MODELING SHS IN POROUS SYSTEMS USING
CELLULAR AUTOMATA APPROACH
B.B.Khina and D.N.Loban
Physico-Technical Institute, National Academy of Sciences, Minsk, Belarus
A new stochastic model of the self-propagating high-temperature synthesis
(SHS), or the combustion synthesis, is developed, which in an explicit form takes into
account the porosity of reacting media. Both conductive heat transfer between solid
particles and radiative heat transfer across the pores are considered. The effect of the
stochastic factor on the SHS wave dynamic behavior is examined through including
into model both the intrinsic stochasticity of solid/solid reactions and random
distribution of the pores throughout the specimen. The model is based on the cellular
automata method. The dynamic behavior of SHS waves propagating through porous
systems is studied numerically, and the transition from the "quasi-homogeneous
regime" to the "relay-race mode" with increasing the overall porosity, which was
previously observed in experiments, is traced by computer simulation.
INTRODUCTION
Self-propagating high-temperature synthesis (SHS), also known as combustion
synthesis or solid-flame combustion, is a cost-effective method for producing highpurity refractory compounds and advanced ceramics, including functionally gradient
composite materials. Being ignited by an external heat source (typically, an electrical
heating coil) on one end, an exothermic reaction propagates like a combustion wave
through a charge mixture of solid reactants (e.g., Ti+C to give TiC or Ti+2B to
produce TiB2) yielding progressively a glowing final product in a short time. Due to
specific conditions in an SHS wave (steep temperature gradient, up to 106 K/cm, high
temperature, up to 4000 K, high heat release rate, and fast accomplishment of
interactions, 1 to 10 s), SHS products possess fine-grain structure and, very so often,
superior properties as compared with the same compounds produced by traditional
synthesis methods (1,2 and others). Since SHS typically proceeds in a relatively
narrow range of parameters, is difficult to control after ignition, and demonstrates
uncommon (from the point of view of Materials Science) structure-forming
mechanisms, which are difficult to examine in situ, mathematical modeling
traditionally plays an important part in the development of SHS-based technologies.
Inherent in SHS waves is dynamic behavior, e.g. oscillating and spin
combustion, which results in non-uniformity of the product structure and degree of
chemical conversion. Recent studies using microscopic high-speed video recording
revealed new dynamic regimes of SHS in porous systems: the so-called relay-race and
quasi-homogeneous patterns and local inhomogeneity of the combustion front
(scintillating regime) in highly porous systems (3-5). New patterns of SHS were also
observed in microgravity on the Russian orbital space station ”Mir” for highly porous
samples (6).
However, the existing models of SHS don't take into account in an explicit form
the effect of porosity on heat transfer in the reaction wave and its dynamic behavior.
Besides, the traditional deterministic approach (i.e., partial differential equations using
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the mean-field approximation of the system parameters) do not consider the stochastic
effects, which reveal themselves in two ways: (i) the intrinsic stochasticity of
heterogeneous reactions resulting from non-uniformity of the surface reactivity and
contact surface area of solid particles, which is important in the narrow "thermal
reaction" zone of an SHS wave, and (ii) random pore distribution affecting local heat
transfer. From the above facts it follows that these effects may be important,
especially in the conditions of unstable combustion. Previously, a stochastic model of
SHS allowing for the former factor was developed for a one-stage (7) and multi-stage
reaction in an SHS wave (8), which permitted us studying the onset of instabilities,
development of spin regime, and transition from steady-state to unstable combustion.
Thus, the objective of this research is to work out a new stochastic model of
SHS in porous systems taking into account local inhomogeneity of heat transfer
connected with random pore distribution, and to examine the SHS wave dynamic
behavior using the “cellular automata” approach. Since the width of the "thermal
reaction" zone is small, a limited amount of particles interact at the same time, and
hence the stochasticity of heterogeneous reactions substantially effects the onset of a
particular dynamic regime (7,8). The anisotropy of heat transfer due to the competition
of heat conduction via solid-state contacts and thermal radiation across pores will
cause a difference in the temperature of particles thereby introducing an additional
instability factor not considered in the existing models.
FORMULATION OF A MODEL
The model considers the following physical factors: porous structure of a
specimen, solid-state reactions, conductive heat transfer in the charge mixture via
solid-state contacts of the particles and radiative heat transfer across the pores. The
elementary particle size is implied to be small enough and the number of contacts of
unlike particles to be large, thus diffusion mass transfer is not considered.
The specimen structure is described as a two-dimensional matrix composed of
equal square cells. Two kinds of cells are considered: cells imitating solid particles
and "empty" cells imitating pores. For this situation, a unit cell can be interpreted as
containing a small portion of both reactants (e.g., Ti+C) in a stoichiometric ratio, the
cell composition being uniform throughout the specimen. Each reacting cell is
characterized by temperature, T, and state or degree of conversion, ;  = 0 for
unreacted and  = 1 for a reacted state. This corresponds to a one-stage chemical
reaction.
Heat exchange between solid cells is described by a discrete analog of the
equation of two-dimensional conductive heat transfer including heat release rate due
to the chemical reaction in a cell:
T/t = a2T + [Q/(c)] /t
(1)
Here Q is heat release of the reaction per unit volume and c is the specific heat, a is
thermal diffusivity, a = /(c), where  is thermal conductivity,  is density.
It is implied that the reaction in a cell follows the first-order kinetics, and the
reaction rate is described by the traditional Arrhenius law. A discrete analog of the
reaction rate equation describes the probability of a reaction in a given cell vs.
temperature:
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P(T) = k exp[E / (RT)]
(2)
where P is the reaction probability,  is a dimensionless time step, R is the gas
constant per mole, E is the activation energy, and k is the preexponential factor.
Thus, the stochastic model includes the reaction probability (or the reaction
threshold) in a cell instead of the unambiguously determined reaction rate, and the
state of an individual cell switches from 0 to 1 with a probability depending on
temperature instead of demonstrating a continuous change of  as in a deterministic
approach.
A specially developed computer code permits generating both chaotic and
ordered distribution of pores throughout a sample with a preassigned average pore size
and total porosity. Without losing generality of the model, the pores are assumed to be
of a rectangular shape, and complex-shaped pores are subdivided into several rightangled pores. To describe radiative heat transfer across a pore, a numerical procedure
is worked out using the integral heat flux balance method (9). The internal pore
surface is treated as an ideal black body. The elementary heat flux between two cells
(denoted by subscripts 1 and 2) having different temperatures, which belong to the
pore surface, is written as
dJr = B (T14  T24)[cos 1 cos 2 / (S2)] dA1 dA2
(3)
where Jr is the energy flux by thermal radiation, B is the Stephan-Bolzmann constant,
S is the distance between the cells, A1 and A2 are the elementary surface areas, 1 and
2 are solid angles at which these surfaces are viewed, T1 and T2 are the temperatures
of cells 1 and 2. The resultant heat flux across a pore is determined by integrating
expression (3) over the whole total pore surface.
RESULTS OF COMPUTER SIMULATION AND DISCUSSION
Computer simulation of the SHS wave propagation has been performed for a
two-dimensional sample with boundary conditions corresponding to a cylindrical shell
of an infinite length. The specimen is ignited on its left edge by the adiabatic
temperature, Tad = T0 + Q/(c), where T0 is the initial temperature, and the
combustion wave propagates from left to right. Figures 1 and 2 present unrollings of a
cylindrical shell with the temperature maps calculated at regular dimensionless time
intervals; the combustion front propagates from left to right.
The Novozhilov criterion for stability against two-dimensional disturbances KII
= 8.91/[1 + 3.1(1)], where  and  are the dimensionless combustion parameters:
=RTad/E, =cRTad2/(QE) is used to estimate the parametric domain of stable
combustion; at KII < 1 the stability loss occurs (10).
The evolution of temperature field (in grayscale, lighter shade denotes higher
temperature) shown in Fig.1 corresponds to a stable combustion regime ( = 0.11,  =
0.16, KII = 1.2) for a porous specimen with porosity =23%. The combustion wave
front is almost planar and propagates with a constant average velocity. However, the
presence of pores results in short-term local distortions of the SHS wave front and the
occurrence of small-sized hot spots whose temperature slightly exceeds the adiabatic
value. These hot spots appear in cells next to pores. This is due to local anisotropy of
heat removal from reacting cells since, according to simple numerical estimates, the
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heat flux across a pore constitutes about 0.10.2 of the flux through solid substance at
the same temperature gradient and maximum temperature in the SHS wave.
Fig. 1. Temperature maps (unrollings of a cylindrical shell) for stable
combustion in a sample with porosity  = 23% and dimensionless
combustion parameters  = 0.11,  = 0.16, KII = 1.2
As the combustion wave advances, the temperature of small-sized hot spots
rapidly levels with that of the surrounding cells, and the local distortions vanish away
so as to appear in another place. In the preheat zone of the SHS wave, i.e. ahead of the
combustion wave front, where the temperature is much below Tad, the pores contribute
additional "thermal resistance" to the heat transfer.
In the parametric domain of unstable combustion, KII <1, porousless specimens
demonstrate spin combustion regime when the exothermic reaction concentrates in a
hot focus with T>Tad following a spiral path over the surface of a cylindrical sample.
In classical deterministic models this hot spot continues its revolutions until the
complete consumption of reactants. Spin combustion was observed experimentally in
a many systems (1,2 and others) and it results in non-uniform product structure. In the
stochastic model (7), the correlation between the temperature and reaction rate in a
cell expressed in Eq.(2) brings about continuous generation of two-dimensional
perturbations in the thermal reaction zone of an SHS wave. This results in
spontaneous origin of spin regime from arbitrary initial conditions, and occasional
disintegration of the hot focus after several revolutions if a sufficiently strong
perturbation occurs; in the latter situation another focus emerges in a different place
(7). In a porous sample (Fig. 2), the spin combustion regime demonstrates specific
features not observed before in computer simulation. A pronounced hot focus with a
superadiabatic temperature originates close to a pore wall because of local anisotropy
of heat transfer. A developed hot focus may cover an area containing several pores.
After having traveled along the wave front for a certain distance, the hot focus may
disintegrate upon "colliding" with a larger pore or a group of pores, however, a new
focus appears on another site of the combustion wave front.
Thus, the porosity of a sample seems to acts as an additional factor of instability
which, irrespective of the combustion regime, contributes to the generation of
distortions on the initially planar combustion wave front, and facilitates the transition
to an unstable regime of combustion under appropriate conditions (at KII < 1).
To examine the effect of overall porosity of the transition from the quasi-
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homogeneous regime to the relay-race mode, the local and average combustion wave
velocities were calculated, following the approach used in experimental works (3,4).
Fig. 2. Temperature maps (unrollings of a cylindrical shell) for unstable
combustion in a sample with porosity  = 19% and dimensionless
combustion parameters  = 0.09,  = 0.12, KII = 0.82
In the mentioned works the location of the SHS wave was estimated by the
maximal gradient of brightness determined using high-speed video recording and
subsequent computer image analysis. We calculate the local position of the SHS wave
front (along the 0x axis) and the corresponding local combustion velocity, u(y,t), using
the field of the conversion degree, (x,y,t), where x is the longitudinal direction and y
is the circumferential coordinate, 0  y  2r, r being the radius of the cylindrical
shell:
u(y, t )  h  (x, y, t  t )  (x, y, t ) / t
(4)
x
where t is the time interval, h is the cell size. The average velocity of the SHS wave
propagation, Ua, is calculated as the amount of cells burned during a certain time
interval:
h   ( x, y, t  t )  ( x, y, t )
Ua 
y
x
2rt
(5)
The histograms in Fig. 3 show the distribution of instantaneous velocities in the
SHS wave front with increasing the specimen porosity, the other combustion
parameters being the same (here F denotes the frequency). It is seen that with
increasing the overall porosity, the most prevalent instantaneous velocity of the wave
front decreases while the difference between the average velocity, Ua, and this value
increases. According to experimental works (3,4), this refers to the transition from the
quasi-homogeneous regime to the relay-race mode of SHS wave propagation. A
decrease of the Ua value is connected with lowering effective thermal diffusivity, a, of
a specimen with a rise of porosity.
The developed model does not permit simulating SHS in samples with open
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porosity, and increasing the total porosity above 45% is impossible. However, the
obtained results support a general idea that the transition from the quasi-homogeneous
regime to the relay-race regime is connected with anisotropy of local heat transfer.
Fig. 3. Distribution frequencies of instantaneous velocities in the SHS wave
front for a sample with dimensionless parameters =0.07, =0.134
(KII = 0.99): (a)  = 25%, (b)  = 35%, (c)  = 45%
CONCLUSION
The developed stochastic model for simulating the dynamic behavior of SHS
wave in porous samples with closed porosity permits studying the novel regimes of
SHS, vis. the quasi-homogeneous and relay-race regimes. The developed approach
opens a prospective of linking the structure of the charge mixture, along with the
classical combustion parameters  and , with the regime of the SHS wave
propagation, which, in turn, affects the structure of the target product.
REFERENCES
1. Merzhanov A G: 'Scientific fundamentals, advances and prospects for the
development of solid-flame combustion processes'. Izvestiya Akademii Nauk 1997
(1) 8-32 (in Russian).
2. Munir Z A, Anselmi-Tamburini U: 'Self-propagating exothermic reactions: the
synthesis of high-temperature materials by combustion'. Materials Science Reports
1989 3 (7/8) 277-365.
3. Merzhanov A G, Mukasyan A S, Rogachev A S, Sychev A E, Hwang S, Varma A:
'Microstructure of the combustion front in heterogeneous gasless systems (on the
example of the combustion of the 5Ti+3Si system)'. Fizika Goreniya i Vzryva 1996
32 (6) 68-81 (in Russian; English translation: Combustion, Explosion and Shock
Waves (USA)).
417
4. Hwang S, Mukasyan A S, Varma A: 'Mechanisms of combustion wave propagation
in heterogeneous reaction systems'. Combustion and Flame 1998 115 (3) 354-63.
5. Rogachev A S, Mukasyan A S, Varma A: 'Microstructure of self-propagating
waves of exothermal reactions in heterogeneous media'. Doklady Akademii Nauk
1999 366 (6) 777-80 (in Russian).
6. Merzhanov A G, Rogachev A S, Sychev A E: 'Self-propagating high-temperature
synthesis: first experiments in outer space'. Doklady Akademii Nauk 1998 362 (2)
217-21 (in Russian).
7. Astapchik A S, Podvoiski E P, Chebotko I S, Khusid B M, Merzhanov A G, Khina
B B: 'Stochastic model for a wavelike exothermal reaction in condensed
heterogeneous systems'. Physical Review E 1993 47 (1) 319-26.
8. Khusid B M, Khina B B, Podvoiski E P, Chebotko I S: 'Stochastic modeling of
multistage combustion of condensed heterogeneous systems'. Combustion and
Flame 1994 99 (2) 371-8.
9. Siegel R, Howell J R: Thermal radiation heat transfer. New York, McGrow-Hill,
1972.
10. Borisova O A, Lidsky B V, Neigauz M P, Novozhilov B V: 'Stability of
combustion of gasless systems against two-dimensional instabilities'.
Khimicheskaya. Fizika 1986 5 (6) 822-30 (in Russian).
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