Modeling Self-Propagating Exothermic Reactions in Multilayer Systems
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Mat. Res. Soc. Symp. Proc. Vol. 481 0 1998 Materials Research Society
Propagation Direction Unreacted Foil
Reacted Foil Spark•
Reaction Zone
Regions where atoms are mixing
/I
B
S~Atomic
Diffusion
II
S..... B
1
Thermal uision
X
Figure 1. Schematic of self-propagating reactions in multilayered systems. FORMULATION Figure 1 shows a schematic of a self-propagating exothermic reaction in a multilayered system. Intermixing between the constituent layers occurs in a narrow reaction band which propagates along the foil. The evolution of the chemical reaction is governed by the equation of motion of the conserved Shvab-Zeldovich variable, C:
d= V.(DVC)
(1)
dt
together with the energy conservation equation:
dT=
A'V2T+
dQ(C)
(2)
where D is the binary mass diffusion coefficient and X is the average thermal diffusivity of the system. The Shvab-Zeldovich variable C is defined such that C = +1 for pure Al, C = -1 for pure Ni and C = 0 for pure NiAl. Q(C) is a coupling function that relates the heat of the reaction to the change in C. In the present study the coupling function was2 assumed to be (i) linear (i.e. Q(C) = PCp(Tf - TO) C) or (ii) parabolic (i.e. Q(C) = pC (Tf - TO) C ). p is the average density and Cp is the average specific heat of the system. We also assume that X is constant and that D has an Arrhenius dependence on T.
564
6
So4 0 0 "7;10 0
0
_
4 00
52o Data for Al-Monel multilayers (w = 0.6nm) [1] 0
o Data for AI-Monel multilayers (w = 1.6nm) [11] 1
80
40
0
0
0
oo
I
100
0
multilayer period - 48 (nm)
multilayer period - 45 (nm)
Figure 2. Comparison between experimental data for Al/Monel multilayers and predictions of analytical model [9]. ANALYTICAL MODELING In order to obtain an analytical solution for the flame speed the coupled equations (1) and (2) must be simplified by making a number of assumptions (Full details of these assumptions are given elsewhere [9]). First, the equation for atomic diffusion (1) is reduced to the steady-state equation: ex(: E d 2C
VX,~°3C
_Aexp .-- Ef -•- =0
A dx
RT
f
(3)
Oý2(3
where v, is the flame speed (dx/dt) and the Arrhenius function Aexp(-E/RI) is used instead of D/A. Note that equation (3) takes C to vary principally in the y-direction and assumes that temperature, T, varies principally in the x-direction. Similar simplifications reduce the energy conservation equation (2) to the following form: CpA-T = _-Vx-Q(C) P dx Xx
(4)
where Q, in equation (4) denotes the average value of Q over y at a given x. Using a substitute variable, F, which is the integral over x of exp(-E/RT), equation (3) can be solved for C(F,y) by separation of variables (see [9] for details). This expression for C can then be used to replace Q(C) in equation (4) using either (i) linear coupling or (ii) parabolic coupling, as discussed previously. Finally, the partial differential aT/lx is re-written in terms of F and then equation (4) is rearranged to give:
v
=od nn=Odd/kY
fTA exp(RT- ) 3-1 n E(fo-To)
565
(5)
x
n=odd/ "n
E(T:o - TO)
for the
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