Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris

Heat Transfer Models for Microwave Thawing Applications S. Curet*1, O. Rouaud,1 and L. Boillereaux1 1 ENITIAA, GEPEA – UMR CNRS 6144, rue de la Géraudière, BP 82225, 44322 Nantes cedex 3, France *Corresponding author: rue de la Géraudière, BP 82225, 44322 Nantes cedex 3, France, [email protected] Abstract: This study deals with numerical and experimental investigations of a microwave thawing process. The heating due to the microwaves is considered as a source term into the heat equation. The microwave power generation is calculated either from Maxwell’s equations and Lambert’s law using COMSOL®3.2. For those two different approaches, the heat source term depends on the dielectric properties of the product. Numerical simulations are compared with experimental results carried out in a microwave guide in fundamental TE1,0 mode working at 2450MHz. The results show the advantages and the disadvantages of each numerical formulation. Although numerical results according to Lambert’s law are comparable to experimental measurements, Maxwell’s equations approach shows that dielectric properties values greatly influence temperature distribution into the product. Keywords: Microwave, thawing, modeling, heat transfer, finite elements

1. Introduction Microwaves are used in many industrial applications where thermal energy is needed. Microwave heating is used in industry for various processes such as sintering, food heating, gluing and for a variety of materials. During a microwave heating process, the energy carried by the electromagnetic wave is converted into thermal energy within a material thanks to its dielectric losses. This process is a good example of a reasonable use of energy comparing to conventional heating systems, especially using fossil energies. The process has also the advantage to have a quick start-up period. However, the most important problem encountered during microwave heating is the heterogeneity of resultant temperature distribution. This phenomenon is due to the strong variations of dielectric properties between frozen and defrosted zones. To avoid hot spots occurring during the microwave thawing,

parameters of the process need to be controlled and mathematical models are required. Modelling microwave heating has been widely used by various authors. Usually the heat generation is modelled using two approaches. The first one consists in solving the Maxwell's equations and the source term is deduced from the electric field (Rattanadecho et al., 2002; Dincov et al., 2004). The second method is known as Lambert's law or similar and consists in using the penetration depth of the microwave inside the product for the heat source term calculation (Ni et al., 1999; Romano et al., 2005). Contrary to the Maxwell's equations approach, it does not require the electric field computation inside the heated materials. Several comparisons between Lambert’s law and Maxwell’s equations are available in the literature (Taher et al., 2001; Liu et al., 2005; Oliveira et al., 2002). Although calculations using Maxwell’s equations are more rigorous, Lambert’s law approach is less complicated and numerical results are comparable with experimental measurements (Liu et al., 2005). If several studies have already compared Maxwell’s equations with Lambert’s law during the heating of cylinders or slabs, few of them have studied microwave thawing of a rectangular piece of material filling the section of a waveguide. In this study, the microwave thawing of a piece of tylose is modelled using COMSOL® Multiphysics 3.2, where the heating due to microwave is introduced in the heat equation by considering Maxwell’s equations and Lambert’s law. Both approaches require the dielectrics properties of tylose, which are considered to be temperature dependent.

2. Method The numerical simulations consist of modelling a sample (86mm×43mm×50mm) which fills a rectangular waveguide.

Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris

3.1 Heat transfer equation Heat transfers are based on the generalised heat equation which depends on thermophysical properties of the product, as follows:

ρ Cpapp

∂T = div.(k∇T ) + Q ∂t

(1)

The effective heat capacity method is ideally suited to analyze thawing in materials that melt over a temperature range (Basak, 1997). Into the general heat equation, Q denotes the internal heat generation source term and quantifies the amount of power which is dissipated into the product by dielectric losses. Figure 1 Model design in 2D

In figure 1 are displayed locations of temperature measurements into the sample during microwave treatment. Thermophysical and dielectric properties of a methyl cellulose gel (tylose) are used for simulations. Microwave generation is a monochromatic wave in the fundamental mode, denoted TE1,0, operating at a frequency of 2.45GHz with an input power fixed at 1000W. Microwave energy is transmitted along the zdirection of the rectangular waveguide (section 86mm×43mm). In order to analyze the process of heat transport due to microwave thawing of a food sample, the following assumptions are introduced: Assumption 1: Since the microwave field is operating in TE1,0 mode, it is propagated in a rectangular waveguide in direction (Oz) with only a component in (Oy) (Jules, 2001). Assumption 2: Variations of electromagnetic field in y direction are not considered. Assumption 3: The product receives the electromagnetic waves by the upper surface. Assumption 4: The product is homogeneous and isotropic. Assumption 5: The thermophysical and dielectric properties are time dependent. Assumption 6: The mass transfer is negligible Assumption 7: The lateral and lower surfaces of the product are perfectly insulated. Assumption 8: The initial temperature of the food sample is homogeneous.

3. Governing equations

3.2 Lambert’s law approach The heat source term calculated from Lambert’s law is obtained without solving the electromagnetic field. The volumetric heating rate according to Lambert’s law can be express using the exponential drop of microwave power from the surface to bottom (Ni et al., 1999):

⎛ z dz ⎞ F0 ⎟ Q( z ) = exp⎜ − ∫ ⎜ ⎟ d p ( z) d ( z ) p ⎝ 0 ⎠

(2)

The penetration depth dp is defined as the distance at which the microwave power represents 1/e of the surface power flux. Mathematically, the penetration depth is linked to the dielectric properties of the product: −1 / 2

2 1/ 2 ⎡ ⎞⎤ ⎛⎛ C0 ⎢ ⎛ ε ' ' ⎞ ⎞⎟ ⎜ (3) ⎜ 2.ε '.⎜ 1 + ⎜ ⎟ dp = − 1⎟⎟⎥ 2.π . f ⎢ ⎟⎥ ⎜ ⎜⎝ ⎝ ε ' ⎠ ⎟⎠ ⎠⎦ ⎝ ⎣ For a sinusoidal wave in TE1,0 mode, the microwave surface flux is expressed by (Akkari et al., 2005):

F0 ( x,0) ≈

⎛ πx ⎞ sin 2 ⎜ ⎟ ab ⎝a⎠

2.Psurf

(4)

The same initial and boundary conditions as Maxwell’s equations approach are applied, excepted for the electromagnetic field which is not solved. 3.3 Maxwell’s equations approach In this approach the electromagnetic field is solved into the product according to the theory of Maxwell’s equations.

Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris

For TE1,0 in plane wave propagation mode, the electric field is solved in COMSOL®3.2 using the following equation:

(

)

σ ⎞ ⎛ −1 ∇ × µr ∇E − ⎜ ε ' r − j ε 0 ⎟k02 E = 0 ω ⎠ ⎝

(5)

The volumetric power absorbed by dielectric material can be calculated from the local electric field strength into the product.

Qabs = ω.E ².ε 0 .ε ' '

(6)

As initial condition, the material is assumed to be at uniform temperature T0≈-20°C. Convective coefficient h=10W.m-2.K-1 is due to natural convective flux at the upper surface. Walls of the waveguide are thermal insulated and perfect electric conductors. Incident electromagnetic wave travelling in TE1,0 mode and 2.45GHz is applied at the top of the product using the “port boundary condition” from Electromagnetic module. The microwave input power is fixed at 1000W.

3. Thermophysical properties Thermophysical properties, i.e. density, thermal conductivity and apparent specific heat were measured in a previous study (Akkari et al., 2005) over temperature range from -20°C to +50°C. Figure 2 displays effective heat capacity of methyl cellulose gel as a function of temperature. In this study, thermophysical properties and dielectric properties are incorporated in COMSOL® Multiphysics using Heavyside functions to model variations as a function of temperature. In figure 3 are displayed dielectric properties of tylose as a function of temperature.

Figure 2 Apparent specific heat of tylose in J.kg-1.K-1 using Heavyside functions

Figure 3 Dielectric properties of tylose as a function of temperature using Heavyside function flc1hs(TTfreezing)

Dielectric properties of methyl cellulose gel are taken from literature in the frozen zone: ε’=6; ε’’=1.5 and ε’=50; ε’’=17 in the defrosted zone (Taoukis et al., 1987).

3. Results and discussion The model has been solved using COMSOL®, release 3.2, Heat transfer equation, Electromagnetic module, 2-D. Time dependent solver is used to model microwave tawing during 45s. In this present work, the direct linear system solver UMF-PACK was used. The quadrangle element was used as the basic element type for the mesh. Mesh refinement is used near the top surface of the product and at the center where temperature gradients are the highest. Maxwell’s equations simulations are very sensitive to mesh size due to numerical computation of electric field into the product. Using mesh refinement, the solution was found to be independent of the grid size. Lambert’s law simulations are independent of the grid size because of the predefined exponential drop of power into the product. In addition to numerical simulations, several experiments are carried out for models validation. Only one of them is presented in this paper in order to compare each numerical approach to experimental measurements. Because of large temperature gradients near the top surface, the first probe T1 is located 5mm under the surface in order to reduce the temperature measurements uncertainties.

Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris

Figure 4 Model and experimental results of temperatures according to Lambert’s law • experimental measurements − numerical simulations

Figure 5 Model and experimental results of temperatures according to Maxwell’s equations • experimental measurements − numerical simulations

Figure 6 Normalized electric field as a function of time and waveguide length 4 shows thermal runaway during microwave thawing with the hottest point situated near the surface of the product and at the centre where the magnitude of electric field is

Figure

maximal. A large temperature gap is observed between the hottest point and the coldest one. On the whole, good agreement is obtained between Lambert’s law and experimental measurements. Small differences between model and experiments are due to dielectric properties value coming from literature (Taoukis et al., 1987). Indeed, literature values do not exactly correspond to dielectric properties of the sample in consideration. To improve accuracy of the model, dielectric properties should have been measured as a function of temperature. Nevertheless numerical results according to Lambert’s law are satisfactory regarding the predicted temperature distribution. As regards the temperature evolution of T1 and T2 in the defrosted zone, Lambert's law provides a linear temperature increase. As a result, Lambert’s law can be used to predict the temperature evolution rapidly because it does not require the electric field computation. In figure 5 are displayed numerical results according to Maxwell’s equations approach. Disagreements between experimental data and numerical model can be observed in the defrosted zone whereas the model agrees with experimental data in the frozen zone. The numerical model developed by solving Maxwell’s equations is very sensitive to dielectric properties value. During a microwave thawing process, a strong variation of dielectric properties is observed during phase transition of the product (figure 3). This phenomenon leads to variation of electric field strength into the product. In this model, dielectric properties of tylose are considered to be constant in the defrosted zone. Regarding differences between model and experimental data, this assumption do not seem to be sufficient to model microwave thawing by Maxwell’s equations. Figure 6 displays electric field computation on a line 5mm under the surface of the product. The normalized electric field is the ratio between the local electric field strength into the product and the maximum initial value of electric field at t=0. First, the electric field is constant with time in the frozen state because the permittivity of tylose does not change. Then, the profile is completely different comparing to the fundamental mode when the dielectric properties change (≈15s). Once the dielectric properties increase, a sharp electric field variation is observed and the product begins to thaw. The changes in dielectric

Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris

properties begin when the temperature reaches 2°C. During phase transition of the product, temperature T1 increases very rapidly due to maximum and non homogeneous local values of electric field near the surface.

4. Conclusion In this paper, we compare different microwave power formulation models dedicated to microwave thawing applications. Lambert’s law is based on the generalised heat equation whereas Maxwell’s equations approach adds the numerical resolution of electromagnetic field. This study proved that COMSOL® is able to model heat transfer with phase change using Lambert’s law or Maxwell’s equations. The dielectric properties need to be known if using Lambert’s law or Maxwell’s equations. However results given by Maxwell’s equations approach are really more sensitive to dielectric properties values comparing to Lambert’s law. In a physical point of view, Maxwell’s equations approach is the most rigorous method to model microwave thawing because it takes into account the variation of electric field into the product. Nevertheless, simulations according to Maxwell’s equations need to be improved by focusing on the dielectric properties values that greatly influence temperature distribution into the product. Disagreement between experimental measurements and numerical results using Maxwell's equations is mainly due to a lack of knowledge on the dielectric properties values. Nomenclature a, b Dimensions of the waveguide (m) C0 Velocity of light in vacuum (m.s-1) Cpapp Apparent specific heat (J.kg-1.K-1) dp Penetration depth of the microwaves (m) E Electric field intensity (V.m-1) f Frequency of electromagnetic radiation (Hz) F0 Microwave power flux at the surface (W.m-2) h Convective heat transfer coefficient (W.m2 -1 .K ) k Thermal conductivity (W.m-1.K-1) L Thickness of sample (m) Pin Incident microwave power (W) Psurf Surface microwave power (W) Q Volumetric heat generation term (W.m-3) T Temperature (°C) T0 Initial temperature of product (°C)

Tf Freezing temperature x, y, z Spatial coordinates dimensions (m)

in

the

three

Greek letters σ Electrical conductivity (S.m-1) ω Pulsation of the microwave radiation (rad.s-1) µr Relative magnetic permeability of the material (H.m-1) ε Complex permittivity ε0 Permittivity of vacuum (8.85×10-12F.m-1) ε’ Dielectric constant (dimensionless) ε’’ Dielectric loss factor (dimensionless) ρ Density (kg.m-3) References Akkari, E., S. Chevallier, and L. Boillereaux, 2005: A 2D non-linear "grey-box" model dedicated to microwave thawing: Theoretical and experimental investigation. Computers & Chemical Engineering, 30, 321-328. Basak, A., 1997: Analysis of Microwave Thawing of Slabs with Effective Heat Capacity Method. AICHE journal, No.7, 43. Dincov, D. D., K. A. Parrott, and K. A. Pericleous, 2004: Heat and mass transfer in twophase porous materials under intensive microwave heating. Journal of Food Engineering, 65, 403-412. Jules, E. J., 2001: Couplages entre propriétés thermiques, réactivité chimique et viscosité des matériaux composites thermodurcissables en relation avec les conditions de leur élaboration fondée sur l’hystérésis diélectrique., Ecole Nationale Supérieure d'Art et Métiers, 244. Liu, C. M., Q. Z. Wang, and N. Sakai, 2005: Power an Temperature distribution during microwave thawing, simulated by using Maxwell's equations and Lambert's law. International Journal of Foof Science and Technology, 40, 9-21. Ni, H., A. k. Datta, and K. e. Torrance, 1999: Moisture transport in intensive microwave heating of biomaterials: a multiphase porous media model. International Journal of Heat and Mass Transfer, 42, 1501-1512. Oliveira, M. E. C. and A. S. Franca, 2002: Microwave heating of foodstuffs. Journal of Food Engineering, 53, 347.

Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris

Rattanadecho, P., K. Aoki, and M. Akahori, 2002: A numerical and experimental investigation of the modeling of microwave heating for liquid layers using a rectangular wave guide (effects of natural convection and dielectric properties). Applied Mathematical Modelling, 26, 449-472. Romano, V. R., F. Marra, and U. Tammaro, 2005: Modelling of microwave heating of foodstuff: study on the influence of sample dimensions with a FEM approach. Journal of Food Engineering, 71, 233-241. Taher, B. J. and M. M. Farid, 2001: Cyclic microwave thawing of frozen meat: experimental and theoretical investigation. Chemical Engineering and Processing, 40, 379-389. Taoukis, P., E. A. Davis, H. T. Davis, J. Gordon, and Y. Talmon, 1987: Mathematical Modeling of Microwave Thawing by the Modified Isotherm Migration Method. Journal of Food Science, 52, 455.