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MICROWAVE MODELING AND VALIDATION IN FOOD THAWING APPLICATIONS Tim Tilford1*, Ed Baginski2, Jasper Kelder2, Kevin Parrott1 and Koulis Pericleous1 1 University of Greenwich, Park Row, Greenwich, London, United Kingdom 2 Unilever Research and Development, Vlaardingen, The Netherlands * [email protected] Developing temperature fields in frozen cheese sauce undergoing microwave heating were simulated and measured. Two scenarios were investigated: a centric and offset placement on the rotating turntable. Numerical modeling was performed using a dedicated electromagnetic Finite Difference Time Domain (FDTD) module that was two-way coupled to the PHYSICA multiphysics package. Two meshes were used: the food material and container were meshed for the heat transfer and the microwave oven cavity and waveguide were meshed for the microwave field. Power densities obtained on the structured FDTD mesh were mapped onto the unstructured finite volume method mesh for each time-step/turntable position. On heating for each specified time-step the temperature field was mapped back onto the FDTD mesh and the electromagnetic properties were updated accordingly. Changes in thermal/electric properties associated with the phase transition were fully accounted for as well as heat losses from product to cavity. Detailed comparisons were carried out for the centric and offset placements, comparing experimental temperature profiles during microwave thawing with those obtained by numerical simulation. Submission Date: 18 December 2006 Acceptance Date: 27 September 2007 Publication Date: 11 January 2008 INTRODUCTION Microwave heating of a typical food load is a complex process. The electromagnetic field distribution within the oven cavity is highly sensitive to the food load it is required to heat. The changes in temperature of the food load lead to variation in electromagnetic and thermophysical properties (see, e.g., [Metaxas and Meredith, 1983]). These changes in turn alter the field distribution and intensity within the cavity and cannot be ignored. Food properties are especially sensitive to phase change, which Keywords: Heat transfer, Maxwell’s equations, microwave heating, moving meshes, numerical simulation, rotation, thawing 41-4-30 can have a highly localized impact. Additionally the food load may rotate during the cooking process, smoothing what would otherwise be a poor quality heating process. Thus, despite the very different thermal and electromagnetic timescales, there is a requirement to strongly couple the electromagnetic and thermal computational analysis. This contribution focuses on coupled electromagnetic-thermophysical simulation of microwave heating of frozen food products. The solution approach utilizes a Finite Difference Time Domain (FDTD) electromagnetic analysis solver linked with a Finite Volume (FV) method multiphysics package. Using separate meshes for each analysis tool, an otherwise complex Journal of Microwave Power & Electromagnetic Energy ONLINE Vol. 41, No. 4, 2007 meshing problem can be solved through general cross-mappings between two separate meshes. Using a thermal analysis mesh confined to the food material (with appropriate surface heattransfer boundary models) allows this mesh to follow the food material as it rotates. This is then automatically mapped to an appropriately refined region within the electromagnetic mesh. There is extensive literature concerned with the computational analysis of microwave heating of food material. The Lambert law approximation can be used to avoid numerical calculation of the field distribution and is accurate in a number of cases (see, e.g., [Zhang and Datta, 2005] and [Liu et al., 2005]). However, the approximation requires the magnitude of the electric field at the surface of the dielectric to be specified and so is unable to deal with surface field variation. Early examples of a coupled approach to microwave heating include, e.g., [Zhang and Datta, 2000; Ratanadecho et al., 2000; Ayappa et al., 2002]. The latter paper predicted temperature profiles in multi-layer slabs by simultaneously solving temperature and Maxwell’s equations using a Galerkin Finite Element Method. Zhang and Datta [2000] described one way coupling between two separate finite element packages (one electromagnetic (EM), one thermophysical). The analysis linked dielectric properties to temperature. The model was applied to heating of foods and to microwave processing of polymers. A very general coupled approach to microwave drying of timber is described by Zhao and Turner [1999]. The method used an unstructured Finite Volume (FV) time domain approach to obtain an EM solution and an unstructured FV heat transfer solver. The model was able to accurately determine the temperature distribution within the load. Numerical models of microwave heating of a phantom gel load were developed by Ma et al. [1995], in which numerical results were contrasted against experimental measurements. The numerical model utilized the FDTD method to solve Maxwell’s equations in order to obtain an EM solution and used an explicit International Microwave Power Institute FV scheme to obtain a thermophysical solution. Liu et al. [1994] simulated microwave heating of a polymer material inside a ridge waveguide. Again, an FDTD scheme was employed to solve Maxwell’s equations and a finite difference scheme was used to solve the heat transfer equations. Kopyt and Celuch [2003] described an approach to coupling FDTD computations to thermal computations based on a single thermal mesh and multiple FDTD meshes for a rotating load. This approach takes the average of the power distributions computed at each load position as input to the thermal computation. The FDTD meshing was conformal and preserved the meshing of the food load during rotation. A general approach to microwave heating without rotation is described in [Kopyt and Celuch, 2005] that couples existing computational fluid dynamics (CFD) and FDTD software using a common mesh. This work successfully linked the two computations dynamically for a variety of CFD discretization. This paper extends the earlier work of Dincov et al. [2004], which described the coupling of a Yee FDTD scheme to the PHOENICS CFD code [PHOENICS, 1974-2007]. The food material was modeled as a porous medium with dielectric properties dependent upon both temperature and moisture content. The coupling required the colocation of the cells of the two domains, i.e the cells, in the two domains had identical vertex and centroid coordinates allowing a direct cell-tocell relationship between the solution domains. This was restricted to non-rotating loads and could not allow for automatic mesh refinement. The approach described in this paper couples a cartesian mesh FDTD solver to an unstructured mesh FV multiphysics solver using a crossmapping algorithm that allows fully independent meshing for each solver. The approach is able to simulate complex product shapes rotating within the oven cavity, with non-linear phase-changes occuring within the rotating load. Another advantage of independent meshing is that the EM mesh can be varied as heating progresses, 41-4-31 e.g., in response to load wavelength changes, to reduce the computational cost. For feasible simulations supporting actual product development, a number of requirements have to be met. First of all, successful interaction with drawing packages is needed to analyze complex product shapes. Secondly, both spatial and temporal accuracy need to be sufficient to capture heating and the phase transition over full heating time, which is on the order of minutes. Finally, problem set-up, computation and results interpretation should take on the order of days. This contribution describes a model which meets these requirements. ELECTROMAGNETIC ANALYSIS The electromagnetic analysis is carried out in a predominantly rectangular domain consisting of a domestic microwave oven cavity excited via a standard waveguide. This allows for the use of a classical Yee FDTD scheme [Yee, 1966] with tensor product meshes [Monk and Suli, 1994], where Maxwell’s equations, in the form shown below, are solved in the time domain with harmonic excitation. ∂H µ = −∇ × E (1) ∂t ε ∂E = ∇×H−J ∂t (2) ∇ ⋅ (ε E) = 0 (3) ∇ ⋅ (µ H ) = 0 (4) where E is the electric field strength, H is the magnetic field strength, μ is the magnetic permeability, ε is the electric permittivity and J is the current density: J = σ eff E 41-4-32 (5) A conventional effective conductivity σeff in (5) generates the microwave power dissipation. In the most general case, σeff can vary both with temperature T and moisture content. In this case, the values of σeff and ε are mapped directly at each thermal time-step from the FV mesh to the FDTD mesh, i.e., the load geometry is captured implicitly as a spatial distribution of material properties via the mapping process. The implicit construction of the load through the use of the cross-mapping process simplifies the simulation of materials that are of non-rectangular shape or are moved within the applicator. The key benefit of the approach is that the FDTD mesh is not dependent upon the geometry or positioning of the load. One single FDTD mesh could be created and used throughout the heating process. The load geometry and position are considered by mapping dielectric properties into the correct spatial location. A sufficient number of cells per wavelength should be provided to maintain numerical accuracy. The automatic mesh generation algorithm outlined below optimizes the FDTD mesh spacing to reduce computational expense while maintaining this accuracy criterion. The drawback to crossmapping is the smearing of the interface between dielectric materials. This may effect reflection and refraction at the interface and adversely affect accuracy. A possible alternative to crossmapping would be to utilize a conformal FDTD mesh that remained fixed within the load as it rotated – a method employed, for example, by Kopyt and Celuch [2003]. The Yee scheme update of the E field is dependent upon only the H field and, likewise, the H field update is dependent upon only the E field. This greatly simplifies the parallelization process. The scheme is second order accurate both temporally and spatially. However, the accuracy of the scheme is dependent upon the resolution of the mesh in terms of the number of mesh cells per wavelength. In the case of frozen foods, the dielectric properties vary significantly during the heating process. The Journal of Microwave Power & Electromagnetic Energy ONLINE Vol. 41, No. 4, 2007 (a) (b) local microwave propagation wavelength is inversely proportional to the loss factor of the material. Consequently, increases in temperature and the loss factor due to microwave heating result in progressive reduction of propagation wavelength during the heating period. The user is required to define a criterion for the number of cells per wavelength needed to ensure numerical accuracy prior to the simulation commencing. The implemented scheme utilizes a tensor-product Cartesian mesh. The locations of the cell vertices in such a mesh are defined by a tensor product of X-, Y- and Z-direction cell distribution vectors. The mesh is re-specified at each thermal time-step to capture any thermally induced wavelength changes and the overall movement of the load by turntable rotation. This is achieved by an automatic mesh generation step which first subdivides the domain in the X-, Y- and Z-directions separately, as illustrated in Figure 1. The algorithm identifies the location of walls and the boundary of the food load in each direction. These locations are then used to subdivide the X/Y/Z-extent of the domain into a number of sub-domains. The minimum wavelength for each sub-domain is determined by assessing the dielectric properties of each cell in the sub-domain. With this minimum wavelength determined, the algorithm is able to define the mesh spacing in each sub-domain in a manner meeting the accuracy criterion. Figure 1 shows the sub-divisions of the EM domain in the XZ-plane with the load in two different positions; one can see that the sub-domains are bounded by walls or load locations. A target of 20 cells per wavelength has been used to accurately capture the spatial distribution of electric field. Time-step length has been determined using a Courant criterion and is therefore a function of grid dimension and wave propagation velocity. The automatic mesh generation process is rapid and has proven numerically stable. Generating new FDTD meshes every time-step incurs minimal additional computational cost since new mappings from the thermal analysis mesh to the EM analysis mesh are required in any case at each time-step to deal with rotation. The time-domain electromagnetic fields are integrated to a time harmonic solution for each thermal analysis time-step. The Fourier transform of the electric field is used to yield the power source term for the heating step. The differences between successive values of absorbed power at successive Fourier transfer analyses are used to determine when a converged time harmonic solution has been obtained. Figure 1. XZ-plane automesher domain sub-division with load in position 1 (a) and 2 (b). International Microwave Power Institute THERMAL ANALYSIS The thermo-physical analysis has been carried out by incorporating the FDTD code into the PHYSICA multiphysics package [PHYSICA, 1996-2007] developed at the University of Greenwich. This software has been created to solve combined stress, heat and fluid transfer 41-4-33 problems and has a flexible programmable interface that allows complete modules with independent meshes to be incorporated. The task of obtaining a solution to the thermal physical problem could be carried out using a wide range of software packages utilizing a number of solution methods and approaches. The PHYSICA package was selected for a number of reasons. Firstly, the package has been co-developed by the authors and is therefore a practical choice. PHYSICA is able to solve partial differential equations using either an unstructured FV solver or using a finite element (FE) solver. Solutions to a number of physical problems (such as structural mechanics analysis) are most efficiently obtained utilizing the FE approach. Problems such as heat transfer and fluid flow are often solved using the FV approach although FE solution is a viable alternative. In this contribution the FV approach has been used to solve the heat transfer problem because the conservative form of the finite volume formulation is more efficient in large non-linear problems than the FE method. Solution of electromagnetic problems using the PHYSICA FE implementation is not viable. As it is suggested by Ehlers et al. [2001], an edge-element formulation may be more suitable for accurate EM solutions while the PHYSICA FE solver utilizes a vertex based approach. However, FDTD is more accurate and more efficient for regular geometries (see, e.g., [Monk and Parrott, 2001]). The implementation of the FV method within PHYSICA used an unstructured co-located (non-staggered) grid and a pressure correction approach by Rhie and Chow [1983]. PHYSICA’s multiphysics modules can obtain solutions to a wide range of physical problems. In this case, the energy equation, in a form allowing for phase change and localized heat sources, has been solved: ( ∂ ρ C pT 41-4-34 ∂t ) = ∇ ⋅ ( k ∇T ) + S + Q (6) where k is the coefficient of thermal conductivity, ρ is the density, Cp is the specific heat capacity and S and Q are the source terms for phase changes and microwave power dissipation respectively. The thermal analysis was carried out on a mesh that was body fitted to the food material alone. Heat losses to the oven space were modeled with appropriate boundary conditions. The energy equation was discretized on a cell-by-cell basis on this unstructured mesh using the FV method with cell average values. These are phase averaged quantities when more than one phase is present in the cell, where the phase values of ρ , Cp and k are functions of temperature; values are calculated from a set of look-up tables prior to each time-step. Power dissipation Q is determined by mapping power values from the FDTD scheme solution: 2 1 Q = σ eff E 2 (7) An additional source term in the energy equation is S, the latent heat source/sink due to phase change. The value of this source is dependent upon the change in liquid fraction in each cell. Foods are complex multi-component materials and so phase change will usually take place over a finite temperature range. A liquid fraction is defined for each cell and approximated as a linear variation of liquid fraction between solidus (Ts) and liquidus (Tl) temperatures, i.e. ( ) f T = 0 , T < Ts (8) f T = 1 , T > Tl ( ) (9) T − Ts (10) ( ) f T = Tl − Ts , Ts ≤ T ≤ Tl The heat source term due to phase change is then: ∂f S = −ρ L (11) ∂t Journal of Microwave Power & Electromagnetic Energy ONLINE Vol. 41, No. 4, 2007 where L is the latent heat and f is the liquid fraction. An iterative approach is used to obtain converged temperature and liquid fraction values. The food material within a computational cell typically undergoes phase change over a number of time-steps. Calculating the liquid fraction directly from a relationship linked to temperature can lead to large oscillations in the solution as a very small change in temperature can cause an element to change from liquid to solid resulting in a substantial latent heat energy release. To eliminate this type of oscillation a Voller-Prakash correction model is used (see, for example, [Voller and Prakash, 1987], [Voller et al., 1989, 1990, 2004] and [Swaminathan and Voller, 1997], where the accuracy and stability of the model are discussed). This model takes a correction approach in which the liquid fraction can take intermediate values between liquid and solid allowing gradual phase change over a number of time-steps. The loss of heat energy from the food load into the oven cavity occurs through convective and radiative heat transfer. Convective and radiative losses, Qconv and Qrad, respectively, were represented using a surface heat loss boundary condition as follows: Qconv = hc Ts − Tamb ( ) (12) ( ) (13) 4 Qrad = ζθ Ts 4 − Tamb where Ts is the surface temperature, Tamb is ambient temperature, ζ is the Stefan-Boltzman constant and θ is relative emissivity. The convective heat transfer coefficient hc used in (12) has been chosen empirically based upon a Rayleigh number approximation. A significant factor in the heating of food loads is the effect of evaporation. In the later stages of the cooking process a substantial amount of the absorbed microwave energy is used to overcome the latent heat of vaporization International Microwave Power Institute and evaporate water from the food. In order to account for this, an evaporative volume sink term has been implemented in the source term of energy equation (6). The approximation is a simplification in that it assumes evaporation only occurs once the food temperature reaches the boiling point of water. That is, the effects of evaporation below the boiling point are not considered. The sink term effectively caps the temperature in a FV cell to a pre-specified boiling temperature. The energy used in the evaporation process is accounted for in a change in water content. The sink term is switched off if no further water remains in the cell. The PTFE bowl containing the food material has not been considered in the dielectric or thermal simulations. The plastic material occupies very little volume and has low dielectric constant and specific heat capacity. Consequently, the authors have assumed the magnitude of the errors generated by this approximation to be small in comparison with system global approximations; this matter will be addressed in their further work. CROSS-MAPPING ALGORITHM The electromagnetic analysis is carried out on a domain consisting of the oven (containing product) and exciting waveguide, with permittivity and conductivity values mapped directly from the thermal analysis FV mesh of the food load at every thermal time-step. This mapping is re-constructed each thermal timestep allowing the food to rotate either placed centrally on the turntable or offset by a distance from the turntable center. Restricting the thermal analysis domain to the food material permits the entire thermal mesh to be moved with the load as it rotates. A second mapping is required to transfer the predicted power distribution from the electromagnetic FDTD mesh directly to the unstructured FV thermal analysis mesh. Both the electromagnetic and thermal analyses use piecewise constant material 41-4-35 properties, thus the mapping is based on an elementary quadrature approach. The mapped cell average value is an integral of the originating values over the intersection of that cell with the originating mesh. In Figure 2, the rectangular area represents an FDTD cell and the three triangular areas represent cells from the CFD domain. The FDTD cell is recursively subdivided into sixteen sub-volumes, each with a sample point located at its center. The pointin-cell algorithm determines which of the CFD cells contains the sample point. This is repeated for each of the sample points. The cell value of the mapped variables (e.g. loss factor or dielectric constant) is then the average of the values determined at each sample point. In practice, around 100 quadrature points per cell are used in both mapping directions using a FV rule based on recursive volumetric subdivision of each cell. The approach is similar in some respects to the use of overset meshes in CFD [Benek et al., 1983] . Generating an efficient set of quadrature points is straightforward for quadrilateral elements, but more complex for tetrahedral elements, using recursive sub-division for the sample points. An efficient point-in-cell search algorithm (see, e.g., [O’Rourke, 1998]) is used to create fast and conservative mappings between the meshes. SOLUTION PROCEDURE The material properties are initialized based upon the initial temperatures. The initial location of the load is determined and the automesher generates an FDTD mesh. The electromagnetic properties of the load are mapped from the FV food material mesh onto the FDTD mesh, and the FDTD scheme is run until a converged EM solution has been obtained. The power distribution calculated on the FDTD mesh is then mapped back onto the FV mesh. The thermophysical solution is marched forward in time by a predefined time-step. The mapped 41-4-36 Φ FDTD CELL = 5 7 4 Φ1 + Φ2 + Φ3 16 16 16 Figure 2. Illustration of cross-mapping algorithm with 16-point rule. power density is used as a source term in solution of temperature and liquid fraction. After a thermophysical solution has been obtained, the electromagnetic and thermophysical properties are updated using the new temperature distribution. The simulation progresses to the following thermophysical time-step. The load location is recalculated and the EM solver is called again. This process is repeated until the thermophysical solution has reached the userdefined simulation end time. Simulations of both the centric and offset cases consisted of 72 thermophysical time-steps each of 5 seconds duration giving a total heating time of 360 seconds. Each thermophysical time-step was comprised of 200 Jacobi preconditioned conjugate-gradient method solver iterations. The food domain mesh was generated by creating a CAD model of the bowl and food material using PTC Pro/Engineer [PTC, 2007]. This CAD model was meshed using the GAMBIT package [GAMBIT, 1988-2007]. Finally, a custom-made filter, developed by the authors, was used to transfer the mesh into a format compatible with the PHYSICA package. The final mesh consisted of 125,000 tetrahedral cells. Figure 3 illustrates the discretized finite volume mesh used by PHYSICA to describe the bowl geometry. The dimensions of the oven used are shown in Figure 4 while load material properties, Journal of Microwave Power & Electromagnetic Energy ONLINE Vol. 41, No. 4, 2007 determined experimentally at the Unilever Research Center, are given in Table 1. The temperature-dependent material properties are determined by linearly interpolating between listed data points. The EM domain is comprised of two sections – a WR340 waveguide and the applicator cavity. The domain is excited by an incident TE10 field, using a total-scattered field method, with an incident field excitation plane located 1⁄4 of the distance along the waveguide in the Z-direction. The excitation plane emits waves in both directions. In order to prevent reflections from the closed end of the waveguide, a Mur type absorbing boundary condition [Mur, 1981] has been implemented. The FDTD mesh was generated automatically for each FDTD solution, using up to 768,000 cells. All simulations were run on a Compaq Alpha ES45 system with 4 processors operating at 1.0 GHz and 4 GB RAM. The solution was obtained after a runtime of approximately 20 hours. The process was accelerated through the use of Open Multi-Processing (Open-MP) parallelization of sections of the FDTD solver. Open-MP is a set of compiler directives enabling parallelization through multithreading. These directives have been implemented in a manner allowing the field updates within the FDTD Figure 3. Typical mesh for the food geometry in the bowl. solver to be processed by a number of CPUs (or CPU cores) simultaneously. Use of Open-MP resulted in a reduction in FDTD solver runtime of approximately 45%. This performance relates to an update rate of 15.36 million cells per second on an obsolete machine. The use of the automatic mesh generation algorithm ensures optimal mesh sizing through assessment of local wave propagation speed and system geometry, which in turn ensures optimal FDTD time-step length reducing the number of FDTD time-steps required to reach steady state. Figure 4. 3D representation of oven and load. International Microwave Power Institute 41-4-37 Table 1. Material Property Data. Thermal conductivity, W/mK Specific heat capacity, J/kg Density, kg/m3 Solidus temp, K Liquidus temp, K T=223 K T=265 K T=271 K T=363 K T=219 K T=265 K T=280 K T=308 K 991.44 271.3 271.3 211.0 84.5 86.0 43.0 1389.0 3813.0 3800.0 3551.0 EXPERIMENTAL STUDIES Temperature measurements were carried out on samples of a commercially made frozen cheese sauce that has a composition of 73.7% water, 16.8% fat, 4.0% protein and 3.5% carbohydrate. 1% sodium alginate (Maugel DMB from International Speciality Products) was added to prevent convection during heating. The sauce was filled into a commercial bowl of 550 cm3, having a diameter of 150 mm and a height of 50 mm. The dielectric properties of the sauce were measured over a temperature range of approximately -20°C to 100°C, using an HP872C Network Analyser with a 14 mm open ended coaxial probe. Specific heat was measured using a Holometrix QTA Adiabatic Calorimeter and thermal conductivity with a Holometrix COM800 instrument. The microwave oven (Panasonic NNT3 553W) was connected to a stabilized voltage supply (Claude Lyons, TS-3) to remove any variation in magnetron output due to changes in main voltage. The effective power output from the microwave oven was measured by monitoring the temperature rise of a 500 g water load in the bowl. To confirm the power deposition in the actual product, the average temperature of the cheese sauce was measured at one minute intervals using a commercial radiometry system (Loma Scientific). Heating of the cheese sauce was temporarily stopped (approximately 10 seconds) while it was placed in the radiometry chamber for temperature measurement. During the microwave heating, internal 41-4-38 temperature measurements of the sauce were made with fibre optic probes (FISO FOT-L) mounted at specific locations in the frozen sauce. A jig was used to prevent probe movement. The probes were connected to the sensor interface (FISO MWS) which was mounted on the microwave oven so that the sensor probes corotated with the turntable. Thermal images of the heated sauce surface were also obtained (FLIR A40M Researcher camera) at one-minute intervals during heating. A total of 8 fiber optic probes were mounted at specific locations in the frozen sauce. These locations are summarized in Table 2 giving the horizontal and vertical displacements of the probe tips from the bowl center. The general layout of the probes is illustrated in Figure 5. RESULTS OF SIMULATION AND MEASUREMENT Figures 6-7 show the temperature evolution as measured and simulated for each of the probe locations. Figures 6(a) to 6(h) relate to the centered rotation case while Figures 7(a) to 7(h) refer to the offset rotation case. Figures 8 and 9 show relative loss factor and temperature contours at a number of heating times. The figures show rapidly increasing temperatures near the edge of the bowl. Prediction and measurement are in satisfactory agreement, especially in the view of the spread in the experimental replicates. Temperatures rise more slowly in the center of the bowl, due to attenuation of the electromagnetic fields (and the resulting reduction in energy deposition) Journal of Microwave Power & Electromagnetic Energy ONLINE Vol. 41, No. 4, 2007 by surrounding material. The agreement is fair when the sauce is frozen, but deteriorates for longer heating times. Comparing the temperature development between the same bowl locations in the centered and offset cases, it is clear that the preferential edge heating occurs in both cases. In addition, the heating uniformity seems better in the centric case. Model and experiment agree best for short heating times when the sauce is still frozen throughout. The analysis of the dielectric properties of the sauce is displayed in Figure 10. The plots show the rapid variation in dielectric properties evident during the phase change. The function used to govern the phase change rate has therefore significant influence upon the temperature traces. The dielectric properties are related directly to the load temperature. Throughout the phase change region the load temperature, and therefore the dielectric properties, are regulated by the Voller-Prakash liquid fraction function. In this contribution a significant approximation has been made in considering the food load to be a pure material with a single specified melting point. The cheese sauce considered is in fact a complex multi-component material which undergoes phase change in a non-linear manner over a temperature range. The effect of dielectric property variation during heating can further be explored performing a Q-factor analysis. The Q-factor relates the energy stored in the system to the energy dissipated in the system per cycle. The variation of Q-factor over the duration of the heating time is shown in Figure 11. This shows that in the first few seconds of heating the load, which is fully frozen and with near zero loss factor, it is almost transparent. After parts of the load undergo phase change, the Q-factor drops rapidly, eventually settling to a level approximately 10% of the initial value. The FDTD solver compares successive values of the discrete Fourier transform of the electric field as a convergence criterion. A high Q-factor increases the number of time-steps required to reach a steady International Microwave Power Institute Table 2. Monitor Probe Positions Relative to Center of Bowl. Probe # Horizontal (mm) Vertical (mm) 1 2 3 4 5 6 7 8 -65.0 -52.5 -35.0 -17.5 0.0 17.5 35.0 52.5 -15 -15 -33 -15 -25 -15 -33 -15 Figure 5. Layout of probe locations. state and therefore increase the simulation time. This has been evident when assessing the convergence of the initial EM solutions. DISCUSSION The results obtained from the numerical model broadly match those from the experimental work and predict the correct trends both in time and as a function of location. While the agreement is satisfactory for short heating times, beyond the melting transition the discrepancy is increasingly worse. For this, several explanations may be given. First, the actual cheese sauce is a nonhomogeneous medium with locally varying properties. Though these variations are small, in view of the strongly non-linear dielectric properties around the melting transition they 41-4-39 (a) (b) (c) (d) (e) (f) (g) (h) Figure 6. Simulated and experimental temperatures at probe position 1 (a), 2 (b), 3 (c), 4 (d), 5 (e), 6 (f), 7 (g) and 8 (h), centric placement. 41-4-40 Journal of Microwave Power & Electromagnetic Energy ONLINE Vol. 41, No. 4, 2007 (a) (b) (c) (d) (e) (f) (h) (g) Figure 7. Simulated and experimental temperatures at probe position 1 (a), 2 (b), 3 (c), 4 (d), 5 (e), 6 (f), 7 (g) and 8 (h), offset placement. International Microwave Power Institute 41-4-41 (a) (a) (b) (b) (c) (c) Figure 8. The loss factor distribution contour plots after 2 (a), 4 (b) and 6 (c) minutes heating time, centered case. Figure 9. Temperature distribution contour plots after 2 (a), 4 (b) and 6 (c) minutes heating time, centered case. 41-4-42 Journal of Microwave Power & Electromagnetic Energy ONLINE Vol. 41, No. 4, 2007 International Microwave Power Institute Dielectric Constant Temperature (Cº) (a) Loss Factor may initiate local “hot-spots” that accelerate the transition into local runaway heating. This “randomness” in the heating may also explain part of the spread in the experimental traces. Secondly, whereas the real sauce has a clear melting temperature range, in the numerical model a melting point was set at -2ºC. In view of the sensitivity of the eventual temperature distribution to the melting transition, this should be accounted for in future modeling attempts. Third, heating was observed to be very nonuniform, with boiling and thawing processes occurring simultaneously in differing locations within the bowl after approximately 300 seconds of heating. This suggests that more detailed attention should be paid to accurately modeling the thermal boundaries, particularly the evaporative contribution, as these have significant influence on temperature distribution in this case. Fourth, in the current simulation a heating time-step of 5 seconds was employed, which may be decreased to better capture the full effect of the rotating turntable. For heating times beyond 360 seconds the experimental traces show a clear periodicity related to the changing distance to the waveguide. Fifth, the cross-mapping process smears the dielectric interface across FDTD cells adjacent to the interface. This smearing may adversely affect the ability of the EM solver to consider focusing effects of the interface and may result in an inaccurate field distribution within the load. This would result in an inaccurate heating solution in the center of the load – a discrepancy evident in the results presented. Finally, the experimental uncertainty should be considered. In addition to the inevitable scatter in the properties data, probe positional accuracy is crucial in view of the strong local property changes. Temperature (Cº) (b) Figure 10. Temperature variation of relative dielectric constant (a) and the loss factor (b) of the considered cheese sauce. CONCLUSION The results presented demonstrate the feasibility of fully coupled EM and thermo-physical simulations of the microwave heating of a frozen product, and emphasise the importance of accurately accounting for the melting transition and evaporative losses. Future work should include a higher spatial and especially temporal resolution, combined with an improved phase change model, both for the melting and evaporation transition. The approach presented in this contribution is applicable to a wide range of applications. In particular it is suited to problems in which dielectric properties change significantly during the heating process and/or in applications in which the load in moved inside the applicator. 41-4-43 Q-Factor Time (s) Figure 11. Variation of the Q-factor during the heating period. The work presented exploits only a small portion of the capabilities of the PHYSICA solver which is capable of solving a wide range of physical problems such as Newtonian and viscoelastic fluid flow, linear/non-linear structural mechanics and simulation of chemical reactions. The ability of independent meshing with cross-mapping to efficiently couple electromagnetic field analysis with the analysis of a wide range of thermophysical problems means that the approach has potential to be used in the simulation of many practical engineering problems. REFERENCES Ayappa, K.G., H.T. Davis, E.A. Davis and J. 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