Multi-source microwave heating temperature uniformity study based on adaptive dynamic programming

Biao Yang; Feiyun Peng; Ziqi Zhang; Zhaogang Wu; Hongbin Huang; Yuyi Shi; Zemin Han

doi:10.1515/htmp-2022-0303

Article Open Access

Multi-source microwave heating temperature uniformity study based on adaptive dynamic programming

Biao Yang , Feiyun Peng , Ziqi Zhang , Zhaogang Wu , Hongbin Huang , Yuyi Shi and Zemin Han

Published/Copyright: December 31, 2023

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From the journal High Temperature Materials and Processes Volume 42 Issue 1

Abstract

In view of the multi-physical field coupling and time-varying characteristics of the microwave heating medium process, how to dynamically plan the state characteristics of multiple microwave sources and optimize the material temperature uniformity becomes the focus of this article. To this end, first, algebraic graph theory is used to construct the multiple microwave sources as a multi-agent system, and a perfect communication topology is established to ensure the transfer and sharing of information. Second, according to the real-time temperature distribution of the material, an event-triggered adaptive dynamic planning algorithm is used to co-operate with the power input of the multiple microwave sources to ensure that no new hot spots are generated during the optimization of the temperature distribution using the self-organizing properties of the medium. Finally, a numerical calculation model for optimizing a mixture of integer and continuous variables is solved using the finite-element method. The experimental and numerical results show that this article improves the temperature uniformity by 32.4–73.5% and the heating efficiency by 14.3–39.4% compared to the generic heating model, and the feasibility of the method is verified by the different shapes of the heated material.

Keywords: microwave heating; multiple microwave sources; intelligent agent systems; event triggering; adaptive dynamic programming

1 Introduction

As a widely recognized and accepted heating method in industry and life, microwave heating has many advantages [1,2,3]. However, the complex internal heat generation mechanism of microwave heating, the stochastic nature of the hot spot, and the difficulty of solving the infinite-dimensional mathematical model lead to the conventional means of accurately obtaining and optimizing the temperature distribution [4,5,6].

In order to optimize the temperature uniformity of materials during microwave heating, many experts and scholars have carried out a lot of research. The first type of research focuses on improving temperature uniformity by adding mechanical structures to the cavity to change the distribution of the electromagnetic field [7,8]. For example, Wang et al. incorporated a controlled rotating column within a cylindrical cavity and discussed the effect of eccentricity and rotation angle on heating efficiency and uniformity, and finally, the rotation angle was selected using the coefficient of variation of energy loss distribution minimization method, and simulation results showed that this method was efficient and feasible [9]. Yi et al. added a mode stirrer and a conveyor belt to the cavity and used the implicit function and level set methods to simulate multiple sets of microwave heating processes with unequal motion speeds, finally concluding that the heating efficiency and uniformity when the mode stirrer and conveyor belt were operated simultaneously were superior to those when they were operated separately [10]. Wu et al. proposed a new microwave heating model by adjusting the geometry of the microwave cavity to optimize the distribution of the electromagnetic field within the cavity, thereby improving the temperature uniformity of the material, and the simulation experiments verified the effectiveness of the proposed method and discussed the important factors affecting the heating efficiency [11]. Carcia-Banos et al. proposed a new approach based on an adjustable position dielectric plate as a near-field focusing lens within a single-mode microwave applicator to provide a specific temperature distribution within a microwave-heated material [12]. Meng et al. proposed the placement of metal patches on a rotating carrier table as a mode stirrer, effectively utilizing the mechanical structure within the chamber [13]. Experimental results showed a significant effect on material temperature uniformity and heating efficiency and indicated that the effect of rotational speed was minimal when the heating time was greater than the rotational cycle.

The second category is mainly the use of multiple microwave sources or microwave properties to improve material temperature uniformity, which is essentially achieved by changing the distribution of the electromagnetic field within the cavity. For example, Kim et al. used reconfigurable diffraction surfaces to suppress unwanted standing waves within the cavity and discussed how diffraction surfaces can be used for microwave absorption and aggregation. Compared to conventional microwave devices, experimental results show a large improvement in the thermal homogeneity of the device [14]. He et al. used two rotating microwave sources for heating and identified an effective radiation port angle by analyzing the heating efficiency at different angles. Finally, the angle of heating was selected using the gradient descent method to improve the homogeneity and heating efficiency of the material [15]. Li et al. addressed the disadvantages of low power density and rapid surface heat dissipation when heating ultrafine structured materials by designing a double-ridge waveguide operating at 2.45 GHz and promoting rapid heating of ultrafine structures by increasing the electric field strength, resulting in higher power absorption of ultrafine materials [16]. Xiong et al. developed a two-port simulation model based on the finite-element method (FEM) and referenced two pins on the heating cavity. Dual frequency heating at 915 and 2,450 MHz was used, and experimental results showed a large improvement in material uniformity [17]. Boonthum et al. designed a reactor with a four-microwave source that could heat the material SiC to over 1,000°C in a relatively short period of time, but the overall temperature uniformity of the material was low [18]. Yang et al. set up six microwave sources as an intelligent body system to classify different microwave sources into different groups by the different heating effects of each microwave source on the material and switch the network topology in real time in order to optimize the uniformity of the material temperature [19]. The aforementioned methods have improved the material temperature uniformity to some extent, but the starting point is to change the electromagnetic field distribution, not to improve the uniformity based on the real-time temperature distribution of the material. Because the input power of each microwave source has different heating effects on different areas of the material, how to dynamically plan the input power of each microwave source is the focus of this research.

In recent years, multi-agent research based on event-triggered adaptive dynamic programming has seen rapid development in intelligent robotics, quadrotor unmanned aerial vehicles (UAVs), and the control of various temperature fields. For example, Song et al. proposed a modular reconfigurable robot with event-triggered control through adaptive dynamic planning, which has the advantages of high environmental adaptability and flexible task completion compared to traditional robots, and the superiority of its algorithm was verified through simulation experiments [20]. Dou et al. designed a formation controller for the formation control problem of quadrotor UAVs using an event-triggered adaptive dynamic planning-based approach. Combining the algorithm with neural networks, it was experimentally concluded that not only the computational effort and actions of the multi-UAV system were reduced, but it was also important for practical applications [21]. Chen et al. proposed an event-triggered adaptive dynamic planning-based method for optimal control of the temperature field of the Roller Kiln, and finally, the effectiveness of the method was verified by simulation results [22].

This article proposes a new intelligent collaborative heating numerical calculation model with multiple microwave sources to improve the uniformity of material temperature in microwave heating. First, the communication topology between multiple microwave sources is constructed through algebraic graph theory; second, the heating model no longer focuses on optimizing the distribution of electromagnetic fields within the cavity but instead uses event-triggered adaptive dynamic programming algorithm to achieve real-time adjustment of the power of multiple microwave sources based on the real-time temperature distribution of the material. Through self-organizing characteristics, the uniformity of temperature field distribution is achieved. Finally, the FEM was used to solve the numerical calculation of the model, and the model was validated by heating the SiC material and different shapes of the material. The results of experiments and numerical calculations show that the method proposed in this article for the microwave heating process of material temperature uniformity has obvious improvement, and at the same time, the heating efficiency has a greater enhancement.

2 Theoretical knowledge

2.1 Microwave heating calculation equation

Microwave heating is a process involving the mutual coupling of oscillating electric and magnetic fields, and the relationship between the two in space–time can be described by a system of Maxwell equations, which can be expressed by the following equation [23,24,25]:

∇ × H = ∂ D ∂ t α + J e , J e = σ E ,

∇ × E = − ∂ B ∂ t − J m , J m = σ m H ,

∇ · D = ρ e ,

(1) ∇ · B = 0 ,

where H denotes the magnetic field strength, E denotes the electric field strength, B denotes the magnetic induction strength, and D denotes the potential shift vector. J e , J m , ρ e denote the current density, magnetic current density, and charge density, respectively, and σ , σ m denote the conductivity and magneto-resistance of the medium corresponding to the loss in the magnetic field, respectively. For a non-linear medium, D and B can be expressed as follows:

(2) D = ε 0 ε r ̿ E B = μ 0 μ r ̿ H ,

where ε 0 , μ 0 are the vacuum permittivity and vacuum permeability, ε r ̿ , μ r ̿ are the complex relative permittivity and complex relative permeability, and ε r ̿ and μ r ̿ can be expressed as follows [9]:

ε r ̿ = Re ( ε r ̿ ) − j Im ( ε r ̿ ) ,

(3) μ r ̿ = Re ( μ r ̿ ) − j Im ( μ r ̿ ) ,

where e ( ε r ̿ ) , Im ( ε r ̿ ) , e ( μ r ̿ ) , Im ( μ r ̿ ) are expressed as follows:

(4) Re ( ε r ̿ ) = ε r xx ′ ε r xy ′ ε r xz ′ ε r yx ′ ε r yy ′ ε r yz ′ ε r zx ′ ε r zy ′ ε r zz ′ , Im ( ε r ̿ ) = ε r xx ″ ε r xy ″ ε r xz ″ ε r yx ″ ε r yy ″ ε r yz ″ ε r zx ″ ε r zy ″ ε r zz ″ ,

(5) Re ( μ r ̿ ) = μ r xx ′ μ r xy ′ μ r xz ′ μ r yx ′ μ r yy ′ μ r yz ′ μ r zx ′ μ r zy ′ μ r zz ′ , Im ( μ r ̿ ) = μ r xx ″ μ r xy ″ μ r xz ″ μ r yx ″ μ r yy ″ μ r yz ″ μ r zx ″ μ r zy ″ μ r zz ″ .

During the heating of materials by microwave, heat is generated by the dielectric loss P e and hysteresis loss P m of the material:

(6) P = P e + P m = 1 2 ω [ ε 0 Im ( ε r ̿ ) | E | 2 + μ 0 Im ( μ r ̿ ) | H | 2 ] ,

where P is the microwave dissipation power and ω is the angular frequency. The heating of the material is accompanied by a solid heat transfer process, and the temperature distribution of the material can be given by the following equation:

(7) P = ρ C ρ ∂ T ∂ t − κ ∇ 2 T ,

where ρ indicates the density of the material, C p is the specific heat capacity, T indicates the temperature, and κ is the thermal conductivity. Ordinary materials for heating will almost not produce hysteresis loss; thermal energy is mainly generated by the dielectric loss of the medium, so you can make P m = 0 , combined with equations (6) and (7) can be obtained:

(8) ρ C p ∂ T ∂ t − κ ∇ 2 T = 1 2 ω ε 0 Im ( ε r ̿ ) | E | 2 .

From equation (8), it can be concluded that if the heated medium is kept constant, the material temperature distribution can be optimized by adjusting the power and thus the electric field distribution and the angular frequency of the electromagnetic waves.

2.2 Symbols and graph theory

A directed graph G = ( ν , ε , A ) is used to describe the communication topology of multi-agent systems, where ν = { 1 , 2 , … , N } denotes nodes, ε ∈ ν × ν denotes edges, and A denotes the collocation matrix. If node i and node j are adjacent, the edges from node i to node j are denoted by ( i , j ) . A graph G is undirected if ( i , j ) ∈ ε ⟺ ( i , j ) ∈ ε can be introduced for any i , j . A graph is said to contain a directed spanning tree if there is a directed path from at least one node to all other nodes in the graph. If the multi-agent systems contain an external system, it is represented by node 0, while the other agents are represented by nodes 1 , … , N . Then, the pull matrix of the communication topology jointly constructed by the external system and the multi-agents is denoted by L = [ l ij ] ∈ R ( N + 1 ) × ( N + 1 ) , and R n × n denotes the set of n × n dimensional real matrices. When i = j , we have l ij = ∑ j ∈ N i a ij ; when i ≠ j , l ij = − a ij and N i is the neighboring node of node i . The collocation matrix A = [ a ij ] ∈ R n × n , a ij = 1 means that i and j are neighboring nodes and can communicate; if a ij = 0 , then no communication is possible. When the graph G contains spanning trees, the pull matrix can be expressed as follows [26]:

(9) L = 0 0 1 × N L ∼ H ,

where H ∈ R n × n is a non-singular M-matrix, satisfying L 1 N + 1 = 0 , L 1 N + 1 = 0 is a N + 1 dimensional vector.

2.3 Event-triggered dynamic planning with multiple microwave sources

For multi-microwave source agent systems, the kinetic equations can be expressed by [27]

x ̇ i ( t ) = A i x i ( t ) + B i u i ( t ) + B wi w ( t ) ,

y i ( t ) = C i x i ( t ) + D i u i ( t ) + D wi w ( t ) ,

(10) z i ( t ) = C mi x i ( t ) ,

where x i ( t ) ∈ R ni , u i ( t ) ∈ R R i , and y i ( t ) ∈ R q denote the state information, control input, and output of the microwave source power, respectively, and i denotes the number of each intelligent agent. z i ( t ) ∈ R q i and w ( t ) are the measured outputs and system perturbations, respectively, and the matrices A i , B i , B w i , C i , D i , D w i , and C m i are constant matrices. Let the system perturbation signal w ( t ) ∈ R l and the power tracking signal r ( t ) ∈ R q be

w ̇ ( t ) = A w w ( t ) ,

(11) r ̇ ( t ) = A 0 r ( t ) ,

where A w ∈ R q × q , A 0 ∈ R l × l . Combining the perturbed and external signals as follows:

(12) v ̇ ( t ) = A v v ( t ) ,

where A v = diag { A w , A 0 } , defining the microwave source power output regulation as e i ( t ) = y i ( t ) − r ( t ) , E i = B wi 0 , F i = D wi − I , then equation (10) can be written in the following form:

x ̇ i ( t ) = A i x i ( t ) + B i u i ( t ) + E i v ( t ) ,

e i ( t ) = C i x i ( t ) + D i u i ( t ) + F i v ( t ) ,

(13) z i ( t ) = C mi x i ( t ) .

The switching topology of each microwave source is represented by G σ ( t ) , where σ ( t ) is the switching signal for the communication topology. For the design of the event-triggered control mechanism, the external signal v ( t ) is observed before the output regulation controller, and the observer is designed as follows:

(14) η ̇ i ( t ) = A v η i ( t ) + μ K ∑ j ∈ n i σ ( t k i ) a ij σ ( t k i ) ( e A v ( t − t k ′ i ) η j ( t k ′ i ) − e A v ( t − t k i ) ) η i ( t k i ) ) + a i 0 σ ( t k i ) ( e A v ( t − t k i ) v ( t k i ) − e A v ( t − t k i ) η i ( t k i ) ) ,

where k' ( t ) = arg max l ∈ N { l |t ≥ t l i } , t ∈ [ t k i , t k + 1 i ) , t k i is the event trigger time, η i ( t ) is the value of the agent for external observations, μ is a constant, and K is the designed gain matrix. The observer is updated according to the change in the communication topology at the current moment. With the trigger mechanism, the agent broadcasts its state information only at the trigger moment, thus reducing the communication load as well as the network loss. Define the state measurement error between the current moment and the previous trigger moment as follows:

(15) η ei ( t ) = e A v ( t − t k i ) η i ( t k i ) − η i ( t ) .

Let v ̅ i ( t ) = η i ( t ) − v ( t ) denote the tracking error, where v ( t ) = e A v ( t − t k i ) v ( t k i ) , after deriving v ̅ i ( t ) , given the constants γ > 0 , 0 < δ < 1 , α > 0 , P satisfies P A v + A v T P − γ PP + α I = 0 . The event trigger condition is

(16) || η ei ( t ) || < δ α ( δ α + 2 μ | | H σ ( t k i ) | | | | P | | 2 ) | | H σ ( t k i ) | | | | z ∼ i ( t ) | | .

The event trigger condition is judged by an embedded microprocessor mounted on the agent. If the event trigger condition is met, the agent transmits the observation of the current moment to the neighboring nodes; otherwise, no information is transmitted. Let ε i ( t ) be the state observations of the agents, K 1 i , K 2 i be the feedback gain matrices, and H ∼ i be the designed gain matrix, then the adjustment output of the microwave source agent can be written as:

(17) e i ( t ) = C i x i ( t ) + D i K 1 i ε i ( t ) + D i K 1 i η i ( t ) + F i v ( t ) = C i ( x ∼ i ( t ) + X i v ( t ) ) + D i K 1 i ( ε ∼ i ( t ) + X i v ( t ) ) + D i K 2 i ( v ∼ i ( t ) + v ( t ) ) + F i v ( t ) = C i x ∼ i ( t ) + D i K 2 i v ∼ i ( t ) + D i K 1 i ε ∼ i ( t ) + ( C i X i + D i U i F i ) v ( t ) .

3 Model design

The heating model consists of a microwave cavity, four microwave source input ports, a heating medium SiC, and a polycrystalline mullite mat. The resonant cavity is set up with an air domain, the waveguide walls and cavity walls are made of copper, and the microwave sources are fed into a standard waveguide of type WR340 in mode T E 10 at 2.45 GHz, and the initial temperature is heated starting at T 0 = 20 °C.

The heating material SiC and the corresponding parameters in the model are designed as shown in Table 1.

Table 1

Relevant model parameters

Material properties	SiC [28]	Copper	Air
Relative permittivity	ε ′ ( T )	1	1
Loss tangent angle	tan δ ( T )	—	—
Conductivity (W·m⁻¹·K⁻¹)	k ( T )	400	0
Constant pressure heat (J·kg⁻¹·K⁻¹)	C p ( T )	3,640	—
Density (kg·m⁻³)	3,100	8,700	—
Relative permeability	1	1	1
Electrical conductivity (S·m⁻¹)	—	5.998 × 10⁷	—

The following equations are used [28]:

tan δ ( T ) = 5 × 10 − 10 T 3 − 9 × 10 − 7 T 2 + 6 × 10 − 4 T + 0.2801 ,

C p ( T ) = 4 × 10 − 7 T 3 + 17 × 10 − 4 T 2 + 2.3729 T + 115.43 ,

ε ′ ( T ) = − 2 × 10 − 4 T 2 + 0.4503 T + 124.26 ,

(18) k ( T ) = 8 × 10 − 5 T − 0.325 T + 326.69 .

When the temperature of the material changes, the dielectric constant also changes, which in turn affects the ability of the material to absorb electromagnetic waves. Therefore, cavity measurement techniques can be used to estimate the complex dielectric constant of a material of arbitrary shape and size [29]. The measured value of the complex permittivity is brought into equation (18) to know the absorption efficiency of the material electromagnetic waves in different regions of the material and hence the temperature optimization.

Considering the copper material of the cavity walls and waveguide walls as perfect conductors, the electromagnetic boundary condition is expressed by [30]

(19) n → × E = 0 n → × B = 0 ,

where n → is the unit normal vector of the surface of the cavity wall. During the heating process, heat is exchanged between the surface of the material and the surrounding air by convection, and the energy transfer expression is

(20) − k ∂ T ∂ n = h · ( T − T a ) ,

where h is the heat transfer coefficient, T a is the air temperature, and ∂ T / ∂ n is the vertical material temperature gradient (Figure 1).

Figure 1

(a) Geometric model of multi-source distribution, and (b) regional model of material distribution.

First, four microwave source intelligent agents are used to input power of 500 W and operate separately to obtain the influence of different microwave sources at different positions on different areas of the material. The heating effect diagram is shown in Figure 2, and the temperature of each area of the material is shown in Figure 3.

Figure 2

Heating effect of different microwave sources.

Figure 3

Temperature variation of each area heated by different microwave sources.

As can be seen from Figure 2, microwave source 1 and microwave source 3 have a greater increase in material heating efficiency, but the temperature uniformity is poor, while microwave source 2 and microwave source 4 are the opposite, so the synergy of each microwave source power on the optimization of material temperature uniformity and heating efficiency is efficient and feasible.

4 Implementation of adaptive dynamic planning for multi-source microwave power

In the previous section, we verified the different heating effects of different microwave sources on different locations of the material, while the material is divided into four regions and different symbols to indicate the average temperature of each region in order to enable the four microwave source agents to realize state tracking and interference suppression of the external system, the system communication topology is set as shown in Figure 4.

Figure 4

System communication topology.

Nodes 1–4 represent the four microwave source agents, and the L node provides reference trajectories for the other nodes, setting the initial values of the four microwave source agents state x 0 = [ 560 570 595 600 ] T . Through Figure 5, it can be seen that each agent can quickly adjust the current state to reach the expected output state when receiving the signal from the neighboring nodes, meeting the requirement of cooperative output adjustment of each microwave source agent.

Figure 5

Synergistic output of multiple microwave source intelligences.

Set the total power of the four microwave sources to 600 W and take the difference between the average temperature of different areas of the material and the overall average temperature of the material as the event trigger moment. As can be seen earlier, when a single microwave source is used for heating, the heating effect of different regions of the material is different, so the dynamic planning algorithm can be used to dynamically control the input power of each microwave source to optimize the temperature uniformity of the material. At the same time, in order to reduce the coupling phenomenon of multiple microwave sources co-heating, the microwave sources are positioned opposite to each other and perpendicular to each other [18]. Set T − T i ≥ 2 °C ( i represents the i th region), reduce the microwave source power that has a greater impact on the heating effect of the region, and increase the microwave source power that has the largest temperature difference in the reverse direction to optimize the uniformity of the temperature distribution, and vice versa. The whole heating process is shown in Figure 6.

Figure 6

Heating flow diagram.

The temperature sensor is used to measure the material temperature, and Figure 7(a) shows the average temperature values of the entire material and each region in the simulation experiment. Meanwhile, Figure 7(b) shows the difference between the overall average temperature and the average temperature of each region.

Figure 7

(a) Overall and regional average temperature values, and (b) overall and regional differences.

In response to changes in material temperature, the microwave source agent interacts with information via the communication topology to change the input power as a percentage of the total power and thus optimize material temperature uniformity. The power share is shown in Figure 8.

Figure 8

The proportion of input power of each microwave source in the simulation experiment.

5 Numerical calculation results and analysis

5.1 Model parameter design

In the numerical calculation of finite elements, the division of the mesh and time step has a significant impact on the accuracy of the calculation results. In order to select the appropriate mesh and time step, the normalized absorbed power is introduced in this article for its calculation. The normalized absorbed power is defined as the ratio of the absorbed power of the heated material to the total input power of the waveguide, which is shown in equation (21) [31]:

(21) NPA = ∫ 0 t ∫ V Q e ( τ ) d V d τ ∫ 0 t P ( τ ) d τ ,

where t is the heating time and Q e ( τ ) and P ( τ ) denote the electromagnetic power loss density at the moment of τ and the total feed-in power, respectively. Theoretically, the finer the mesh division and the smaller the time step, the more accurate the calculation, but the corresponding computational burden is too heavy and the heating time too long. Therefore, the selection of the grid and time step is very important. After several experimental comparisons, the grid size and time step selected are shown in Figure 9.

Figure 9

NPA values for different numbers of grids and time steps.

5.2 Model validation

To verify the accuracy of the model building, the numerical calculation model was compared with real experiments for heating results. The heated material SiC was first cut into cubic blocks of 50 mm × 50 mm × 20 mm and marked for temperature measurement points. The microwave application unit and the distribution of temperature measurement points are shown in Figure 10(a) and (b), respectively.

Figure 10

(a) Experimental equipment, and (b) distribution of SiC sampling points.

Based on the real-time temperature variation of the material, Figure 11(a) shows the average temperature of each area of the material in the actual heating with the overall. In response to the temperature variation, each microwave source is dynamically planned for the input power ratio, and the results are shown in Figure 11(b).

Figure 11

(a) Overall and area temperature averages in the experiment, and (b) percentage of input power for each microwave source.

By comparing Figure 11 with Figures 7 and 8, it can be seen that due to the time delay of temperature measurement, the actual heating results show a corresponding decrease in average temperature compared to numerical calculations. Due to different event triggering times, the change time of input power between microwave sources is also different. However, according to experimental data, dynamic planning of the input power of the microwave source can adjust the temperature distribution in different regions of the material, thereby improving the temperature uniformity of the material. Figure 12 shows the comparative relationship between simulated data and experimental data.

Figure 12

Comparison of experimental and simulated data average temperature in each region with COV.

5.3 Numerical calculation results and analysis

By analyzing the experimental results in the previous section, it is feasible to study the temperature uniformity of multi-source microwave heating based on adaptive dynamic programming. Therefore, under the premise of inputting the same total power of 600 W, the same model is used to heat a single microwave source (Port 2) with constant power, a single microwave source (Port 2) with variable power, a multi microwave source with constant power, and a multi microwave source with variable power. The heating effect is shown in Figure 13.

Figure 13

Heating effect of different models.

From the experimental results, it can be seen that compared to a single microwave source of constant power or variable power, multiple microwave sources have obvious advantages in heating efficiency and temperature uniformity. For the constant power of multiple microwave sources, although the method in this article does not have obvious advantages in heating efficiency, the effect of optimizing the temperature uniformity of the material is significant.

For quantitative analysis, the temperature uniformity is measured by introducing the coefficient of temperature variation (COV), while the heating efficiency is measured using the ratio of the reflected power to the feed-in power | S 11 | . The corresponding equation is as follows [9]:

(22) COV = n − 1 ∑ i = 1 n ( T i − T a ) 2 T a − T 0 ,

(23) | S 11 | = P r P i = P i − ∫ 0 t ∫ V Q e ( τ ) d V d τ P i ,

where T a , T 0 are the average and initial temperature of the selected area and P r , P i are the reflected and incident microwave power, respectively. Smaller values of COV indicate a more uniform overall temperature of the material, and the corresponding results for the various models are shown in Table 2. The equation R a = 1 − | S 11 | can be used to represent the absorbed power ratio of the sample and the corresponding results are shown in Table 3.

Table 2

COV by region under different heating models

Heating models	Zone 1	Zone 2	Zone 3	Zone 4	Overall material
Single source constant power heating	0.0076	0.0119	0.0085	0.0067	0.0087
Single source variable power heating	0.0035	0.0063	0.0052	0.0043	0.0048
Multi-source constant power heating	0.0071	0.0135	0.0148	0.0098	0.0113
Multi-source variable power heating	0.0025	0.0029	0.0037	0.0031	0.0031

Table 3

R a for each region under different heating models

Heating models	Zone 1	Zone 2	Zone 3	Zone 4	Overall material
Single source constant power heating	0.1197	0.1198	0.1175	0.1162	0.4732
Single source variable power heating	0.1129	0.1137	0.1122	0.1112	0.4500
Multi-source constant power heating	0.1311	0.1343	0.1350	0.1306	0.5310
Multi-source variable power heating	0.1340	0.1347	0.1349	0.1344	0.5380

By analyzing the heating effects under different models as well as COV and R a , it can be seen that the proposed method is significantly better than other heating models in terms of temperature uniformity and heating efficiency. In particular, for the universal heating model (single source constant power heating), not only the temperature uniformity is improved by 32.4–73.5%, but also the heating efficiency is improved by 14.3–39.4%.

5.4 Calculation results for changing the shape of the material

The material was heated in different shapes with the same experimental material, initial temperature, and other conditions, and the heating effect was analyzed as shown in Figure 14.

Figure 14

Heating effect of different shapes of materials.

From the heating effect diagram and the COV calculation results shown in Figure 15, it is clear that the method in this article has significant advantages in heating efficiency and temperature uniformity compared to other heating methods.

Figure 15

Different shapes of material COV.

6 Conclusion

For the study of microwave heating temperature uniformity and heating efficiency, this article proposes a multi-source microwave heating method based on adaptive dynamic planning. A more complete communication topology is first established. A new multi-source dynamic planning heating numerical calculation model is provided for microwave heating. Second, according to the real-time temperature distribution of the material, the microwave source input power is controlled in real time by the event-triggered adaptive dynamic planning algorithm, so as to achieve the improvement of the optimization of the material temperature uniformity. Finally, the FEM is used to accurately calculate the heating efficiency and temperature uniformity, and its validity is verified by heating the material SiC, and the feasibility of the method is verified by heating different shapes of materials. Numerical calculations and experimental results show that the study of temperature uniformity of multi-source microwave heating based on adaptive dynamic planning proposed in this article is executable. Compared with the general heating model, the proposed method improves the temperature uniformity by 32.4–73.5%. Meanwhile, the heating efficiency is improved by 14.3–39.4%.

Acknowledgement

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (62363019, 61863020).

Funding information: This work was supported by the National Natural Science Foundation of China (62363019, 61863020).
Author contributions: Biao Yang: conceptualization, methodology, formal analysis, writing-original draft, writing-review & editing, funding acquisition. Feiyun Peng: conceptualization, methodology, formal analysis, writing-original draft, writing-review & editing. Zhaogang Wu, Hongbin Huang, Yuyi Shi, Zemin Han: formal analysis, writing-review &editing.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All authors can confirm that all data used in this article can be published in High-Temperature Materials and Processes.

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Received: 2023-07-07

Revised: 2023-11-08

Accepted: 2023-12-03

Published Online: 2023-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/htmp-2022-0303

Keywords for this article

microwave heating; multiple microwave sources; intelligent agent systems; event triggering; adaptive dynamic programming

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