Combustion synthesis in the Ti-C-Ni-Mo system: Part II. Analysis
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I.
INTRODUCTION
IN the companion article,tq the micromechanisms (e.g., phase and structural transformations, capillary spreading, product formation and coarsening, etc.) of self-propagating high-temperature synthesis in a Ti-C-Ni-Mo powder mixture, using the quenching technique developed by Rogachev et al.,[2~ were examined. The utility of this technique for examining physical processes on the microscale which occur within the three general regions (i.e., unreacted, partially reacted, and fully reacted regions) of a combustion wave has been previously demonstrated (and advocated).t2 ~0] However, this technique does not allow quantification of reaction rates, degree of conversion, and temperature as a function of time or space within these regions. Information such as this further enhances the understanding of the physical processes and their relative importance to the propagation of a stable combustion wave. t") ~3~ The primary focus of this article is to rationalize the observations reported in the companion articlet~l and to correlate them with the micromechanisms. Calculations leading to quantification of the parameters responsible for the key features of the mechanism of spherule formation are presented. II.
APPARENT ACTIVATION ENERGY DETERMINATION
The companion articletx] contains information on powder preparation, as well as on the combustion-wave velocity and temperature measurement technique and data. An expression for the velocity of propagation of a stable planar combustion wave in terms of the adiabatic temperature, activation energy for the reaction, and thermophysical properties has been previously derived, t14,15`L61 The equations which describe the propagation of a planar "gasless" corn-
J.C. LaSALVIA, Postdoctoral Fellow, Institute for Mechanics and Materials, and M.A. MEYERS, Professor, Department of Applied Mechanics and Engineering Sciences, are with the University of California, San Diego, CA 92093. Manuscript submitted October 24, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS A
bustion wave traveling in an isotropic, homogeneous medium are [161
~T Ot
--
--
Ot
=
=
a -
K o (1
02T Ox2
-
Q 07 Cp Ot
[1]
"o)ne (E/RT)
[2]
+
- - -
-
-
where T is the temperature, a is the thermal diffusivity, Q is the heat of reaction, Cp is the heat capacity, K o is the reaction-rate constant, n is the order of the reaction, R is the universal gas constant, and E is the activation energy for the reaction. Equation [1] also assumes constant thermophysical properties and negligible heat losses to the surroundings. Based upon some simplifying approximations (e.g., "narrow" zone), Eqs. [1] and [2] can be solved to obtain the following expression for the velocity of a stable planar combustion wave: t~6]
U=~/~r. Koa
(----~--
exp ( ~ ) [ 3 ]
where U is the combustion-wave velocity, ~r. is a constant which depends upon the order of the reaction (e.g., for n = 0, 1, and 2, o-. = 2, 1.1, and 0.73, respectivelytl6]), and T.a is the adiabatic temperature. The apparent activation energy E for the react
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