Microwave heating of ceramic composites

The microwave heating of a ceramic composite is modelled and analysed. The composite consists of many small ceramic particles embedded in a ceramic cement. The composite is assumed to be well insulated, and each particle is assumed to be in imperfect thermal contact with the surrounding cement. Based on these two assumptions an asymptotic theory exploiting the small Blot number and small non-dimensional contact conductance is developed. Our asymptotic theory yields a set of nonlinear partial differential equations which govern the temperature in the composite. These are reduced to a set of coupled nonlinear ordinary differential equations in which the surface area of each particle enters as a parameter. Recent experiments with such composites have shown that the steady-state temperature of the composite is strongly dependent upon the radii of the embedded particles. Our model captures this effect. In fact, our analysis shows that the assumption of imperfect thermal contact between the particles and the ceramic cement is essential for this trend to be established

A multiscale derivation of a new parabolic equation which includes density variations

AbstractA new parabolic equation is obtained from the acoustic equation by the multiscale method. The new equation incorporates the effects of a variable ocean density. The density can be smooth or piecewise smooth. Thus, the new formulation alleviates the need for interfacial conditions when the density is stratified in a piecewise constant fashion. It also reduces to the standard P.E. when the density is constant. The new equation has the same conservation law as the P.E. A difference equation is presented which has a discrete version of the same law

An Illustrative Model Describing the Refraction of Long Water Waves by a Circular Island

The refraction of small shallow water waves by an idealized island is studied in this paper. The island\u27s shoal is modeled by a quartic polynomial in the radial variable. This particular model allows the explicit construction of the rays (wave orthogonals) and the determination of several important features of the wave motion. The various shortcomings of the particular profile are discussed

SCATTERING BY PENETRABLE ACOUSTIC TARGETS

An acoustic target of constant density pt and variable index of refraction is imbedded in a surrounding acoustic fluid of constant density pa. A time harmonic wave propagating in the surrounding fluid is incident on the target. We consider two limiting cases of the target where the parameter e = pa/p, + 0 (the nearly rigid target) or E + ~0 (the nearly soft target). When the frequency of the incident wave is bounded away from the 'in-vacua' resonant frequencies of the target, the resulting scattered field is essentially the field scattered by the rigid target for E = 0 or the soft target if E + a). However, when the frequency of the incident wave is near a resonant frequency, the target oscillates and its interaction with the surrounding fluid produces peaks in the scattered field amplitude. In this paper we obtain asymptotic expansions of the solutions of the scattering problems for the nearly rigid and the nearly soft targets as E + 0 or E + co, respectively, that are uniformly valid in the incident frequency. The method of matched asymptotic expansions is used in the analysis. The outer and inner expansions correspond to the incident frequencies being far or near to the resonant frequencies, respectively. We have applied the method only to simple resonant frequencies, but it can be extended to multiple resonant frequencies. The method is applied to the incidence of a plane wave on a nearly rigid sphere of constant index of refraction. The far field expressions for the scattered fields, including the total scattering cross-sections, that are obtained from the asymptotic method and from the partial wave expansion of the solution are in close agreement for sufficiently small values of E

Modern microwave methods in solid state inorganic materials chemistry: from fundamentals to manufacturing

No abstract available

An Illustrative Model Describing the Refraction of Long Water Waves by a Circular Island

The refraction of small shallow water waves by an idealized island is studied in this paper. The island\u27s shoal is modeled by a quartic polynomial in the radial variable. This particular model allows the explicit construction of the rays (wave orthogonals) and the determination of several important features of the wave motion. The various shortcomings of the particular profile are discussed

Microwave Heating and Joining of Ceramic Cylinders: A Mathematical Model

A thin cylindrical ceramic sample is placed in a single mode microwave applicator in such a way that the electric field strength is allowed to vary along its axis. The sample can either be a single rod or two rods butted together. We present a simple mathematical model which describes the microwave heating process. It is built on the assumption that the Biot number of the material is small, and that the electric field is known and uniform throughout the cylinder's cross-section. The model takes the form of a nonlinear parabolic equation of reaction-diffusion type, with a spatially varying reaction term that corresponds to the spatial variation of the electromagnetic field strength in the waveguide. The equation is analyzed and a solution is found which develops a hot spot near the center of the cylindrical sample and which then propagates outwards until it stabilizes. The propagation and stabilization phenomenon concentrates the microwave energy in a localized region about the center where elevated temperatures may be desirable