtr
ą
dius is expressed in terms of porosity and specific surface
2
of the particles S0 as follows [18]:
where � = RTliq/E < 1, tr = D2(Tliq)/RA, and t0 is
ą
determined by Eq. (11). Substituting the solution ob-
rc H" 2m/S0 H" 2mRA/3(1 - m).
tained into Eq. (10), we find the estimate of the spread-
Condition (15) takes the form
ing length in the case of the solid-phase reaction:
3
RA > 27D(Tliq)(R" - RB)2�(1 - m)3/4�m3. (16)
a Tliq� a Tliq�
lą H" - ln 1 - H" .
u Tliq - T0 u Tliq - T0
In the case of the opposite sign of the inequality, prop-
agation of the combustion zone is controlled by the
The time of melt spreading to the length lą is found
liquid-flow velocity in pores. The main difference of
by the formula
manifestation of the limiting action of stages in com-
2
tc H" 2�lą/rc�. (13)
bustion is the temperature sensitivity [17].
From these estimates of the spreading scale, there fol- For tr < t0 < tc, the scale of spreading is lc H"
ą
lows that the product ląu is constant, which has a clear l0 t0 /tc. This is the case of a slow flow. The spreading
ą
 Roughness of the Gasless Combustion Front 279
Fig. 2. Schematic representation of the times of separate stages versus the relative size of the fusible
particle for t0 tr (a) and t0 < tr (b).
ą ą
time is found from formula (13) with the substitution layer to another. This regime of combustion was called
lą = lc. the relay-race [23]. Its main difference from the con-
The regions of different flow regimes for t0 > tr ventional regime of combustion is alternation of flashes
ą
are shown in Fig. 2a (� = RB/RA). The kinetic and and depressions. Fast chemical interaction in a sepa-
capillary regimes correspond to the regions �1 < � < �2 rate layer occurs during flashes, and comparatively slow
and �2 < � < �3, respectively. The value of �2 is found heating of the substance in the layer takes place during
by comparing the times tc (4) and tr (8) and value of depression periods [7, 23]. Within the framework of the
�3 is found by comparing the times tc (4) and t0 (11). model of [20], it was shown that the burning rate in the
ą
The coordinate �1 is shown schematically. It limits the limiting case is independent of the kinetic parameters
region � < �1, where there is no volume spreading, and and is determined only by the thermophysical charac-
the scale of diffusion is determined by the size of both teristics of the heterogeneous medium.
reagents. The value of �1 is close to unity. Complete We analyze a large-scale  roughness (l0/x" 1)
spreading is observed in the region �1 < � < �3, and for the case close to the relay-race regime, on the basis of
the spreading length is lc in the region � > �3. For the analogy between the relay-race burning and subse-
t0 < tr (Fig. 2b), spreading toward the initial mixture quent ignition of separate layers, which was revealed by
ą
is limited by heat removal and is possible only in the Merzhanov [20]. Since in the situation considered, there
region �1 < � < �2. is no special stratification of the heterogeneous medium
As the cell size increases, conditions (9) and (14) into individual layers, as in the experiments of [23], it
are violated, and spreading toward the initial mixture is is necessary to determine the characteristic size of the
limited. The scale of spreading is determined by lą or lc, layers that burn down during a flash. We consider the
and the time of spreading is calculated by formula (13). process of reaction in an individual cell after the pre-
The reaction in an individual cell acquires an oscillatory vious cell burned down. During a certain time theat,
character caused by periodicity of spreading stages, heat the cell is heated, the component B located in the mid-
release from the chemical reaction, and heating. dle of the cell melts, and the melt spreads. The melt
With further increase in the cell size (fusible par- spreads to all sides, except for the front direction, to
ticles and porosity), the size of  roughnesses further the maximum depth l0 and toward the initial mixture,
increases. The reaction rate is significantly affected by to a distance smaller than l0. After spreading, chemi-
the heating time of the cell and the melting time of cal transformation occurs, the substance is additionally
the particle B in an individual cell. For example, in heated, and the melt spreads toward the initial mixture
calculations of combustion of a chain of reaction cells, up to the cell boundaries. The cell size, which is ev-
taking into account melting of the fusible reagent [19], idenced by the above examples, corresponds, in order
an oscillatory dependence of the front coordinate on of magnitude, to the size of the large fusible reagent.
time was obtained (see [19, Fig. 1]). An even more It follows from here that the main delay is observed at
limiting situation was considered in [20 22], where it the first stage during which the main part of the cell is
was assumed that the limiting stage of propagation of heated and reacts. Thus, the cell size should be taken
the reaction front may be the heat transfer from one as the characteristic scale. The total time of burning of
280 Smolyakov
one cell is small-scale  roughness, we use simplified phenomeno-
logical concepts of heat transfer in a heterogeneous
tŁ H" theat + tr + tc, (17)
system, taking an identical temperature for the entire
where
heterogeneous medium, and distinguish the large-scale
(l/x" 1) and small-scale (l/x" < 1)  roughnesses.
theat H" c"�V/ą"S (18)
Small-Scale  Roughness of the Combustion
is the heating time, V is the cell volume, ą" is a co-
Front. This is the case of a weak deviation from the
efficient that characterizes heat exchange between the
model of thermal homogeneity. It corresponds to a
burned sections of the specimen with a capillary-porous
large, though finite, number of particles and reaction
structure and granular structure in the heated cell,
cells in the combustion wave. For the one-dimensional
S is the surface through which heating is performed,
approximation considered, we have to find the velocity
c" = c + CBQm�(T - Tliq) is the heat capacity of the
for a planar stationary combustion front, corresponding
medium [c is a constant, Qm is the melting heat of the
to the macroscopic velocity of a nonplanar front whose
reagent B, and �(T - Tliq) is the delta function]. The
curvature is caused by convective heat transfer in the
heat-transfer coefficient may be estimated by the for-
cells. The macroscopic burning rate is understood here
mula
as the ratio of the representative length of a powder
ą" H" Nu"/2RA H" "/RA, (19) medium including the statistically credible set of struc-
tural elements to the time of combustion of the medium.
where " is the thermal conductivity of the gas. The
Determination of the velocity of the planar front, which
quantity theat, in accordance with the choice of the heat
is equivalent to the velocity of the curved front, closes
capacity of the medium, includes the melting time of the
the scheme for obtaining interrelated estimates of the
particle B.
small-scale  roughness in the one-dimensional approx-
imation.
We assume that the change in the burning rate, as
ESTIMATES OF THE BURNING RATE
compared to the value obtained in the model of thermal
homogeneity, is small and is related to the increase in
As a result of spreading of the melt formed by a
heat transfer toward the initial mixture by the flowing
large fusible particle, the scale of heterogeneity (mass-
melt. We introduce the effective thermal conductivity
transfer length for the chemical reaction) decreases dra-
" =  + ". The correction " is chosen as the con-
matically in the finely dispersed component. It is
vective heat transfer multiplied by the measure of the
about R" - RB prior to spreading; after spreading, the
 roughness :
characteristic scale of the chemical reaction is H"RA.
For instance, for systems  transitional metal carbon
" = c�vl"(l"/x"). (20)
(boron), which are extensively used in practice, we
In a coordinate system moving from right to left
have RA H" 10-7 10-6 m, and the size of metal par-
with the burning velocity u, the heat-conduction equa-
ticles may be H"10-3 m. The effect of refinement of
tion has the form
the solid component in contact interaction with the liq-
dT d dT d�
uid component should also be noted [24]. In systems
c"�u = " + Q�u . (21)
dx dx dx dx
with both reagents melting, the decrease in the scale
The boundary conditions are
of heterogeneity is favored by instability of the inter-
phase boundary [25]. Thus, spreading of the melt de-
x = -": T = T0, � = 0;
creases the scale of heterogeneity existing in the initial
mixture. However, the  cost of this structural trans-
x = ": dT/dx = 0, � = 1.
formation is the appearance of local curvatures of the
Kinetic Regime of Combustion. In the case of the
leading part of the front. In other words, the hetero-
limiting action of the chemical reaction, using Eqs. (6)
geneity of the mixture, as its qualitative characteristic,
and (21), taking into account Eqs. (3) and (20), and ig-
does not disappear during spreading but is transformed
noring the dependence of viscosity of the melt on tem-
into the  roughness.
perature (� = const), we obtain an estimate of the burn-
In the general case, to describe heat propagation
ing rate for a narrow reaction zone [16]:
in a heterogeneous medium with the scale of inhomo-
geneities (cell and particle sizes) comparable with the
u2 H" u2(1 + rc�l"u/4a2�). (22)
0
size of the examined domain, one has to consider heat-
Here
conduction equations for each component of the mix-
2 2
ture. To evaluate the burning rate in the case of a u0 H" (2RTf D(Tf )/QE�rd)1/2
 Roughness of the Gasless Combustion Front 281
is the burning rate in the approximation of thermal ho- u H" u"(1 + u"rc�l"/8a2�) (29)
mogeneity and
to be valid, inequalities (14) and (25) should be satisfied
Tf = T0 + (Q - CBQm)/c (23) for l" = l0.
Note that similar results for small deviations from
is the burning temperature. Since we seek the correction
the model of thermal homogeneity may be obtained if
to the burning rate u0 [i.e., the second term in brack-
the thermal conductivity l" is replaced by the effective
ets in Eq. (22) should be smaller than the first term],
heat flux
expression (22) may be represented as
dT l"
q H"  + vc�Tliq .
u H" u0(1 + rc�l"u0/8a2�). (24)
dx x"
It should be noted that, in the general case, the burning
The corresponding corrections [second terms in (24) and
rate u0 may depend on the uniformity of spreading of
(29)] differ by a factor of Tliq/(Tf - T0).
the melt in the cell, i.e., on structural factors  particle
The postulate on the increase in heat transfer,
size and porosity [5].
which was used to obtain the estimates, corresponds
For estimate (24) to be correct, the following in-
to the general notions on propagation of curved fronts
equality should be satisfied:
with the heat- and mass-transfer scale smaller than the
combustion-wave length [8, 16].
u < 4�a2/rc�l". (25)
Large-Scale Manifestation of Thermal Het-
Violation of this inequality a posteriori indicates that
erogeneity of the Mixture. We evaluate the burn-
the initial assumption of the small-scale  roughness in
ing rate of the cell in the case considered, taking into
invalid for prescribed values of the parameters. The
account the small corrections to the chemical reaction
spreading condition (14), i.e., the choice of l" in the
and spreading times. Using the expressions for the cor-
formulas obtained, has also to be verified. If inequali-
responding times, from (17), we obtain
ties (14) are valid for l" = l0 and u(l0), then the spread-
2
c"�V ą"S rd 2(R" - RB)2�
ing proceeds over the entire cell. Simultaneous satis-
tŁ H" 1 + + ,
ą"S c"�V D(Tliq) rc�
faction of inequalities (14) and (25) for l" = l0 allows
us to consider the  roughness as small-scale and use
where the reaction temperature is chosen as the melting
formula (24) to estimate the burning rate. If one of
point at which the time tr is maximum. The burning
conditions (14) and (25) is not satisfied, a combustion
rate of one reaction cell of size 2R" is
regime with a larger  roughness is formed.
2R"ą"S
u H"
Capillary Regime. To evaluate the burning rate in
c"�V
the case, where the limiting stage is the increase in the
2
interphase reaction surface (spreading), we consider a
ą"S rd 2(R" - RB)2� -1
� 1 + + . (30)
system of equations consisting of the heat-conduction
c"�V D(Tliq) rc�
equation (21) and the equation of capillary spread-
In the limiting case, where the reaction and spreading
ing (3). We define the depth of transformation as the
times are negligibly small as compared to the heating
ratio of the path covered by the liquid in the capillary
time, relation (30) corresponds to the formula obtained
and the maximum possible value of the spreading length
in [22]. Solution (30), due to the assumptions adopted,
(R"-RB), which follows from the ratio of reagents in the
is valid for (tc + tr) < theat.
initial mixture. Thus, the role of the kinetic equation
Owing to the sequence of stages (heating from the
in the case considered belongs to the following equation
neighboring cell, and then heat release due to the chem-
derived from Eq. (3):
"
ical reaction), the burning temperature of the cell Tf
d� rc�
is greater than the adiabatic value (23). The maxi-
u = . (26)
2
dx 4�l0�
mum excess is the value of heating from the initial tem-
perature to the melting point of the particle B, i.e.,
Integrating approximately system (21), (26), we obtain
"
Tf H" (Tliq - T0) + Tf . Preliminary heating of the cell
u2 H" u2(1 + rc�l"u/4�a2), (27)
"
is important for the possibility of melt spreading. In
cases close to the relay-race regime, the temperature is
where u" is the burning rate in the approximation of
roughly identical over the entire cell (T H" Tliq), and
thermal homogeneity:
melt spreading is not limited by conditions of the type
2
u" H" (3rc�(Tf - T0)/2�Q�l")1/2. (28)
of (9) and (14).
For formula (27) and the corresponding estimate of the In the case of a large-scale  roughness, the notion
burning rate of the burning rate has to be clarified. For a perfectly
282 Smolyakov
mixed mixture with spherical particles of components of
a monofractional composition, local inhomogeneities of
the leading part of the combustion front have the form
ordered in a certain manner and repeated in the time
interval H"tŁ. In actual gasless mixtures, there are al-
ways some deviations in the shape and size of particles
and homogeneity of mixing, which leads to stochastic
manifestation of the effect of  roughness ( flickering
of the combustion wave) and to a certain averaged burn-
ing rate.
Let f(R") be a function characterizing the size dis-
-
tribution of cells from the smallest (R" ) to the great-
+
est (R" ) values. By definition,
+
R"
f(R") dR" = 1. Fig. 3. Relative  roughness as a function of di-
mensionless dispersion of the fusible component:
-
R"
segments 1 and 2 are plotted by formulas (31)
and (32), respectively; the dashed segment conven-
Therefore, the averaged macroscopic burning rate is
tionally shows the transitional region.
R+
"
u H" u(R")f(R") dR".
where � = [CB(1 - m)]-1/3 and �1 = rc�RAu0/8a2�.
-
For the above values and a H" 10-5 m2/sec, u0 H"
R"
10-2 m/sec, and rc H" RA, we have � H" 1.5 and
It should be noted that, in the case of a wide distribu-
�1 H" 10-2, i.e., for the chosen values, estimate (31)
tion function, the values of u(R") may be determined
is valid for 1 < � < 102.
by different regimes of reaction.
In the case of a large-scale  roughness, for S/V H"
1/R" and ą" determined from Eq. (19), we obtain the
estimate
DISCUSSION
l0 2"(� - 1)�
H"
x" 
Based on the estimates obtained, we may trace the
effect of the basic parameters varying in an actual ex-
2 2
"a rc 2RA(� - 1)2��2 -1
periment on the front  roughness.
� 1 + + , (32)
2
RA�� D(Tliq) rc�
From a comparison of the time of capillary spread-
ing (4) over the entire cell (l = l0) with the time of the
which is valid within the interval 103 < � < 106.
chemical reaction of the formation of the liquid product
The main parameters determining the possibility of
(rd H" rc) at the melting point (8), it follows that the
appearance of  roughnesses (necessary conditions), as
capillary regime of combustion occurs under the condi- it follows from relations (31) and (32), are the porosity
tion
and the relative size of the fusible reagent. An increase
-2
in any of these parameters leads to an increase in the
2mRA 3 1 �
2
RB > - 1 .
spreading length necessary for mixing of the reagents
3(1 - m) CB(1 - m) 2D(Tliq)�
[the size of the reaction cell (1)] and also in the size of
For typical values of the parameters � H" 1 N/m,
 roughnesses. This conclusion is in agreement with the
� H" 10-3 Pa � sec, D(Tliq) H" 10-8 m2/sec, m H" 0.5,
known experimental data. Based on relations (31) and
and CB H" 0.7, the size of the fusible reagent should
(32), Fig. 3 shows the dependence of the  roughness
be greater than the size of the refractory reagent by
on the relative size of the particles B, similar to that
more than two orders. For example, for RA H" 10-6,
obtained in [6]. Using Eqs. (31) and (32), one can plot
we should have RB > 10-4 m. This circumstance
the  roughness versus the porosity. In this case, one
makes the possibility of the small-scale  roughness in
should take into account the influence of the ratio be-
the regime of capillary spreading little probable. In the
tween the melt volume and the volume of pores formed
kinetic regime, the relative size of  roughnesses is
by the refractory reagent on the depth of transforma-
tion, temperature, and effective kinetics of interaction,
l0 RAu0(� - 1)
H" �[1 + (� - 1)�1�], (31)
i.e., take into account the porosity [5].
x" a
 Roughness of the Gasless Combustion Front 283
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