where Pand O are the mass and thermal diffusivities, respectively, indicating that heat conduction occurs much faster than mass diffusion. As a result, mass transfer by diffusion at the macroscopic scale may be neglected, and an average concentration of the reactants in any local region of the heterogeneous mixture may be used. Thus, the physical and thermal properties (e.g., density, thermal conductivity, heat capacity) are the average of reactant and product values. In this case, the combustion synthesis process is controlled only by heat evolution from the exothermic reaction and heat transfer from the reaction zone to the unreacted mixture

Propagation of combustion wave through the reactant mixture with a velocity U can be described by the set of energy continuity and chemical kinetic equations (Ref 18):

where f\ Cp, and Aare respectively the density, heat capacity, and thermal conductivity of the reaction mixture. The term represents the reaction rate, and x is the spatial coordinate moving with the reaction front. Assuming that the temperature (7) and conversion ('/) profiles do not change with time (t), i.e. constant-pattern propagation, then the time derivatives can be set equal to zero. Then Eq 10 takes the form:

along with the boundary conditions:

The solution of Eq 11a, 11b, 12a, and 12b was first obtained in the context of gas flames (Ref 19) and was later adapted for solid mixtures (Ref 20, 21) to yield the following formula for reaction wave velocity:

where E is activation energy and R is a gas constant. This expression can be solved for various types of kinetics. For example, for zero-order reaction, then:

This formula is widely used to determine effective kinetic constants from experimental data, because:

Thus by varying Tc (by dilution or by changing initial temperature, T0) and plotting ln(U/Tc) vs. 1/Tc, the activation energy can be obtained readily. In this manner, the effective values of E have been measured for various SHS systems (Ref 5).

While providing a simple method for analyzing the redistribution of energy in the combustion wave, the model discussed above does not account for the local structural features of the reaction medium. Microstructural models account for details such as reactant particle size (d) and distribution, product layer thickness, and so on, and correlate them with the characteristics of combustion to give expressions of the form:

The first microstructural models were developed independently and essentially simultaneously (Ref 18, 21, 22, 23). For these models, the elementary reaction cell, which accounts for the details of the microstructure, consists of alternating lamellae of the two reactants (A and B), which diffuse through a product layer (C), to react (see Fig. 2a). Assuming that the particles are flat allows one to neglect the change in reaction surface area during synthesis. The characteristic particle size, d, is equivalent to the layer thickness, and the relative thicknesses of the initial reactant layers are determined by stoichiometry.

W AW* aw-CM (b)A(i)+BH) -cm (a)jifl)+B(i) -cm + -en]

Fig. 2 Geometry of the reaction cells considered in the theoretical models

W AW* aw-CM (b)A(i)+BH) -cm (a)jifl)+B(i) -cm + -en]

Fig. 2 Geometry of the reaction cells considered in the theoretical models

A simple analytical solution for the case of diffusion-controlled kinetics has been developed, and yields f(tl) OO \/cl (Ref 21). A more accurate expression, based on the same physical geometry, was developed using the following kinetic function (Ref 22, 23):

which describes linear (m = 0, n = 0), parabolic (m = 0, n = 1), cubic (m = 0, n = 2), and exponential (m > 0, n = 0) kinetic dependencies. For example, the solution of Eq 11a, lib, 12a, and 12b for power-law kinetics (i.e., m = 0, n >0) can be written as:

For the next two types of theoretical models, the elementary reaction cell consists of a spherical particle of one reactant surrounded by a melt of the other reactant. In the first case, the product layers (C) grow on the surface of the more refractory particles (B) due to diffusion of atoms from the melt phase (A) through the product layer (see Fig. 2b). At a given temperature, the concentrations at the interphase boundaries are determined from the phase diagram of the system (Ref 24, 25). The second model (Fig. 2c) assumes that upon melting of reactant A, a layer of initial product forms on the solid reactant surface. The reaction proceeds by diffusion of reactant B through this layer, whose thickness is assumed to remain constant during the reaction (Ref 26, 27). The final product (C) crystallizes in the volume of the melt after saturation. Based on this model, an analytical expression for the combustion wave velocity has been reported (Ref 28):

where iis the thickness of the initial product layer.

For the case where both reactants melt in the preheating zone and the liquid product forms in the reaction zone, a simple combustion model using the reaction cell geometry presented in Fig. 2(d) has been developed (Ref 29). After both reactants melt, their interdiffusion and the formation of a liquid product occur simultaneously. Numerical and analytical solutions were obtained for both kinetic- and diffusion-controlled reactions. In the kinetic-limiting case, for a stoichiometric mixture of reactants (A and B), the propagation velocity does not depend on the initial reactant particle sizes. For diffusion-controlled reactions, the velocity may be written as:

where bis a constant that depends on the mixture composition.

The microstructural models described above represent theoretical milestones in gasless combustion. Using similar approaches, other models have also been developed. For example, in Ref 30 the solid-liquid model (Fig. 2c) was used to determine the combustion velocity as a function of stoichiometry, degree of dilution, and initial particle size. Calculations for a variety of systems compared favorably with experimental data. In addition, an analytical solution was developed for diffusion-controlled reactions, which accounted for changes in A, P. and Cp within the combustion wave, and led to the conclusion that U OO Mel (Ref 31).

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