Optimal Control Of Microwave Heating And Thawing

Microwave heating is a kind of heating method that can be converted into heat energy by absorbing microwave energy from an object.As a new type of energy carrier,it has been widely used in many fields such as food heating and thawing,biological sterilization,chemical synthesis and metal smelting.Compared with the traditional heating mode,microwave heating has the characteristics of short heating time,high efficiency and small thermal inertia.Hence,the automatic control skill is able to be used in microwave heating process.However,due to the internal heat energy generated by microwave,the heating of some heated stuff may be uneven or even ”runaway heating”.Therefore,many scientists and technologist study microwave heating control and optimal control problem.Based on the analysis of microwave heating mechanism and the mathematical modelling,the optimal frequency control of microwave uniform heating is studied theoretically.In this paper,we assume that control frequency only depends on position,does not depend on time.Two cases,the dielectric function and electric conduction function depends and does not depend the temperature,are considered.The mathematical model of optimal frequency control is established.Under the reasonable assumptions,the well-posedness of controlled system is proved.Moreover,the new existence theorem of optimal is given and the optimization conditions of the problem is obtained.The content includes the following aspects:1.Mathematics modelling for optimal frequency control of microwave uniform heating.The microwave heating process can be described by the coupled system of Maxwell equations and heat conduction equations.The alternation of electromagnetic field in the object will lead to the collision between molecules and the generation of internal heat source.2.Assume that dielectric function and electric conduction function of controlled system do not depend on temperature.In this case,the controlled system can be described as a linear Maxwell’s equations and weak coupled system composed of strongly nonlinear heat conduction equation.By applying he Lax-Milgram theorem and the theory of monotone operators,we show that there is unique solution for coupled system.Moreover,the regularity of solution is also proved.By the definition of control-to-state operator and the weakly compactness of the operator,we obtain there is a optimal control for the optimal frequency control problem.Furthermore,the optimality conditions for this optimal frequency control is derived by introducing the Fr′echet derivatives of the state to frequency.3.Assume that the dielectric function and electric conduction function depend on the temperature.In this case,the controlled system can be described as a strongly coupled system composed of the nonlinear Maxwell equations and the strongly nonlinear heat conduction equations.According to the existence and uniqueness results of the solution of the weakly coupled controlled system,the existence of the solution of the strongly coupled system is proved by using Schauder’s fixed point theorem.By deducing the energy estimation inequality of the solution,the existence of the solution of the optimal control problem is proved.Furthermore,the optimal condition of the optimal frequency control of the strongly coupled system is derived by proving the Fr′echet derivatives of the state relative to the control.4.Microwave uniform thawing has phase transition process.We get the heat transfer equation for the corresponding temperature by using the enthalpy method.We introduce a function to obtain the parabolic equation of Stefan problem for two-phase flow.We first prove the existence of optimal control of heat source in the heat conduction process of two-phase flow.By considering the existence of a heat source,the electric field intensity.At the same time through the study of Maxwell’s equations,we prove the existence of the border control solutions.The innovation of this paper lies in that it is different from the description of the process of realizing uniform microwave heating through numerical simulation mostly.From the perspective of optimal frequency control theory,this paper studies the feasibility(existence of solution)and optimal condition of realizing optimal control strategy of uniform microwave heating thawing.These studies are the basis of numerical calculation of optimal frequency strategy.This paper assumes that the selection of frequency is only related to location and has nothing to do with time.By proving the regularity of the coupling system of Maxwell equation and heat conduction equation under certain conditions,this paper can show that under this condition,microwave heating will not produce thermal runaway phenomenon.These conclusions can provide theoretical basis for the design of microwave heating device.Because the heat conduction coefficient in the heat conduction equation considered in this paper is related to temperature,the heat conduction equation is a strongly nonlinear equation,which makes it difficult to study the fitness of the solution of the corresponding initial boundary value problem of the coupled system.We use the theory of strong monotone operator to introduce proper workspace and overcome this difficulty.In addition,the paper studies the control variable for microwave frequency,cause state of system control in the system of partial differential equations of coefficient function,brings to the existence of the optimal control solution to prove a lot of difficulties,fortunately,after careful analysis we found equations of some function coefficient is on frequency linear change,thus proves the existence of the optimal control solution;The derivation of variational inequality in the condition of optimal involves a coupled system,which is difficult to obtain.We obtained the variational inequality satisfied by the optimal frequency by analyzing the Fr′echet derivatives of control-to-state and deriving the equation satisfied by Fr′echet derivatives.Finally,a simple summary and working plan in the future is presented.