The use of dielectric heating (microwave and radio frequency, or RF) can greatly accelerate the thawing process. Microwave and RF are both part of the electromag-netic spectrum, with RF having frequencies below 300 MHz while microwaves have frequencies from 300 to 3000 MHz (these boundaries can be flexible). In RF heating an alternating electric field is generated by plate electrodes on each side of the product, while in microwave heating microwave travels from a separate genera-tor to the product enclosure via a waveguide. Because of its lower frequency and longer wavelength, RF has greater penetration depths (typically tens of cm for RF compared with a few cm for microwave) and can give more uniform heating, but the equipment is more costly. With microwave thawing there is a great risk of un-even absorption and runaway heating, because of higher absorption of microwave in liquid water compared to ice. Any part of the product that thaws first will con-tinue to heat up more quickly and may become cooked before thawing is complete in the rest of the food. This may be remedied by using an intermittent microwave source to give time for the temperature profile to become more uniform. Micro-wave tempering, where the product is brought to a temperature just below melting point in order to soften it for cutting and slicing or before completing the thawing process by conventional means, is much easier to control and hence widely used.
An important objective of the modelling of microwave thawing is therefore to pre-dict the occurrence of local heating.
The heat transport equation can be solved numerically. The key question is how to model the source term Sq to accurately represent the heat generated by the mi-crowaves. There are two approaches (Ayappa et al. 1991; Budd and Hill 2011):
Lambert’s Law and Maxwell’s equations.
6.8.1.1 Lambert’s Law
Lambert’s law states that equal thicknesses of a material absorb equal fractions of incident power, and therefore the power flux I varies exponentially with distance x:
(6.64)
2
( ) 0 x
I x =I e−β
where I0 is the flux at the surface and β is the attenuation coefficient. The inverse of 2β is called the penetration depth. The power absorbed per unit volume is therefore:
(6.65)
Lambert’s law is valid for a semi-infinite solid where there is little reflection on the opposite surface, and is a good approximation when the thickness L satisfies the criterion (Ayappa et al. 1991)
(6.66) where L0 = 0.0008 m. The critical size is several times greater for a cylinder (Olivei-ra and Franca 2002). When the above criterion is not satisfied, Maxwell’s equations must be used.
6.8.1.2 Maxwell’s Equations
Maxwell’s equations describe the interactions between the electric and magnetic fields in space (Feynman 2014). Food is a non-magnetic, electrically neutral me-dium and may be considered non-conducting in a microwave or RF field. For such a material, Maxwell’s equation leads to (Zhang and Datta 2000):
(6.67)
where E is the electric field μ0 the permeability of free space and ε the permittivity of the medium. For an electromagnetic wave with angular frequency ω this leads to:
(6.68)
ω is a complex angular frequency whose real and imaginary parts depend on the dielectric properties ε′ and ε" (see below) of the medium. Zhang and Datta (2000)
(6.70) where ε0 is the permittivity of free space. ε′(or κ′) measures the material’s ability to store electrical energy and ε″ (or κ″) its capacity for energy dissipation. The power dissipation per unit volume, i.e. the microwave heat source term, is given by (Ayappa et al. 1991):
(6.71)
During food thermal processing and especially thawing, food dielectric properties change strongly with temperature (at 2.45 GHz microwave penetration depth is almost three orders of magnitude higher in ice than in water) and therefore the elec-tromagnetic and heat transfer equation are very strongly coupled. A realistic model must take this coupling into account. The property parameters ε′ and ε″ (or κ′ and κ″) for many foods have been measured at several frequencies and listed by Ayappa et al. (1991). A recent review of data can be found in Sola-Morales et al. (2010).
Prediction methods for foods were recently reviewed by Gulati and Datta (2013).
6.8.2 Analytical Solutions
Ayappa et al. (1991, 1997) gave analytical solutions of Eq. 6.69 for a multilayer slab and a cylinder respectively for both Lambert’s law and Maxwell’s equation.
Lambert’s law predicts that power absorption falls smoothly with distance from the surface, while Maxwell’s equation predicts a sinusoidal pattern superimposed on the decreasing trend with distance.
6.8.3 Numerical Solutions
Oliveira and Franca (2002) carried out finite element calculations for the heating of slabs, cylinders and foods of irregular shapes. Maxwell’s equations was discretized and solved along with the heat conduction equation. Changes of dielectric proper-ties with temperature were not taken into account, i.e. Eq. 6.69 was used. Equation 6.71 was used to calculate the heat generation. Good agreement with the analytical solution for a slab was obtained. Oliveira and Franca also simulated the effect of product rotation and intermittent microwave power and found that, when applied in combination, they gave more uniform heating than each separately.
For more realistic predictions, the effect of temperature on dielectric properties and the effect of the design of the microwave equipment and placement of the prod-uct in the cavity on the electromagnetic field must be taken into account. Zhang and Datta (2000) built such a model using two independent commercial FEM packages, EMAS for electromagnetic field calculations and NASTRAN for heat conduction in
0( )
i i
ε ε= +′ ε′′=ε κ′+ κ′′
1 2 q 2
S = ωε′′E
a solid product. The two software packages were coupled at the computer’s operat-ing system level. First the electromagnetic field is calculated (Eq. 6.67), then the absorbed power during a time interval (Eq. 6.71). The temperature profile at that point in time is then calculated by solving the heat conduction equation in NAS-TRAN, then the dielectric properties ε′ and ε″ are calculated for each node and fed to EMAS to compute the magnetic field in the next time step.
Newer FEM packages have multiphysics capability, which means that they can solve several field equations simultaneously so that manual coupling of different software is not necessary. Salvi (2011) compared two multiphysics FEM packages, Comsol and Ansys, on a 3-D problem involving the continuous flow microwave heating of a liquid (fresh water and 1.5 % salt solution), a fairly demanding problem that involves the solution of Maxwell’s equation, heat transfer and fluid flow. For electromagnetic problems, the element size Lel requirement was decided based on the Nyquist criterion Lel < λ/2 where λ is the wavelength. Results of the comparison were:
• Agreement of predictions: the power absorption predicted by Comsol was up to 15 % higher than that by Ansys. Predicted temperature profiles were quali-tatively similar (same hot spot location) but there were discrepancies of up to 25 K for the maximum temperature in the salt solution. The predicted average temperature rise was the same for both packages.
• Ease of use: Comsol was found much easier to use. Ansys required the user to specify two separate grids, one for electromagnetism and another for fluid and heat transport, and the results for each were exported to a separate file.
• Coupling of equations: in Comsol the electromagnetism equations were solved first then heat and fluid transport. Comsol uses more memory and full coupling of the equations was not possible with the computer available (3 GHz Xeon pro-cessor with 3 GB RAM). In Ansys full coupling was carried out with the three equations solved iteratively, resulting in possibly more accurate solution.
• Computation time: 15 min for Comsol, two hours for Ansys (due to the one-way coupling of physics in Comsol and full coupling in Ansys).