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##### Export article ### The Application of MPC in Microwave Heating Process Based on Model Constructed by Lambert’s Law Combined with Temperature

Jianshuo Li, Kai Wang, Xin Shi, Shan Liang, Min Gao
Published: 25 October 2015
Mathematical Problems in Engineering , Volume 2015, pp 1-11; doi:10.1155/2015/610490

Abstract: Microwave heating has been the research hot spot for many years. It has the capability of volumetric heating, which makes it save energy and time compared to traditional heating by conduction. A lot of work has been done to easily and exactly describe power distribution in the heating material. Maxwell’s equations and Lambert’s law are the most common ways. Maxwell’s equation is complicated and hard to apply, while Lambert’s law ignores the temperature influence. For material thickness less than penetration depth, only Maxwell’s equation can accurately solve power distribution. For large thickness material, Lambert’s law combined with regional temperature proposed in this paper can be more precise than only Lambert’s law. But there also exist some differences. To precisely control the heating process and make the whole process safe, this paper proposes the use of model predictive control (MPC) algorithm to make the maximum temperature follow a preset reference trajectory. The simulation results demonstrate that the algorithm can well control the heating process with little difference between the reference trajectory and the practical output.1. IntroductionMicrowave heating has been widely used in our lives, from domestic utilized microwave oven to industrial high-power applications. Compared with traditional heating method, microwave heating saves energy and time and is easier to control. But there exist two main problems holding back its wide applications. Firstly, during the heating process, due to the microwave’s inhomogeneous heating, temperature distribution nonuniformity will make the heating results unsatisfied. Secondly, at some parts of the heating material, sharp temperature rising may happen. Then it may lead to thermal runaway, which is very dangerous in practical applications. It may cause reactants burning or even blow them up.Maxwell’s equation and Lambert’s exponential law form the basis for modeling microwave assisted heating process. According to Lambert’s law, microwave power decays exponentially by the penetration depth into the material. The correctness to calculate power distribution for large thickness material has been proved. While Maxwell’s equation is based on space and time, its calculation characterizes the exact microwave propagation behavior. But the calculation process is complex and needs exact temperature distribution.Maxwell’s equation is commonly used to theoretically explain some phenomenon or validate proposed models, for example, the simulation analysis of the heating process in microwave ovens [1, 2], the relationship between the emerging of resonance and material thickness [3, 4], the temperature field distribution during the thawing process , and the verification of impulse microwave on making heating process temperature distribution uniform ; those researches help people know microwave heating characteristic. But, in practical applications, the exact microwave distribution is hard to obtain. Microwave field distribution is influenced by material permittivity, which directly relates to temperature. The usual way to obtain temperature is by optical fiber thermometer, infrared thermometer, or infrared camera . Those methods are suitable for detecting surface or point temperature. Another way to detect temperature is by ultrasonic sensors, which can obtain the temperature field in the heating material. But there are some problems unsolved to be applied in practical applications . Because of the incomplete temperature field, the exact microwave distribution is basically unknown during the heating process. So, Maxwell’s equation cannot be used to solve power distribution in real time practical applications.Lambert’s law is the common way to calculate power distribution in actual applications, such as food heating, thawing, and drying process [9–13]. But it disregards the influence of temperature which directly relates to permittivity and makes the power distribution quite different with reality. This paper uses Lambert’s law combined with material temperature to analyze power distribution. It makes up the shortcoming of ignoring material regional permittivity. Although Lambert’s law combined with temperature can have a better solution to solve power distribution, there still exist some differences. So, to make the heating process safe and avoid thermal runaway, some control methods have been proposed by former researchers, for example, the global linearizing control of multiple-output-multiple-input microwave assisted thawing process  and the automatic control of microwave heating process . But because of the lacking of power distribution information, some problems remain unsolved to exactly control the heating process.Model predictive control (MPC), an advanced model based control strategy, is well dedicated to solve this constrained problem of ordinary differential equation systems [16–18]. MPC or receding horizon control is used to predict and optimize process performance. The main idea is to solve the manipulated variable input value over a finite prediction horizon at each sampling time. The procedure is reiterated at the next sampling time with the updated process measurements and model parameters. Today, MPC has become an advanced control strategy widely used in industry.According to the above, to analyze microwave assisted heating process and guarantee its safety, MPC algorithm based on Lambert’s law combined with temperature is proposed in this paper. This paper is organized as follows. In Section 2, it mainly deals with the difference among the three ways (Maxwell’s equation, Lambert’s law, and Lambert’s law combined with local temperature) to calculate power distribution. Section 3 introduces the way how control algorithm can be applied in the proposed model to control the heating process. At last, the simulation results of this model and the effect of the control algorithm are revealed.2. Model BuildingThe power distribution model applied in this analysis is shown in Figure 1. The heating material is placed in free space, with permittivity and permeability , which are varying with temperature. is the free space dielectric constant, and is the free space permeability. Assuming that a plane, time harmonic electromagnetic wave of frequency impinges normally upon an isotropic material which fills the region , the transverse electric (TE) wave propagates along the -direction, with perpendicular electric () along the -direction and magnetic () along the -direction. The wave satisfies the first order Mur absorbing boundary condition. The electromagnetic field satisfies the following formulas:Figure 1: Wave propagation model.To solve the electric field within the sample, the solutions of following coupled Maxwell’s equations are required. The heating sample may be made by material with different permittivity. Let us set it to have layers, so each layer can be represented bywhere and . Here denote the interface position. is the propagation constant which depends on dielectric constant and dielectric loss . is the material permeability, which equals one for nonmagnetic heating material. Let us assume each layer has a constant dielectric property; therefore, the solutions of (2) can be represented by propagation and reflection wave asThe electric and magnetic field are continuous at the interface between different layers. That is as follows:By using the interface conditions, the general solution coefficients can be obtained via solving the set of algebraic equations:Let us denote by the incident power, which is the control variable in actual applications. Then the electric amplitude at can be calculated bywhere is the velocity of light.Based on (5) and (6), the absorbed power in the th layer, calculated by Poynting vector theorem, iswhere the superscript “” denotes the complex conjugate.2.1. Case of Lambert’s LawLambert’s law is valid for semi-infinite material, and the power absorbed by the material per unit volume can be represented by where represents the penetration coefficient calculated by the sampling temperature on material surface and represents attenuation factor:2.2. Case of Lambert’s Law Combined with Regional TemperatureThe main characteristic of Lambert’s law combined with regional temperature is that it considers the regional permittivity and attenuation factor. The algorithm uses electric amplitude attenuation instead of power attenuation. The eclectic amplitude in each layer can be calculated by (10) and (11). The penetration depth of each layer is calculated by the corresponding permittivity, and the power dissipation in the material can be solved by (12). The use of this equation needs to sample material temperature. Space distribution of temperature sensors can be set as demanded: 2.3. The Comparison among the Three Methods2.3.1. Material with Constant PermittivityThe permittivity and thermophysical properties of the heating material are referred to in Table 1. The microwave frequency used in this analysis is chosen to be 2.45 Ghz. The initial material temperature is set to be 280 K. The surrounding material is filled with air. The penetration depth can be calculated by , equal to 3.64 cm. For the entire material with the same temperature, the calculation of Lambert’s law and Lambert’s law combined with temperature has the same value. So Maxwell’s equation and Lambert’s law are utilized to calculate the difference in power distribution. As Figure 2(a) shows, for small thickness material, power distribution calculated by Lambert’s law has a huge difference with Maxwell’s equation. That means Lambert’s law cannot be utilized to material thickness less than penetration depth. For thickness a little larger than penetration depth, as shown in Figure 2(b), Lambert’s law can approximately describe power distribution. But it cannot reveal the phenomenon of power fluctuation. For thickness much larger than penetration depth, seeing Figur
Keywords: MPc / wave propagates / optical fiber / Model predictive control / power distribution / heating / temperature / Thawing / Lambert / dielectric loss

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