A Mathematical Model for Microwave-Assisted Chemical Vapor Infiltration
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A MATHEMATICAL MODEL FOR MICROWAVE-ASSISTED CHEMICAL VAPOR INFILTRATION James W. Evans and D. Gupta Lawrence Berkeley Laboratory and Dept. of Materials Science and Mineral Engineering, University of California, Berkeley CA 94720 ABSTRACT
A model for microwave heating of a ceramic preform is presented. Microwave power and external cooling are manipulated to obtain desired temperature profiles for effective chemical vapor infiltration (CVI) of the preform. A mass transport and chemical reaction model is developed to demonstrate preform densification. Conditions affecting matrix deposition are evaluated. INTRODUCTION CVI for synthesis of ceramic-ceramic fiber composites requires a suitable vapor-phase precursor, such as trichloromethylsilane (TMS) for silicon carbide. Vapors infiltrate the fiber preform, the precursor reacts on the fiber surface and gaseous products diffuse out of the pores [4,5]: CH3SiCl3 = SiC + 3HCl
(in H2)
The result is deposition of SiC matrix. High temperature (1400-1500K) is required for an appreciable reaction rate. Isothermal CVI invariably leads to preferential deposition of SiC at the pore mouth. Thus, sealing of the preform occurs before deposition deep within pores. This problem might be overcome by employing a temperature gradient with temperatures inside the preform higher than outside. Such temperature gradients can be generated by microwave heating coupled with external cooling.
MATHEMATICAL MODEL Microwave heating: The model treats a ceramic preform subjected to transverse electromagnetic waves Eoe--jl(z-L)
at z=O and EoejIl(z-L)
notation). at z=2Lat (phasor z=1 paorntto).
j
is I
the propagation constant in free space (Fig. 1), xi=W V#/ -,0c0, w is angular frequency of the EM wave, co and #o are permittivity and permeability of free space. Part of
the power carried by incident waves
I
Microwave P.ower per
unit
Ceramic
PD
ra.
Gaseous
Reactants
Preform
0
Pore
(initial radius Ro)
0 Environment
temperature 298K
L
Heat transfer coeficient,. H
Symmetry
transmitted is environment/ceramic rest is reflected.
the across boundary; the At the second
Fig. 1: Pore geometry
ceramic/environment
boundary,
subjected to microwaves.
the
about z-.
wave is again partly reflected and partly transmitted.
Mat. Res. Soc. Symp. Proc. Vol. 189. 01991 Materials Research Society
and
preform
102
Theoretically, the process undergoes an infinite number of such reflections. Nonetheless, the electric field inside the ceramic preform can be conveniently written as E = ae-jX2(z-L) + ýejK2(z-L) ; c' and 0" are dielectric constant and loss where X2 = W ,,oeo(C'-jf") factor of the ceramic material. Application of electromagnetic boundary conditions at z=O and z=2L yields
S2Eon 2esJ
L
( 2- 71) eJ- 2 L ] ) eJ K 2L -_ [(72+7)2 eJ 2K2L - (n7 -n)2 e- 2 2 L]
and where
(1
i= a =
oe '2=
o/
(2)
-F"
The power dissipated in the preform is given by the real part of
V.[ Ex(" )*] dV where (-)*
is the complex conjugate of the magnetic field associated with E
n12
[1], where overbars
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