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182D DISCRETE MODEL OF THE MULTICOMPONENT SHS PROCESS V. G. ProkofevflA
aTomsk State University, Tomsk, 634050 Russia bTomsk Scientific Centre, Tomsk, 634021 Russia e-mail: pvg@ftf.tsu.ru
DOI: 10.24411/9999-0014A-2019-10134
Theoretical study on formation of the macrostructure of products in the wave of heterogeneous combustion of gas-free systems is an important part of the targeted combustion synthesis of materials. Combustion synthesis of inorganic materials or SHS includes heterogeneous exothermic reactions. However, most theoretical concepts of high-temperature synthesis are based on homogeneous approximation.
In [1], an analytical review of modern heterogeneous models of combustion of SHS systems was reported. The authors compare the simulation results with known experimental data. A two-dimensional model of the combustion of a heterogeneous "fuel-inert component" system, represented as a set of square cells of the same size with a uniform and stochastic distribution of fuel in cells. was proposed in [2]. The reaction rate was described by a step function of temperature. The thermophysical properties of all cells were considered to be constant. The heat transfer between the cells was determined according to Newton's law, which implied the presence of an air gap at the cell boundaries. The model of the stochastic spatial structure was developed in [3], where the concentration and thermal limits of the propagation of the combustion wave were investigated. The significant role of fluctuations in the spatial distribution of reagents was revealed for the limit of propagation of the combustion wave, and the behavior of the system under critical conditions was explained based on the percolation theory. The effect of cell-particle sizes on the transition of combustion from the quasi-homogeneous to heterogeneous relay mode was considered for the "fuel-inert component" system in [4]. The microstructure of products was experimentally studied during the transition from the stationary (quasi-homogeneous) to scintillation (relay) combustion mode with an increase in the particle size of the Ni-Al mixture [5].
In multicomponent SHS systems, several thermally-coupled highly and slightly exothermic reactions that differ in both kinetics and thermal effects can occur simultaneously. A.G. Merzhanov proposed to call the first type reactions the donor, and the second one the acceptor. This method of forming a mixture of reactive components in the form of a layered composition in order to synthesize material using a slightly exothermic or endothermic mixture is called a "chemical oven".
In the paper, a two-dimensional discrete model of combustion of a multicomponent gas-free mixture consisting of two types of reaction cells, donor (DC) and acceptor (AC), such as Ni + Al and Ti + Al is proposed and numerically investigated. Granular mixtures, particles of one reagent coated with a layer of another one, can be analogous to cells. The ratio of DC and AC volumes was constant and assumed to be 1:1 in all calculations, which was related to the chosen spatial distribution of cells in the model. All cells have a square shape and one size, determining the heterogeneous system scale that is one of the main parameters of the problem. The adjoint boundary conditions of heat transfer are set at the boundaries between the cells. The rate of exothermic reaction is determined by the Arrhenius temperature dependence. Both regular system elements with a uniform distribution of cells and irregular ones with a random distribution of system elements are considered (Fig. 1).
368
V. G. Prokof'ev
ÏSHS2019
Moscow, Russia
Y
«mi RESSSSss
w
4
Y
w
4
Fig. 1 Scheme of the sample with (a) regular and (b) irregular volume generating the cells.
The mathematical model of the combustion of a multicomponent gas-free mixture includes the dimensionless heat conduction equation and the equation of chemical kinetics:
ee ex
e f ee" + S e r
a, A,
54 V 54, i,u ■ ew 1, V
+
W (e, n)
f = r, wy (e, n)
W (e, n) = exp M
° je
Are
(1) (2) (3)
All cells have a square shape with a side d = ^ - 4i-1 or d = % - where d is the characteristic cell size which determines the heterogeneous scale of the system. Scale variables are determined by the composition of the DC donor cell. The known stoichiometric Ni-Al SHS mixture was chosen as the donor mixture. Parameters with subscripts (i, j) are determined according to the type of the cell as follows: S, = 1, A, = 1, Q, = 1, r, = 1, and a, = 1 for DC cells and S, = Sa, A, = Aa, Q, = Qa, Tj = ja, and a, = ga for AC cells. The parameters of the composition of acceptor cells were varied. The above-mentioned system is complemented by boundary conditions at the outer edges of the rectangular sample and by initial conditions.
ee(o, w, x) „ ee(L, w, x) „
x<x^: e(0,w,X) = 0, x >x- : ( ,w ) = 0; ( ,w ) = 0;
lgn
54 54
- Bi [9(4,0, x)-00 ] = 0;
+ BI [e(4,Y, x)-e0 ] = 0;
e(4, w, 0) = e0; n(4, w,0) = 0.
(4)
(5)
(6) (7)
where i and j are the cell numbers along the axes 4 and y, respectively.
The mathematical formulation of the problem (1)-(7) uses dimensionless variables and
c RT
parameters as follows: yD = D *
2
A A = X A X D C CDPD , SA = Ca PA , 4
Bi = ŒX$ XD t x = —, x = . ' Ign t* t '"ign t*
QdED x
Xjjs
, L = L0
Xjk
y A =
Ka(T.) Kd(T.)
y D , Ar =
RT* e = ( T - t* ) ED
RT2
y
cDRT.
w = — , t* = - ,
x* qdedkd (T* )
x* —
X Dt*
Y d
, Y = Y0, d = d0
X * X*
« A =
VcdPD QaKa (T. ) Cd QdKd (T. ) Ca .
e0 ^,
y D
C A =
Ea Er
a
b
0
L
0
L
D
One of the goals of the numerical solution of equations (1)-(7) was to determine the average combustion rate of the system under adiabatic conditions at Bi = 0, depending on the scale of heterogeneity d (Fig. 2). The combustion rate was determined as the ratio between the linear dimension of the system L and the time of complete combustion of all cells in the volume. The scale of heterogeneity d can significantly affect the combustion rate of the system (Fig. 2, curve 2). For this curve, the relative parameters of the acceptor mixture were chosen close to the Co + Ti system that was the main composition of the acceptor mixture. With a small cell size d = 4, the combustion front is uniform and the temperature pulsations are insignificant. Combustion of systems with large cells is accompanied by high temperature pulsations along the front line, which leads to a decrease in the propagation velocity of the combustion wave with an increase in the scale of heterogeneity by 30%. With the increase in the scale of heterogeneity, the maximum combustion temperature of a DC cell increases from 9max = 0.15 at d = 4 to 0max = 1.28 at d = 32. This effect is explained by a significant difference in the heat-conducting properties of AC and DC. In the classical theory of gas-free combustion of homogeneous systems, the stationary combustion rate, as a rule, is higher than the average non-stationary combustion rate. A possible (hypothetical) change in the properties of acceptor AC cells with respect to the basic composition of DC changes the dependence U(d) (Fig. 2, curve 1). A decrease in the activation energy (oa = 0.5) and an increase in the thermal conductivity coefficient (Aa = 0.46) result in equalizing the combustion rates of AC and DC and leveling the effect of the cell size on the combustion rate of the heterogeneous system.
Fig. 2. Combustion rate as a function of the scale of heterogeneity for the regular (1, 2) and irregular (3) systems: 1 oa = 0.5, Aa = 0.8, Sa = 0.5, Qa = 0.68, 80 = -6, Ar = 0.1, L = 512, Y = 128; 2, 3 oa = 0.92, Aa = 0.46, Sa = 0.6, Qa = 0.68, 80 = -6, Ar = 0.1, L = 512, Y = 128.
To set the random volume distribution of particles. the Random procedure in the Delphi7 compiler was used. For each set of parameters, 20 packings with a random volume distribution of initial composition AC and DC particles were generated and the average combustion rate was calculated with respect to the number of generations (Fig. 2, curve 3). In the case of a random distribution of cells, clusters can be formed from weakly reacting acceptor AC cells which overlap the sample across its entire width Y for sufficiently large cells. With such generating the cells, the combustion rate drops sharply when the combustion wave passes through the cluster.
The combustion rate of an ordered system is higher than that of a system with a random arrangement of elements in the region with a small heterogeneity scale d < 20 (Fig. 2). During the combustion of a system with large cells d > 20, the effect of fluctuations in the spatial distribution of the parameters of the donor and acceptor mixtures increases, which leads to the occurrence of maxima and minima on the U(d) curve. However, the size of the cell becomes
370
V. G. Prokof'ev
ISHS 2019 Moscow, Russia
comparable to the width of the sample, and the calculated combustion rates of the regular and irregular systems are close. It is worth noting that in [2] the average combustion rate of a regular system with the presence of inert particles far from the concentration limit of combustion is 2 and more times higher than that of a disordered (random) system.
Numerical simulation of the gas-free combustion of SHS systems with a cellular structure has established the dependence of the average combustion rate on the size of a cell. The dependence is determined by the ratio of thermophysical and formal kinetic characteristics of donor and acceptor mixtures. The combustion rate of a system with a random distribution of cells is lower than that of a structurally-ordered system with the same ratio of system components. The thermal limit of propagation of a combustion wave weakly depends on the size of cells: the region of stable combustion under conditions of an external heat sink expands with increasing the size of cells.
The research was supported by the Russian Foundation for Basic Research, (project no. 19-0300081).
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