Microwave-Heating Temperature Profiles for Thin Slabs Compared to Maxwell and Lambert Law Predictions SHERYL A. BARRINGER, EUGENIA A. DAVIS, JOAN GORDON, K. GANAPATHY AYAPPA, and H. TED DAVIS ABSTRACT Slabs of agar gel were heated in a microwave oven. Temperatureswere measured at various depths into the sample to experimentally determine the internal temperatureprofile. These were compared to power and tem- perature profiles predicted from Lambert’s law, Maxwell’s field equa- tions and a Combined equation. Lambert’s law and the Combined equa- tion predicted a much slower heating rate than found experimentally, while Maxwell’s field equations gave a much more accurate prediction. Becauseof the internal standing waves that are created, a small variation in sample thickness could make a large difference in heating rate for thin samples. Key Words: agar gel, heat penetration, Lambert’s law, Maxwell’s Field, microwave INTRODUCTION WITH INCREASING MICROWAVE OVEN usethereis aneedtopre- diet the temperature profile in a microwave heated food sample. Heat generation can be modeled using either Lambert’s law or the more rigorous solution of Maxwell’s equations to determine the microwave power term. Both sets of equations have been reported in heating profile predictions (de Wagter, 1984; Jia and Jolly, 1992; Ohlsson and Bengtsson, 1971; Padua, 1993; Stuchly and Hamid, 1972). Lambert’s law predicts an exponentially decaying absorption of energy as a function of depth into the sample. It is valid for semi-infinite samples only since all reflections are neglected. An approximation of the percent radiation transmitted at the air- sample interface has to be determined experimentally. This is generally done by measuringthe heating rate of a beaker of water and calculating the power absorbedby the sample (Ohlsson and Bengtsson, 1971; Stuchly and Hamid, 1972; Mudgett, 1986). This calorimetric determination introduces errors as it is based on the assumption that all transmitted energy is absorbed, which may not be true for thin samples. Maxwell’s equations (Maxwell, 1881) predict the absorption of energy, incorporating reflections at the front and back sur- faces and all internal interfaces. Wave interactions are included, so that internal standing waves are calculated, and the electric field distribution and absorbedpower are consistently evaluated. Lambert’s law is a simplification of the electromagnetic inter- actions and is frequently used in predictive models becausethe calculations are much simpler than for Maxwell’s field equa- tions. The penetration depth is defined as the depth at which the power has decayed to e-l of its initial value at the front interface Author Barringer, formerly with the Univ. of Minnesota, is now with the Dept. of Food Science & Technology, The Ohio State University, 122 Vivian Hall, 2121 Fyffe Road, Columbus, OH 43270. Authors E.A. Davis and Gordon are affiliated with the Dept. of Food Science & Nutrition, Univ. of Minnesota, 1334 Eckles Ave., St. Paul, MN 55108. Author H.T. Davis is with the Dept. of Chemical Engineering & Materials Science, Univ. of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455. Author Gana- pathy Ayappa, formerly with the Univ. of Minnesota, is now with the Dept. of Chemical Engineering, India Institute of Science, Ban- galore 560012, Karnataka, India. Address inquiries to Dr. Joan Gordon. and is the inverse of Eq. (2) (Theory section). For example, the penetration depth of water is 3.6 cm. For samples thicker than several penetration depths, internal reflections become negligible because all transmittedradiation is absorbed. Ayappa et al. (199 1) compared the power distributions predicted by Maxwell’s equa- tions and Lambert’s law and showed that the two formulations predicted identical power profiles for samples thicker than 2.7 times the penetration depth. For thinner samples, Maxwell’s equations predicted a very different profile from Lambert’s law because the radiation reflected back through the sample from the far interface increased the overall power absorption. Theo- retically, neglecting theseback reflections for thin slabs, as hap- pens with Lambert’s law, introduces large errors (Fu and Metaxas, 1992). Experimentally, the inclusion of wave interac- tions was necessary to explain the heating rates of small cylin- ders of water (Barringer et al., 1994) and emulsions (Barringer et al., 1995). The theoretical depth at which Lambert’s and Maxwell’s equations converge is 7.8 cm for agar. Few foods heated in the microwave oven are that thick. Therefore two sample thick- nesses reasonable for an entree in the microwave oven were tested to determine how different the predictions were for such samples. The objective of our research was to determine experimen- tally if Maxwell’s field equations predicted heating rates signif- icantly better than Lambert’s law for thin samples. Measured temperatures were compared to those predicted by Lambert’s law, Maxwell’s equations and a Combined equation. The Com- bined equation was a limiting form of Lambert’s law derived from Maxwell’s equations. It neglects internal reflections but accounts for reflections at the incident face of the sample. The Combined equation was used to test whether a calculated trans- mission value would give a more accurate approximation than Lambert’s law without introducing the complexities of internal reflections. THEORY Power from Lambert’s law Lambert’s law predicts that the power absorbed per unit vol- ume, P”(z) at a given depth z is: P’(z) = 21,pe-2pz (1) where It is the transmitted power intensity and /3 is the attenuation factor: . . P= 43 (2) A, is the wavelength in free space, or 12.24 cm at 2450 MHz. The dielectric properties are the real portion, K’, and imaginary portion, K”, of the relative complex permittivity, K*. Lambert’s law is strictly valid for semi-infinite samples and requires an estimate of the transmitted power flux. Power from Maxwell’s equations for a slab Maxwell’s equations can be solved for the absorbed micro- wave power, P”(z). For a slab of thickness L exposed to radi- Volume 60, No. 5, 1995-JOlJRNAL OF FOOD SCIENCE-i 137