Multi-Physics Field Computation for Microwave Heating of Multi-Mobile Components Based on Transformation Optics and Implicit Function

Abstract

This paper proposed a method based on transformation optics, implicit function, and level set methods to solve the challenge of multi-physics simulation of a microwave heating cavity with two different motion modes. A 3D computation model with a rotating turntable, a lifting support rod, and a sample is proposed as a detailed demonstration. Based on the theory of transformation optics, the rotating turntable is surrounded by two circles with a time-varying, inhomogeneous and anisotropy medium, and the electric field in the moving region is rotated by controlling the two mediums. The implicit function and level set methods compute the lifting motion by setting the properties of the lifting region as a function of space and time. The correctness of the proposed method is verified by comparing the proposed method’s results with the discrete position’s results, and then its accuracy is further verified by experiment. Subsequently, compared with the implicit function and level set methods only, the proposed method is more accurate. Finally, the effects of lifting motion, rotating motion and lifting motion (i.e., spiral motion) on microwave heating uniformity and heating efficiency were analyzed, respectively.

1. Introduction

Microwave energy has been used in a variety of applications, including radar, the food industry and other fields [1,2,3,4]. Microwave heating is a new heating method due to its high efficiency, environmental protection and selective heating [5,6,7]. However, the nonuniformity of microwave heating limits the application of microwave energy in the food industry [8]. Moving elements, such as turntables and mode stirrers, are usually introduced into the microwave cavities to solve the inhomogeneity of microwave heating [9,10,11,12]. Unfortunately, due to the dynamic movement of the elements in the heating process, the mathematical modeling is complex, so it is still a challenge to accurately compute the microwave heating process in the cavity with moving elements [13].

There is a rotation model designed by Chatterjee et al. [14], and the microwave heating process of liquid rotating with a cylindrical cavity was studied based on Lambert’s law. This brings about the fact that this method is challenging for computing a small penetration depth inside of the sample, and the distribution of electromagnetic field in the microwave oven without load cannot be estimated [15,16,17]. For instance, Plaza-Gonzalez et al. [15,16] proposed a two-dimensional microwave cavity with a laminate mode stirrer, and the finite-element method (FEM) was used to compute the effects of the mode stirrer on the electromagnetic field distribution and heating pattern of the sample inside the cavity. Geedipalli et al. [17] computed the turntable’s rotation inside a microwave cavity by rotating each angular degree of π/12 using FEM. However, this method does not consider that the sample’s dielectric properties will vary with the temperature during the heating process. Subsequently, Pitchai et al. [18] updated the sample’s dielectric properties after each temperature computation step to overcome this defect. This also means the mesh needs to be redivided with each rotation angle, which leads to the mesh not being able to be matched well.

Recently, Ye et al. [19] proposed a more accurate method based on the implicit function and level set methods to compute the microwave heating process of a heating cavity with a turntable. According to this method, the moving region’s properties are a function of space coordinates and time, and the change in parameters represents the elements’ movement. Compared with the traditional method, this method reduces the grid density by 71% and the computing time by 81%. A method based on transformation optics was proposed by Zhu et al. [20], which utilized both a 2D and 3D model with turntables that were computed by FEM. In this method, the moving region is surrounded by a layer of a time-varying anisotropic medium, which can control the electric field. The anisotropic region’s variations represent the rotating motion of elements. However, when computing the 3D rotating turntable model, influenced by the mesh density, they divided a circle into four discrete positions, which means that the mesh needs to be redivided after each π/4 rotation, and deviation might occur.

This paper proposed a method based on transformation optics, implicit function and level set methods to compute the continuous microwave heating process with spiral motion. In the proposed method, the properties of the lifting region vary with the movement equations of space and time. Meanwhile, the rotating region is surrounded by two cylindrical regions of a time-varying anisotropic medium. Compared with one region [20], the two-region transformation medium can solve the problem of continuous rotation computation well and avoid the need for re-meshing with several discrete positions. Namely, this method completely avoids the division of the mesh. Moreover, by using the coordinate transformation method, the electromagnetic field and temperature field can be coupled. In Section 2, a 3D model with a turntable and a support rod is presented, and the theoretical basis is also studied. In Section 3, the computation results based on the FEM of the proposed model are presented, and experiments are carried out to validate the accuracy of the proposed method in the 3D model. Meanwhile, according to this method, the influence of lifting motion, rotating motion, and spiral motion on the heating effect of the sample is analyzed. In Section 4, the conclusions of this paper are summarized.

2. Materials and Methods

2.1. Model Description

This paper built a 3D microwave oven model using COMSOL Multiphysics software 6.0 (COMSOL Inc., Stockholm, Sweden), and Figure 1 shows the computation model’s geometry. The microwave cavity size is 200 mm × 200 mm × 100 mm. A standard BJ26 waveguide is used for power input. Inside the cavity are a glass turntable, a polyethylene support rod and the sample placed on the turntable. The 2.45 GHz electromagnetic wave feeds the waveguide in TE₁₀ mode. The input power is set to 100 W. During the microwave heating process, the support rod can control the rotation of the turntable clockwise or counterclockwise and make it lift in the height range of Δh = 10 mm to 70 mm. The support rod’s lifting speed is v_p mm/s, and the turntable’s rotational rate is ω₀ rad/s. The input parameters of the model are shown in

Table 1. Input parameters.

Property	Applied Domain	Value	Source
Dielectric constant (ε′)	Air	1	COMSOL build-in
	Potato	−6.4 × 10−3 T2 + 2 × 10−1 T + 56.8	[13]
	Turntable	4.2	COMSOL build-in
	Support rod	2.3	COMSOL build-in
Dielectric loss (ε″)	Potato	−10−4 T2 − 1.08 × 10−1 T + 16.1	[13]
	Others	0	COMSOL build-in
Density (kg/m3)	Potato	1050	[13]
Thermal conductivity (W/(m3·K))	Potato	0.64	[13]
	Turntable	1.4	COMSOL build-in
	Support rod	0.38	COMSOL build-in
Specific heat capacity (J/(kg·K))	Potato	4180	[13]

.

2.2. Governing Equations

2.2.1. Governing Equations of Physical Field

The electric field distribution in the microwave heating cavity can be obtained by solving Maxwell’s equation [21]:

$$\nabla \times {\mu }_r^{-1}\times \left(\nabla \times E\right)-k_0^2\left(\epsilon -\frac{j\sigma }{\omega {\epsilon }_0}\right)E=0$$

(1)

where E is the electric field, ${\epsilon }_0$ is the dielectric constant of vacuum, σ is electrical conductivity, ${\mu }_r$ is relative permeability, ${\epsilon }_r$ is relative permittivity and k₀ is the wave number in free space.

In order to realize the coupling of the electric field and heat transfer in the microwave heating process, the microwave dissipative power should be analyzed after computing the electric field. The microwave loss power Q_e(t) in the sample as the heat source can be obtained by the following formula [17]:

$$Q_e=\frac{1}{2}\omega {\epsilon }_0{\epsilon }^{″}{\left|E\right|}^2$$

(2)

where ω is the angular frequency microwave, ε″ is the sample’s imaginary part of dielectric relative permittivity and |E| is the magnitude of the electric field.

The temperature field of the sample is computed by the heat transfer formula [17]:

$$\rho C_p\frac{\partial T}{\partial t}-\nabla \cdot \left(k\nabla T\right)=Q_e$$

(3)

where ρ is the density of the sample, C_p is the heat capacity and k is the thermal conductivity.

2.2.2. Boundary Conditions and Initial Values

In this study, except for the microwave feed port, the cavity walls are defined as perfect electric conductors. The tangential component of the electric field is continuous at the interface. The boundary conditions are as follows [18]:

$$n\times E=0$$

(4)

where n is the normal unit vector of the corresponding wall.

In the heating process, the surfaces of the potato, which exchange heat with the surrounding air by convection, are expressed as [18]:

$$-k\frac{\partial T}{\partial n}=h\cdot (T-T_{air})$$

(5)

where T_air is the temperature of the air, h is the heat transfer coefficient, of which the value is 10 W/(m²·K) and T_air is the temperature of the air. The initial temperature of the sample and the environment are both 293.15 K (20 °C).

Since the heating time is only 20 s, the thermal boundary condition between the sample and the turntable is defined as insulation.

2.2.3. Governing Equations for Lifting and Rotating Motion Computation

In the proposed computation model, the motion of moving components can be decomposed into two modes, rotating motion and lifting motion, and we used two methods to set the governing equations of these two modes of motion separately.

First, we noted that the transformation optics method has a good advantage in computing microwave heating processes with rotating motion [20]. Based on the method, the coordinate transformation is carried out according to the motion mode of the sample, and the tensor parameter of the transform optical region is computed to control the electromagnetic field in the transform optical region. Meanwhile, the implicit function and level set methods can accurately compute the microwave heating process with regular motion modes, such as lifting motion [22], using this method to solve the computation of lifting motion. In this research, the two methods are combined to solve the computation of spiral motion during microwave heating.

To compute the rotating motion, we set up two transform optical regions outside the motion region, as shown in Figure 2a. The coordinate transformation of transform optical regions 1 and 2 is defined as:

$$r^{\prime }=r$$

(6)

$${\theta }^{\prime }=\{\begin{array}{cc}\theta +{\theta }_0& r<a\\ \theta +\frac{1}{2}{f}_1(r)& a<r<b\\ \theta +\frac{1}{2}{f}_2(r)& b<r<c\\ \theta & r>c\end{array}{\theta }^{\prime }=\{\begin{array}{c}\theta +{\theta }_0\\ \theta +\frac{1}{2}{f}_1(r)\\ \end{array}$$

(7)

where:

$$f_1(r)=(r-b)\frac{1}{a-b}\cdot {\theta }_0$$

(8)

$$f_2(r)=(r-c)\frac{1}{b-c}\cdot {\theta }_0$$

(9)

The above equations imply that the turntable region is rotated at the angle ${\theta }_0$ and transforms optical region 1 and region 2, each bearing a rotating angle of $0.5{\theta }_0$.

Based on the coordinate transformation equations, the transform optical region’s tensor parameters can be obtained from [20]:

$$\stackrel{=}{\epsilon }={\epsilon }_{}\stackrel{=}{\eta }$$

(10)

$$\stackrel{=}{\mu }=\mu \stackrel{=}{\eta }$$

(11)

where:

$$\stackrel{=}{\eta }={\stackrel{=}{J}}^T\cdot \stackrel{=}{J}/\det \stackrel{=}{J}$$

(12)

$\stackrel{=}{J}$ is the Jacobian tensor:

$$\stackrel{=}{J}=\left[\begin{array}{ccc}\partial x^{\prime }/\partial x& \partial x^{\prime }/\partial y& \partial x^{\prime }/\partial z\\ \partial y^{\prime }/\partial x& \partial y^{\prime }/\partial y& \partial y^{\prime }/\partial z\\ \partial z^{\prime }/\partial x& \partial z^{\prime }/\partial y& \partial z^{\prime }/\partial z\end{array}\right]$$

(13)

Combining Equations (6)–(13), the $\stackrel{=}{\eta }$ of regions 1 and 2 can be obtained:

$${\stackrel{=}{\eta }}_i=\left[\begin{array}{ccc}{\eta }_{11i}& {\eta }_{12i}& 0\\ {\eta }_{12i}& {\eta }_{22i}& 0\\ 0& 0& 0\end{array}\right](i=1,2)$$

(14)

where:

$${\eta }_{11i}=1+2{\xi }_{i}k+{\xi }_i{}^{2}n$$

(15)

$${\eta }_{12i}=-{\xi }_i{}^{2}k-{\xi }_i(m-n)$$

(16)

and

$$m=\frac{{x^{\prime }}^2}{r^2}$$

(17)

$$n=\frac{{y^{\prime }}^2}{r^2}$$

(18)

$$k=\frac{x^{\prime }{y}^{\prime }}{r^2}$$

(19)

$${\xi }_i=\frac{1}{2}{\theta }_0(t)f_i(r)(i=1,2)$$

(20)

where ${\overline{\overline{\eta }}}_1$ and ${\overline{\overline{\eta }}}_2$ denote the transform tensor of regions 1 and 2, respectively, and ${\theta }_0(t)$ is the rotation angle of the turntable.

Next, we set the implicit function of the lifting motion. As shown in Figure 3, the moving region can be divided into four regions, namely, the air region, the turntable region, the potato region and the rod region. Using the implicit function and level set methods, the lifting region’s dielectric coefficient is set as [22]:

$$\begin{array}{l}\epsilon ={\epsilon }_p\cdot H\left(D_p\left(x,y,z,t\right)\right)+{\epsilon }_t\cdot H\left(D_t\left(x,y,z,t\right)\right)\\ +{\epsilon }_r\cdot H\left(D_r\left(x,y,z,t\right)\right)+{\epsilon }_{air}\cdot H\left(D_{air}\left(x,y,z,t\right)\right)\end{array}$$

(21)

where ${\epsilon }_p$, ${\epsilon }_t$, ${\epsilon }_r$, ${\epsilon }_{air}$ are the dielectric coefficient of the sample, the turntable, the support rod and air, respectively. H is the side function. D_p (x, y, z, t), D_t (x, y, z, t), D_r (x, y, z, t) and D_air (x, y, z, t) are the implicit functions of potato region, turntable region, rod region and air region, respectively. Specific equations for implicit functions of lifting motion in each region are listed in the literature.

2.3. Computation Procedure

In the process of computing the lifting motion by using the implicit function and level set methods, the dielectric constant in the lifting region will be changed. And in the process of computation, it is also necessary to compute the thermal field. Therefore, in the computation, it becomes essential to relocate the dissipated power to the initial position. Meanwhile, since the transform optical regions have realized the sample’s rotating motion, the relocating of the dissipated power pertains only involves the coordinate transformation of the Z-axis, which can be expressed as:

$$\{\begin{array}{l}{x}_1=x_0\hfill \\ y_1=y_0\hfill \\ z_1=z_0+v_p\cdot \Delta t\hfill \end{array}$$

(22)

where (x₀, y₀, z₀) is the initial coordinate of the sample and (x₁, y₁, z₁) is the next time step position of the sample. Following a time step ($\Delta t$), the computation of the electric field distribution is performed. Subsequently, the dissipated power is relocated to the initial position for the computation of the temperature field. Then, the updating of the implicit function and parameters of the transform optical regions, which is performed to move the sample to the next position for the next time step computation, is carried out. The computation process is shown in Figure 4.

2.4. Mesh and Time Step

To obtain the mesh size in the simulation, the normalized power absorption (NPA) is often used to determine the number of elements, which can be obtained from [21]:

$$NPA=\frac{P_{absorbed by the processingmaterials}}{P_{fed into the system}}$$

(23)

When the NPA almost does not change with the number of elements, the computation is considered accurate because the increase in the number of elements does not have much effect on the computation result. Figure 5 illustrates the correlation between the number of elements, NPA and time step. According to this study, the mesh of 1,498,036 elements (including 86,528 swept mesh elements in the sample region, 808,555 tetrahedral mesh elements in two transform optical regions and 602,953 tetrahedral mesh elements in other cavity regions) was determined to be used for computation. Meanwhile, according to Figure 5, the time step is set to 0.5 s.

2.5. Experimental Setup

In order to further verify the correctness and accuracy of the proposed method, an experimental system was established, as shown in Figure 6a. The internal structure of the cavity is shown in Figure 6b. The proposal is validated by comparing the experimental results with the computation results. The microwave is fed into the waveguide through a coaxial line from a microwave solid-state source with an output power of 100 W at 2.45 GHz. The circulator and water load absorb the reflected microwave, protecting the microwave source. A microwave power meter (AV2433) is connected to a dual-directional coupler to measure the effective input and reflected power. The turntable and support rod in the cavity are controlled by the motor to rotate and lift, and a slice of potato is placed on the turntable. There is a cut-off waveguide on the upper surface of the microwave cavity. This setup ensures the prevention of microwave leakage, allowing the use of an optical fiber thermometer to measure the point temperature of the potato slice. The surface temperature image of the potato slice is recorded using a thermal imaging camera (VarioCAM hr inspect 500, InfraTec, Dresden, Germany) with an accuracy of ±0.03 K. The total time of the microwave heating process is 20 s, and during the heating process, the lifting speed and the rotation angular speed are set to 3 mm/s and π/10 rad/s, respectively. Two groups of experiments were carried out. One group used potato with size 25 mm × 25 mm × 10 mm, and the center of the potato was placed 2.5 cm away from the center of the turntable. The other group used a potato with a size of 35 mm × 35 mm × 10 mm and placed the potato in the turntable’s center. During the heating process, a fiber optic thermometer was used to record the temperature at the point we set in the potato. After heating, remove the potato from the cavity and measure its upper surface with the thermal imaging camera.

3. Results

3.1. Model Validation

3.1.1. Electromagnetic Field Validation

To validate the correctness of the proposed method, the electromagnetic field distribution at different times and the port reflection coefficient computed by the proposed method are compared with the computation results of the conventional discrete position method.

The turntable lifting speed is v_p = 0.3 cm/s, the angular velocity of rotation is ω = π/10 rad/s and the electric field distribution after 10 s motion is shown in Figure 7. As shown in Figure 7a, h₀ is the initial height equals 10 mm, the rotation angle of the sample is π and the lifting height h is equal to 3 cm. The electric field plane shown in Figure 7a is the center section plane of the sample, and the center of the sample is located at (0.025, 0) (Unit: m). A position sketch of the sample is shown in Figure 7d. Clearly, the electric field outside the transform optical region computed by the proposed method is consistent with that at the discrete position. Inside the two transform optical regions, due to the medium, the parameters change into time-varying tensor form, and the electric field of region 1 and region 2 rotates with an angle, which, as we know from Equation (22), is π/2. Compared to the electric distribution at a discrete position, the electric field computed by the proposed method rotates with an angle of π, and the electric field distribution of the two inside the potato matches very well. Moreover, in Figure 7a, the discrete position shows the positions of the elements after movement and the modeling situation, while the positions of all elements inside the microwave cavity remain static; that is to say, when using the proposed method to compute, the mesh is only needed once. Furthermore, this method eliminates the correlation between the electric field distribution and the position of the physical domain but becomes a functional form related to the coordinates and time. The results show that the proposed method is feasible.

To further verify the accuracy of the proposed method, the electric field distribution in the central section of the sample at different times is shown in Figure 8. In Figure 8a, the electric field distribution was computed at discrete positions, and Figure 8b is the electric field distribution computed by the proposed method. It should be noted that with the change of time, the discrete positions method needs to reconstruct the model of different time steps and redivide the elements. However, the proposed method only needs to divide the elements once. Moreover, in Figure 8, the electric field distribution computed by the proposed method is in good agreement with the discrete method. Meanwhile, the comparison of port reflection coefficient S₁₁ computed by the proposed method and at discrete positions at different times is also plotted, which is shown in Figure 9. The result shows that the S₁₁ computed by the proposed method fits well with those computed by the discrete positions.

These results show that the electric field distribution of the microwave cavity with moving elements can be accurately computed by the proposed method.

Adjust the sample size to 35 mm × 35 mm × 10 mm and move it to the center of the turntable where the x-coordination and y-coordination are 0. The lifting speed and angular frequency are still v_p = 0.3 cm/s and ω₀ = π/10 rad/s, respectively. The electric distribution at t = 10 s is shown in Figure 10b,c. In Figure 10b, the electric field distribution of the potato central section after 10 s of motion is shown. It is worth noting that the electric field within the transform optical regions computed by the proposed method was rotated. In addition, in Figure 10c, the electric field at the discrete position after rotating 180° counterclockwise is consistent with the electric field in the potato central section computed by the proposed method. Similarly, the port reflection coefficients in this motion case are plotted in Figure 11, from which it can be seen that the S₁₁ of the proposed method matches the discrete positions well. From Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, the proposed method is very consistent with the results of discrete positions, which can verify the correctness of the proposed method.

3.1.2. Experimental Validation

To verify the accuracy of the proposed method, we compared the computation results with the experimental results and the results of the implicit function and level set methods only.

Firstly, the transient temperature at two points of the potato was measured by using an optical fiber thermometer. The initial position of the potatoes is shown in Figure 10d. Figure 12 shows the comparison of results between the two computation methods (the proposed method and implicit function and level set methods only) and the experiments. The two points of the temperature measurement are located on the upper surface of the potato, and their positions are indicated in Figure 12a,b. The results show that the points’ temperature computed by the proposed method is more consistent with the experiment, and the error computed by the implicit function and level set methods is only within an acceptable range. Through the comparison of points temperature, the accuracy of the proposed method is verified, and the results show that the proposed method’s accuracy is better than that of the implicit function and level set methods only.

Second, we compared the computed temperature distribution of the potato with the experimental results. The temperature distribution on the potato’s upper surface is shown in Figure 13 and Figure 14, and we compared the surface temperature distribution of the potato’s upper surface with the proposed method, the implicit function and level set methods only and the experiment.

In Figure 13, the temperature distribution on the potato’s upper surface obtained in the experiments is x-y axisymmetric, which is consistent with the results computed by the proposed method. However, the temperature distribution on the potato’s upper surface computed only by the implicit function and level set methods do not present such a symmetrical pattern. Moreover, it is noted that the temperature at the center of the temperature distribution on the potato’s upper surface computed by the proposed method is lower than the surrounding area, presenting a circular area of low temperature. The experimental results also show that the temperature at the center is lower, but the cold spots are not too concentrated compared with the proposed method. The reason for this may be that it takes about 5 s for the microwave-heated potatoes to be taken out of the microwave cavity to take a picture of the temperature distribution with a thermal imaging camera during the experiment, which causes the potato temperature distribution to diverge. From the temperature value, the results of the experiment are lower than the temperature value of the computation results. The possible reasons for this phenomenon in the experiment are mainly due to the following reasons: (1) the initial temperature of the potatoes during the experiment being slightly lower than the computation temperature; (2) the fact that the thermal convection during the experiment is greater than the thermal convection parameter set for the computation and (3) there is a time delay in taking pictures of heated potatoes with a thermal imaging camera, which results in the lowering of the temperature of their surfaces.

In Figure 14, the temperature distribution of the computation results and experimental results are concentrated in the center of the upper surface of the potato, and it is obvious that the computation result of the proposed method is more consistent with the experimental result. Likewise, the temperature distribution of the experimental result is more divergent than that of the computation, and the temperature value is slightly lower than that of the computation, which is also caused by the three main influencing factors mentioned above that affect the results of the experiment.

Generally, the computed results of the proposed method are consistent with the results of the experiment, demonstrating the accuracy of the proposed method. Furthermore, the computation accuracy of the proposed method is further improved compared to that of the implicit function and level set methods only, which can more accurately compute the microwave heating process with spiral motion.

4. Discussion

4.1. Comparison of Computation Details of Conventional Method

Since time discontinuity can exist in electric field computation, time must be continuous during temperature computation. Therefore, the conventional FEM is not able to compute the electro-thermal coupling computation. The transformation optics method is limited by the derivation of the formula for spiral motion upward. Therefore, in this section, the comparison between the proposed method and the implicit function and level set method is focused on. In the proposed method, the rotating motion is changed by controlling the electromagnetic field in the transform optical region, which means that there’s no need to consider the parameter changes of the sample due to rotating. In addition, based on the sample being a regular cube, the lifting motion region can be divided into a regular cube, and the lifting motion region can be divided using the sweeping shape elements, as shown in Figure 15a. Meanwhile, the implicit function and level set method needs to change the parameters of the entire rotating and lifting motion region; if the entire region is mesh-segmented using the sweeping shape elements, the cylindrical motion channel boundaries will be mesh-constructed incorrectly. Therefore, only the motion region can be meshed in free tetrahedral form, as shown in Figure 15b. This method brings a problem that the regular cube samples will have parameter motion due to the rotating motion, and furthermore, there is a non-smooth of the sample boundaries during the electric field computation, as shown in Figure 16, which needs to be improved by increasing the number of elements.

In this paper, a computer model DELL XPS8950 is used, which has an Intel Core i9-12900K Processor (3.9 GHz, 16 Cores), 64 GB of DDR5 memory running at 4400 MHz, a 1 TB PCIe SSD for storage, and an 8 GB NVIDIA GeForce RTX 3060 Ti graphics card. The operating system used was Windows 11 (21H2). The Root Mean Square Error (RMSE) is commonly used to compare the errors between methods; we use the RMSE of S₁₁ to compare the accuracy between the proposed method and the RMSE is defined as [23]:

$$RMSE=\sqrt{\frac{{\displaystyle \sum _{a=1}^m{\left|S_{11}{}_{{}_{discrete}}-S_{11}{}_{{}_{method}}\right|}^2}}{m}}$$

(24)

where m is the total number of S₁₁ at different times during the movement in a circle. The smaller RMSE implies a smaller computation error. The comparison of computational details of the proposed method with the implicit function and level set method under this computer as shown in

Table 2. Computational details of the heating process.

Method	Elements	Time (s)	RMSE
Proposed method	1,498,036	2937	0.1321
Implicit function and level set method 1	1,492,215	2041	0.2128
Implicit function and level set method 2	2,826,543	7984	0.1330

. It can be seen that when both have the same number of elements, the computational efficiency of the proposed method is slightly lower, but the RMSE of S₁₁ obtained with the discrete position method is lower. Moreover, when the RMSE of the two methods is close, the element number of the proposed method is decreased by 1,328,507, and the computation time is decreased by 62.8% compared with the implicit function and level set method.

4.2. Microwave Heating on the Sample with the Lifting Motion

During the experiments, potatoes were heated in a spiral motion at the same time. These two movements could potentially have distinct effects on the energy absorption and temperature distribution of the potato, so it is necessary to study the two movements separately. In this section, the effect of only lifting motion on the potato is analyzed in detail.

Figure 17a–c shows the temperature distribution of the potato’s upper surface, lower surface and central section. The potato (25 mm × 25 mm × 10 mm) was placed in the turntable’s center at different heights. Figure 17d,e shows the temperature distribution of the potato lifting motion only at different lifting speeds. The average body temperature (T_a) is often used to evaluate the heating efficiency of the sample, T_sa is the surface average temperature and the COV is the covariance of variation in the potato temperature. The COV is defined as [19]:

$$COV=\frac{\sqrt{\frac{1}{N}{\displaystyle \sum _{j=1}^N{\left(T_j-\overline{T}\right)}^2}}}{\overline{T}-T_0}$$

(25)

where $T_j$, $\overline{T}$, $T_0$ the point temperature of the selected region, average temperature and initial temperature. In general, a lower COV means better heating uniformity.

As can be seen from Figure 17a–c, when the potato and turntable are at different heights, the average temperature and the potato’s COV after heating for 20 s are different. When the potato is placed at a height of Δh = 30 mm, the potato has a relatively lower average temperature compared with that in Figure 17a,c. When Δh = 70 mm, the potato gains a relatively higher average temperature, as shown in Figure 17c. From here, we see that the heating efficiency at a fixed height is different. Compared to fixed-height heating, lifting motion enables the sample to traverse the entire height range during microwave heating, enabling the heating efficiency to be controlled within a reasonable and effective range without selecting the optimal heating height. Obviously, as shown in Figure 17c,d, the sample’s T_a after lifting motion is between the maximum and minimum value of the temperature heated at a fixed height.

Because of the different height positions, the pattern of the internal electric field in the sample varied, as well as the COV after being heated for 20 s. Generally, the lifting motion can help improve the heating uniformity, as shown in Figure 17, and the value of the potato’s COV when heated after the lifting motion is generally lower than that of the potato when heated at a fixed height. The range of decrease in COV is 7.0% to 21.1%. Only when compared with the corresponding heating height in Figure 17c is the negative optimal heating uniformity (−18.2–21.2%) achieved by the lifting motion. In addition, it can also be seen from Figure 17d,e that the speed of lifting motion has little influence on the distribution of temperature field, average temperature and heating uniformity.

4.3. Microwave Heating on Sample with Rotating Motion

The potato’s temperature distribution after microwave heating with a rotating motion for 20 s is shown in Figure 18b–e. Compared with Figure 17a,b, the rotating motion can significantly decrease the COV (the range of the decrease in COV is 1% to 54.8%). Meanwhile, we found that the sample rotation in one circle decreased the COV better than the sample rotation in the half circle, as shown in Figure 18b–e. At the same height, the decrement in COV in one circle compared to the rotation in the half circle is 29.9% (Δh = 10 mm) and 22.0% (Δh = 70 mm). However, whether the sample rotates half a circle or one circle, the range of decrease in COV is 30.2% to 51.0% and 30.3% to 45.6% compared with the sample being stationary, which is a significant improvement. In addition, the rotating motion can increase the average temperature of the sample to some extent. Comparing Figure 18a and Figure 18f with Figure 18b,c and Figure 18d,e, respectively, the range of T_a of the sample is 5.25% to 7.71% and 6.89% to 6.95%. It can be found that the temperature distribution of the sample is symmetrical in microwave heating with rotating motion, which might significantly improve the heating uniformity.

4.4. Microwave Heating on the Sample with Spiral Motion

From the previous section, we found the sample’s heating efficiency is affected by height. By controlling a reasonable lifting motion, the heating efficiency can be improved. Meanwhile, the sample’s heating uniformity can be improved by rotating motion. This section analyses the influence of microwave heating on spiral motion. Figure 19a–f shows the temperature distribution of the horizontal sections of the potato after heating for 20 s at different rotating speeds and lifting speeds. It can be seen from the figure that under the premise of the same lifting motion range, the temperature distribution and heating efficiency of the sample are similar. The use of the spiral motion is proven to be an effective heating method for decreasing the COV (the range of decrease in COV is 9.1% to 61.5%) and improving the average temperature (the range of improvement in heating efficiency is −3% to 15.6%). Compared with Figure 18a–f, in most cases, the spiral motion can further improve the heating uniformity compared with the rotating motion only, and it can be seen from the temperature distribution images that the hot spots in Figure 19 are not as prominent as those in Figure 18. Moreover, the temperature distribution of the sample after the half-circle rotation motion is shown in Figure 19a,d. It is worth noting that a half-circle rotating motion does not increase heating uniformity much more than one circle motion, which is consistent with the conclusion in the previous section. In general, we should not expect a good heating efficiency for the sample at a constant height. Therefore, high heating efficiency and good heating uniformity can be obtained by Spiral motion.

5. Conclusions

This paper proposed a method that combines the implicit function and transformation optics to compute the heating cavity with two types of moving elements and solves the challenge of the multi-physical field in the microwave heating process with spiral motion. The rotating motion of samples was computed by the transformation optics method, and the lifting motion of the sample was computed by the implicit function and level set methods. A 3D computation model was proposed, and the proposed method’s correctness was verified by computing regular samples. The outstanding contributions of this paper are that (1) we propose to use two transform optical regions for computing the rotating motion of the sample, and we propose to couple the transformation optics method with the implicit function and level set method. The results show that the two methods are well coupled. (2) The accuracy of the proposed method was verified through experiments, and the computation accuracy of the proposed method was better than that of the implicit function and level set method only. Moreover, we compared the computational details between the proposed method and the implicit function and level set method, and the results show that the proposed method takes longer but has a lower RMSE when both methods have the same number of elements. When the two methods have similar RMSE, the proposed algorithm takes 62.8% less computational time. (3) We used the proposed method to compute and analyze the effects of fixed-position heating, only rotating heating, or only lifting heating and lifting rotary heating on the heating uniformity and heating efficiency of samples. The results show that although the heating efficiency of the sample may be higher in a few locations, the hot spot distribution is only on one side, and the rotating motion can improve the heating uniformity of the sample more significantly. Therefore, the two motion modes can be combined in the heating process so that the spiral motion can further improve the heating uniformity and heating efficiency of the sample. However, the proposed method is currently only able to compute the spiral heating motion of a regular sample due to the limitation of the need to characterize the boundaries of the sample as a function.

The proposed method combines the transformation optics method with the implicit function and level set methods and provides a case for solving other microwave heating cavities with different moving elements. In the future, it is hoped that the proposed method can be applied to the multi-physical field computation of complex samples.