Modeling of Microwave Heating of Particulate Metals P. MISHRA, G. SETHI, and A. UPADHYAYA Recent studies have shown that metal powder compacts can be heated to high temperatures using microwaves. While microwave heating of ceramics is well understood and modeled, there is still uncertainty about the exact mechanism and mode of microwave heating of particulate metals. The current study describes an approach for modeling the microwave heating of metal powder compacts using an electromagnetic-thermal model. The model predicts the variation in temperature with time during sintering. The effect of powder size, emissivity, and susceptor heating on the heating rate has also been assessed. These predictions have been validated by the experimental observations of the thermal profiles of Sn-, Cu-, and W-alloy compacts, using a 2.45 GHz multimode microwave furnace. I. INTRODUCTION MICROWAVE heating has been a unique contribution in the field of sintering of particulate materials. [1] As com- pared to conventional heating, microwave heating is more rapid, which results in reducing the overall sintering time. [2–5] In addition to the cost efficiency, the faster heating rate achieved in a microwave furnace minimizes micro- structural coarsening and abnormal grain growth. [2,6] For powder compacts, sintering is usually associated with den- sification as well as concomitant microstructural coarsen- ing. [7] If the latter is restricted, the mass atomic transport is enhanced due to the availability of more grain bounda- ries. [7,8] Recent studies on various systems have shown higher sintered density and mechanical property enhance- ment in microwave sintered compacts as compared to their conventionally sintered counterparts. [6,9,10] Despite the use of microwaves in sintering ceramics, it has only recently been shown that metals too can couple with microwave and get heated provided they are in powder form. [11] This has resulted in widespread interest in sintering of these com- pacts in microwave furnaces. Despite its potential for industrial applications, the phenomenology of microwave heating of metals remains to be fully understood. Although the large surface area in powders makes coupling of metal powders with microwave more amenable, yet a clear under- standing of microwave interaction and its absorption lacks in terms of modeling of the heating process. This article describes a new approach to explain the microwave sintering of particulate metals using a two- dimensional (2-D) coupled electromagnetic-thermal model. As part of the current study, the temporal temperature dis- tribution was simulated using 2-D finite difference time domain (FDTD) calculations to obtain the electromagnetic field parameters. The model also incorporates the effect of particle size, emissivity, and susceptor on heating. The validation of predicted thermal profiles was done on tin, copper, and tungsten-alloys powders through careful mea- surements of the temperature variation with time in a multi- mode-cavity microwave furnace. II. MODELING APPROACH Modeling of microwave heating involves solving the Maxwell’s equations of electromagnetism simultaneously with the heat-transfer equation. [12] To determine the electro- magnetic fields in a cavity, the FDTD technique was used by Yee [13] and Zhang et al. [14] Zhang et al. ’s [14] results showed that the numerical methodology can be used in proper designing of the microwave cavities for various applications. It was indicated that by appropriate selection of the field points, a set of finite difference equations can be formulated for a boundary condition involving perfectly conductive surfaces. These solutions were used for cal- culating power absorption, flow fields, and temperature patterns in nonmetallic systems. [14,15,16] Subsequently, a quasi-analytical model was proposed by Lasri et al., [17] to examine energy conversion during the microwave sintering of a ceramic that is surrounded by a susceptor. Elsewhere, a simulation package based on 2-D modeling of microwave heating of ceramics was developed by Craven et al., [18] who used a coupled electromagnetic-thermal model. This approach was further extended by Alpert and Jerby [19] for microwave heating of temperature-dependent dielectric media. In the current study, the calculated power absorbed is correlated with the temperature rise in the compact. The latter requires solving the heat-transfer equation for the boundary conditions defined by the dimensions of the sam- ple and its physical, thermal, and dielectric properties. The dependency of these parameters on temperature necessi- tates the need for a dynamic simulation. Figure 1 shows the modeling approach adopted in this study through a schematic flowchart. A. Electromagnetic Power Microwaves penetrate and propagate through a dielectric material, such as SiC. This generates an internal electric field (E) within a specific volume, which in turn induces polarization and movement of charges. The resistance to these induced motions due to internal, elastic, and frictional forces attenuates the electric field. These losses result in volumetric heating. The resulting electromagnetic power absorbed per unit volume (P EM ) by a material is given by [20] P. MISHRA, Undergraduate Student, and A. UPADHYAYA, Associate Professor, are with the Department of Materials and Metallurgical Engi- neering, Indian Institute of Technology, Kanpur 208016, India. G. SETHI, Graduate Research Assistant, is with the Center of Innovative Sintered Products, The Pennsylvania State University, University Park, PA 16802. Contact e-mail: anishu@iitk.ac.in Manuscript submitted September 1, 2005. METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 37B, OCTOBER 2006—839