CN113015279B - Temperature control method for microwave heating of silicon carbide ceramic based on sharp point mutation model - Google Patents
Temperature control method for microwave heating of silicon carbide ceramic based on sharp point mutation model Download PDFInfo
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Abstract
The invention relates to a temperature control method for microwave heating of silicon carbide ceramic based on a cusp mutation model. Secondly, factors influencing the stability of the system are comprehensively considered, whether sudden jump occurs in the temperature at a certain moment can be judged through numerical calculation of a finite element method, and the critical temperature of thermal runaway is obtained. Then, by adopting a variable index approach law sliding mode variable structure control method and designing a reasonable sliding mode switching surface and control law, the heating process can be accelerated and the materials can be prevented from being burnt. The invention can quantitatively research the condition of thermal runaway occurrence, quantitatively analyze the temperature change condition in the microwave heating process, meanwhile, the sliding mode control method adopting variable index approaching law can change continuously according to the system state purposefully, has high response speed and strong robustness, and can realize dynamic control of the heating process.
Description
Technical Field
The invention relates to a temperature control method for microwave heating of silicon carbide ceramic based on a cusp mutation model, and belongs to the technical field of microwave heating control.
Background
Microwave heating is a clean and efficient heating mode, and is widely applied to industrial production because of the advantages of integral heating, selective heating, easy control of heating, high energy utilization rate and the like. However, in the microwave heating process, the material has different microwave absorption capacity, so that the temperature distribution of the heated material is not uniform, a thermal runaway phenomenon occurs, and the whole process of microwave heating is seriously influenced. The mutation theory solves the discontinuous jump phenomenon in a nonlinear system by describing the mutation phenomenon into seven basic elementary mutations in a mathematical mode, is widely applied to solving the mutation phenomenon in the fields of rock and soil, mining industry, slope, traffic and the like, and can provide theoretical basis and research basis for the research of the mutation phenomenon in other fields. The traditional PID control method is widely used for industrial process control due to simple algorithm, good robustness and high reliability. However, the method has certain disadvantages in practical application, for example, for a complex nonlinear system, when the internal parameters of the system change or are disturbed by the outside, the normal operation of the system is greatly affected, the requirement of a high-performance system cannot be met by adopting the PID control, and the PID control process needs to repeatedly adjust the parameters to select more ideal parameters, thereby increasing the complexity of the control process. For the microwave heating process, as the system has the characteristics of time-varying property, nonlinearity, strong coupling property and the like, an accurate mathematical model is difficult to establish, and the parameter setting result is poor. Therefore, it is difficult to precisely control the microwave heating system using the conventional PID control method. The sliding mode variable structure control method is a special nonlinear control method, can dynamically adjust the sliding mode variable structure along a sliding mode switching surface according to the current state of a system, and finally generates an expected state track.
Disclosure of Invention
The invention provides a temperature control method for microwave heating of silicon carbide ceramics based on a sharp point mutation model, which expresses the temperature mutation phenomenon in the microwave heating process by using the sharp point mutation model through a mutation theory, carries out numerical calculation on the heating process by combining a finite element method, and purposefully controls the temperature of a material to change along the expected temperature rise trend by adopting a variable index approach law sliding mode variable structure control method, thereby realizing stable and rapid heating of the silicon carbide ceramics.
In order to solve the problem that the microwave heating thermal runaway phenomenon is difficult to quantitatively describe, the method adopts a mutation theory to express the phenomenon as a cusp mutation model and obtain a steady-state criterion of a microwave heating system. According to the temperature change condition of microwave heating, the stability judgment of the microwave heating system is realized and the critical temperature value of thermal runaway is obtained by numerical calculation and combination of steady-state criterion. And by combining the variable structure control idea, a proper sliding mode switching surface and a proper control law are designed, and the temperature of the material is controlled to rise stably and quickly.
The technical scheme of the invention is as follows: a temperature control method for microwave heating of silicon carbide ceramic based on a sharp point mutation model comprises the steps of firstly, constructing a potential function of a microwave heating system and carrying out quantitative analysis on a heating process by combining a finite element method to judge whether thermal runaway occurs or not; and then a sliding mode switching surface and a control law are designed according to the sliding mode variable structure control idea to control the heating process.
As a further scheme of the invention, the method comprises the following specific steps:
step 2, finite element analysis microwave heating process; the part comprises two parts: firstly, establishing a microwave heating silicon carbide ceramic geometric model, and considering the influence of constitutive relation, initial conditions, boundary conditions, physical properties of a medium and electromagnetic properties on system stability in the construction process; secondly, calculating according to the material temperature change value to know which time periods are subjected to thermal runaway and which time periods are not subjected to thermal runaway in the heating process, and further obtaining the critical temperature of the thermal runaway;
step 3, sliding mode variable structure control: selecting a variable index approximation law to control the temperature of the material according to the time-varying and nonlinear characteristics of the microwave heating process;
designing a control law of a sliding mode to enable the state of the microwave heating system to reach an equilibrium position within a limited time; sliding mode control sets two temperature thresholds, namely threshold T close to critical temperature in microwave heating process 1 And a maximum temperature threshold T for preventing phase change of the material 2 (ii) a The variable index approach law can improve the temperature rise speed of the material in the initial heating stage, and when the temperature is close to T 1 The time control law acts to make the temperature rise steadily until the temperature approaches to the temperature T 2 。
As a further scheme of the present invention, the specific steps of step 1 are:
according to the action of silicon carbide ceramic and microwave in the microwave heating processBy mechanism, the potential function V is used for representing the change trend of the system state, then the state variables T are respectively derived and the result is 0, and the equilibrium surface equation V' and the cusp temperature expression T of the system are obtained t ；
Potential function:
equation of equilibrium surface:
the sharp point temperature:
wherein f is the frequency of the microwave source; e is the electric field strength; sigma 0 Is the electrical conductivity in vacuum; t is 0 Is the peak temperature; gamma characterization of microwave penetration depth,
R z The height of the material on the z axis is the thickness of the material; t is a unit of e Is ambient temperature; epsilon (T) e ) And σ (T) e ) Dielectric coefficient and conductivity at ambient temperature, respectively; sigma 0 And k are respectively the electric conductivity and the heat conduction coefficient in vacuum;
taylor's expansion of equation (2) with respect to the temperature at the cusp and truncation to a cubic term results:
the standard form of the equilibrium curved surface can be obtained by carrying out variable substitution on the formula (4):
x 3 +ux+v＝0 (5)
in the formula:
u＝3T t γm-1 (7)
v＝3(1+T t γm) (8)
as can be seen from the formula (9), m is a coefficient related to material properties and microwave heating parameters;
in the formula (5), the variables forming the system equilibrium surface equation are divided into a state variable and a control variable, wherein x is the state variable, and u and v are the control variables;
and (3) obtaining an odd point set by differentiating the state variable by the balance surface, and eliminating the state variable by combining the balance surface equation and the odd point set equation to obtain a bifurcation set equation of the mutation model:
Δ＝4(3T t γm-1) 3 +243(1+T t γm) 2 (10)
the combination formula (7) can obtain the steady state criterion for judging the system stability in the heating process:
formula (5) is a standard one-dimensional cubic equation, and by combining formula (11) and the root-finding formula of the equation, it can be known that when u is less than 0 and Δ =0, formula (11) has three real roots, including a double root, and accordingly, the potential function has a minimum value and an inflection point, at this time, the system is unstable, the material temperature is liable to jump, and the jump variable is
A rate of change of temperature of
Therefore, the expression for the critical temperature value of thermal runaway can be obtained:
as a further scheme of the present invention, in the step 2, the system steady-state criterion and the critical temperature expression are derived according to the step 1, wherein the critical temperature expression is a cusp mutation model, the cusp mutation model constructed in the step 1 is selected as a constitutive model of finite element analysis, a geometric model of microwave heating silicon carbide ceramic is established through a COMSOL Multiphysics software platform, and according to a simulation result, the temperature jump time in the heating process can be judged by numerical calculation in combination with the steady-state criterion of microwave heating and the thermal runaway critical temperature expression, and the critical temperature at the time is obtained.
As a further scheme of the present invention, in the step 3, the critical temperature value of thermal runaway obtained by finite element analysis in the step 2 is used as the expected temperature, and the actual temperature T and the expected temperature T are compared e The temperature error of the material is known as e; the temperature error e and the first derivative thereof of the material
As a state variable of a variable index approximation law sliding mode variable structure control strategy, and designing a sliding mode surface switching function based on the deviation amount of the state into
C is Hurwitz matrix C = [ C = 1 ,c 2 ,c 3 …c n-1 ,1] T In sliding mode control, parameter c 1 ,c 2 ,c 3 …c n-1 Are all greater than 0, write the satisfying polynomial p n-1 +c n-1 p n-1 +…+c 2 p+c 1 P is a Laplace operator, and n-1 is a positive integer; and selecting a control law, and dynamically adjusting and transforming parameters of a microwave heating system to change the temperature along an expected track on the premise of not burning the silicon carbide ceramic so as to realize the purpose of rapid heating.
According to the mutation theory and the microwave heating principle, in order to avoid thermal runaway caused by local temperature overheating in the heating process, a sharp point mutation model is adopted to quantitatively research the temperature change condition in the heating process. The critical temperature value of the heated material is obtained according to a finite element method, and the system parameters are adjusted according to the current state (temperature deviation, derivatives of various orders and the like) of the system by combining a sliding mode variable structure control method, so that the control of the material temperature can be realized.
The invention constructs a cusp mutation model of the system and obtains a thermal runaway critical temperature value. Firstly, constructing a potential function according to the total energy of the system in the microwave heating process, then respectively carrying out primary derivation and secondary derivation on the state variable in the potential function and making the result be 0, thus obtaining a balance curved surface and a singularity set equation of the system, simultaneously establishing the equation obtained by the two-time derivation, eliminating the state variable, and obtaining sufficient conditions and necessary conditions for judging the stability of the system, namely system steady-state criterion. And finally, obtaining a thermal runaway critical temperature expression according to the equilibrium surface equation and necessary conditions.
The invention obtains the thermal runaway critical temperature value through finite element analysis. The microwave heating process of the material is simulated through COMSOL Multiphysics finite element software, and the maximum value of the heating temperature can be obtained according to the temperature change condition. And (4) carrying out numerical calculation by combining constitutive relation, initial conditions, boundary conditions, physical properties of the medium, electromagnetic characteristics and the like to obtain the thermal runaway critical temperature value.
The sliding mode variable structure control strategy changes the rising trend of the material temperature. And taking the thermal runaway critical temperature value obtained by finite element analysis as the expected temperature of sliding mode control, and comparing the actual temperature value with the actual temperature value to obtain a temperature error. In the sliding mode control, a material temperature error and a first derivative thereof are taken as state variables to design a sliding mode switching surface, and a proper approach law is selected to design a control law, so that the temperature rise speed of the material is accelerated while the material is not burnt.
The invention provides a stability judgment method of a microwave heating system based on a cusp mutation model, and the control of microwave heating temperature is realized through a sliding mode variable structure control method. The mutation phenomenon is difficult to describe by adopting the traditional calculus method, and the microwave heating process can be expressed into a potential function form through a mutation theory, so that a balance curved surface, an odd point set, a bifurcation set and a critical temperature expression of the model are obtained. Secondly, the maximum temperature value of the silicon carbide ceramic in the heating process can be obtained through simulation by adopting a finite element method, and the thermal runaway critical temperature value is obtained according to a steady-state criterion. Finally, according to the error between the critical temperature value and the actual temperature, a proper control law is designed, and then the microwave heating system is dynamically adjusted and transformed, so that the temperature of the silicon carbide ceramic is changed along an expected track, and the purpose of stable and rapid heating is realized.
The invention has the beneficial effects that:
1. the invention describes the microwave heating process by using a cusp mutation model, can realize the quantitative research of the heating process, can obtain the steady-state criterion of the system and the critical temperature expression of thermal runaway according to the analysis, and provides a comparison basis for the subsequent numerical calculation.
2. The invention adopts a finite element method, carries out simulation heating research on the silicon carbide ceramic through a COMSOL Multiphysics software platform, can judge whether the system is out of control thermally at the moment according to numerical calculation and combination with steady-state criterion, and further obtains the critical temperature value in the heating process.
3. The invention adopts a variable index approach law sliding mode control method, uses the temperature value obtained by finite element analysis as the control basis, controls the temperature of the material to stably rise near the critical value by designing a reasonable sliding mode switching surface and a control law, and finally realizes rapid heating on the premise of not burning the material.
Drawings
FIG. 1 is an overall flow chart of temperature control for microwave heating of silicon carbide ceramic based on a cusp mutation model;
FIG. 2 is a flow chart of the construction of a system cusp mutation model.
Detailed Description
Example 1: as shown in fig. 1-2, a temperature control method for microwave heating of silicon carbide ceramic based on a cusp mutation model includes the steps of firstly, constructing a potential function of a microwave heating system and carrying out quantitative analysis on a heating process by combining a finite element method to judge whether thermal runaway occurs or not; and then a sliding mode switching surface and a control law are designed according to the sliding mode variable structure control idea to control the heating process.
As a further scheme of the invention, the method comprises the following specific steps:
step 2, carrying out finite element analysis on a microwave heating process; the part comprises two parts: firstly, establishing a microwave heating silicon carbide ceramic geometric model, and considering the influence of constitutive relation, initial conditions, boundary conditions, physical properties of a medium and electromagnetic properties on system stability in the construction process; secondly, calculating according to the material temperature change value to know which time periods are subjected to thermal runaway and which time periods are not subjected to thermal runaway in the heating process, and further obtaining the critical temperature of the thermal runaway;
step 3, sliding mode variable structure control: according to the time-varying and nonlinear characteristics of the microwave heating process, selecting a variable index approach law to control the material temperature;
designing a control law of a sliding mode to enable the state of the microwave heating system to reach an equilibrium position within a limited time; sliding mode control sets two temperature thresholds, namely threshold T close to critical temperature in microwave heating process 1 And a maximum temperature threshold T for preventing phase change of the material 2 (ii) a The variable index approach law can improve the temperature rise speed of the material in the initial heating stage, and when the temperature is close to T 1 The temperature of the temperature sensor rises steadily until the temperature approaches the temperature T under the action of the time control law 2 。
As a further scheme of the present invention, the specific steps of step 1 are:
according to the action mechanism of silicon carbide ceramic and microwave in the microwave heating process, the potential function V is used for expressing the change trend of the system state, then the state variables T are respectively derived and the result is 0, and the equilibrium curved surface equation V' and the cusp temperature expression T of the system are obtained t ；
Potential function:
equation of equilibrium surface:
the sharp point temperature:
wherein f is the frequency of the microwave source; e is the electric field strength; sigma 0 Is the electrical conductivity in vacuum; t is 0 Is the peak temperature; gamma represents the penetration depth of the microwave,
R z The height of the material on the z axis is the thickness of the material; t is e Is ambient temperature; epsilon (T) e ) And σ (T) e ) Dielectric coefficient and conductivity at ambient temperature, respectively; sigma 0 And k are respectively the electric conductivity and the heat conduction coefficient in vacuum;
taylor's expansion of equation (2) with respect to the temperature at the cusp and truncation to a cubic term results:
the standard form of the equilibrium curved surface can be obtained by carrying out variable substitution on the formula (4):
x 3 +ux+v＝0 (5)
in the formula:
u＝3T t γm-1 (7)
v＝3(1+T t γm) (8)
as can be seen from formula (9), m is a coefficient related to material properties and microwave heating parameters;
in the formula (5), the variables forming the system equilibrium surface equation are divided into a state variable and a control variable, wherein x is the state variable, and u and v are the control variables;
and (3) obtaining an odd point set by differentiating the state variable by the balance surface, and eliminating the state variable by combining the balance surface equation and the odd point set equation to obtain a bifurcation set equation of the mutation model:
Δ＝4(3T t γm-1) 3 +243(1+T t γm) 2 (10)
wherein the singularity set equation is
The steady state criterion for judging the system stability in the heating process can be obtained by combining the formula (7):
formula (5) is a standard one-dimensional cubic equation, and by combining formula (11) and the root-finding formula of the equation, when u is less than 0 and Δ =0, formula (11) has three real roots, wherein the three real roots include a double root, and accordingly, the potential function has a minimum value and an inflection point, at this time, the system is unstable, the material temperature is easy to jump, and the jump variable is
A temperature change rate of
Thus, a critical temperature value expression for thermal runaway can be obtained:
as a further scheme of the present invention, in the step 2, the system steady-state criterion and the critical temperature expression are derived according to the step 1, wherein the critical temperature expression is a cusp mutation model, the cusp mutation model constructed in the step 1 is selected as a constitutive model of finite element analysis, a geometric model of microwave heating silicon carbide ceramic is established through a cmos Multiphysics software platform, and according to a simulation result, the temperature jump time in the heating process can be judged through numerical calculation by combining the steady-state criterion of microwave heating and the thermal runaway critical temperature expression, and the critical temperature at the time can be obtained.
As a further aspect of the present invention, in step 3, the critical temperature value of thermal runaway obtained by finite element analysis in step 2 is used as the expected temperature, and the actual temperature T and the expected temperature T are compared e The temperature error of the material is known as e; the temperature error e and the first derivative thereof of the material
As a state variable of a variable index approximation law sliding mode variable structure control strategy, and designing a sliding mode surface switching function based on the deviation amount of the state into
C is the Hurwitz matrix C = [ C ] 1 ,c 2 ,c 3 …c n-1 ,1] T In sliding mode control, parameter c 1 ,c 2 ,c 3 …c n-1 Are all greater than 0, satisfy polynomial p n-1 +c n-1 p n-1 +…+c 2 p+c 1 P is a Laplace operator, and n-1 is a positive integer; and selecting a control law, and dynamically adjusting and transforming parameters of a microwave heating system to change the temperature along an expected track on the premise of not burning the silicon carbide ceramic so as to realize the purpose of rapid heating. The sliding mode control is used for setting two temperature thresholds in the microwave heating process: threshold temperature T for approaching critical temperature and preventing thermal runaway 1 And a maximum temperature threshold T for preventing phase transformation of the silicon carbide ceramic caused by overhigh temperature 2 . Correspond toThe whole heating process is divided into two stages: initial heating stage, i.e. heating the material from initial temperature value to T 1 At this time, the variable index approach law can make the material temperature reach T rapidly 1 (ii) a Steady rising phase, i.e. controlling the temperature of the material at T 1 The temperature of the heating furnace is increased smoothly, and the final heating temperature is not higher than T along with the heating 2 。
While the present invention has been described in detail with reference to the embodiments, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.
Claims (1)
1. A temperature control method for microwave heating of silicon carbide ceramic based on a sharp point mutation model is characterized by comprising the following steps: firstly, constructing a potential function of a microwave heating system and combining a finite element method to carry out quantitative analysis on a heating process so as to judge whether thermal runaway occurs or not; further designing a sliding mode switching surface and a control law according to the sliding mode variable structure control idea to control the heating process;
the method comprises the following specific steps:
step 1, constructing a potential function of a microwave heating system to describe the change trend of the system state in the process of microwave heating of silicon carbide ceramic, and deducing to obtain a system steady-state criterion and a critical temperature expression;
step 2, finite element analysis microwave heating process; the device comprises two parts: firstly, establishing a microwave heating silicon carbide ceramic geometric model, and considering the influence of constitutive relation, initial conditions, boundary conditions, physical properties of a medium and electromagnetic properties on system stability in the construction process; secondly, calculating according to the material temperature change value to know which time periods are thermally out of control and which time periods are not thermally out of control in the heating process, and further obtaining the critical temperature of the thermal out of control;
step 3, sliding mode variable structure control: selecting a variable index approximation law to control the temperature of the material according to the time-varying and nonlinear characteristics of the microwave heating process;
designed to be in sliding modeA control law, which enables the state of the microwave heating system to reach a balance position within a limited time; sliding mode control sets two temperature thresholds, namely threshold T close to critical temperature in microwave heating process 1 And a maximum temperature threshold T for preventing phase change of the material 2 (ii) a The variable index approach law can improve the temperature rise speed of the material in the initial heating stage, and when the temperature is close to T 1 The temperature of the temperature sensor rises steadily until the temperature approaches the temperature T under the action of the time control law 2 ；
The specific steps of the step 1 are as follows:
according to the action mechanism of silicon carbide ceramic and microwave in the microwave heating process, the potential function V is used for expressing the change trend of the system state, then the state variables T are respectively derived and the result is 0, and the equilibrium curved surface equation V' and the cusp temperature expression T of the system are obtained t ；
Potential function:
equation of equilibrium surface:
the peak temperature:
wherein f is the frequency of the microwave source; e is the electric field strength; sigma 0 Is the conductivity in vacuum; t is 0 Is the peak temperature; gamma represents the penetration depth of the microwave,
R z The height of the material on the z axis is the thickness of the material; t is a unit of e Is ambient temperature; epsilon (T) e ) And σ (T) e ) Dielectric coefficient and conductivity at ambient temperature, respectively; sigma 0 K is the electrical conductivity and the heat conduction coefficient in vacuum respectively;
the equation (2) is Taylor expanded relative to the temperature at the cusp and truncated to a cubic term, which is simplified as follows:
the standard form of the equilibrium curved surface can be obtained by carrying out variable substitution on the formula (4):
x 3 +ux+v＝0 (5)
in the formula:
u＝3T t γm-1 (7)
v＝3(1+T t γm) (8)
as can be seen from formula (9), m is a coefficient related to material properties and microwave heating parameters;
in the formula (5), the variables forming the system equilibrium surface equation are divided into a state variable and a control variable, wherein x is the state variable, and u and v are the control variables;
and (3) obtaining an odd point set by differentiating the state variable by the balance surface, and eliminating the state variable by combining the balance surface equation and the odd point set equation to obtain a bifurcation set equation of the mutation model:
Δ＝4(3T t γm-1) 3 +243(1+T t γm) 2 (10)
the combination formula (7) can obtain the steady state criterion for judging the system stability in the heating process:
formula (5) is a standard cubic unitaryEquation, combining equation (11) and the root-finding formula of the equation, it can be known that when u is less than 0 and Δ =0, equation (11) has three real roots, including a double root, and accordingly, the potential function has a minimum value and an inflection point, at this time, the system is unstable, the material temperature is liable to jump, and the jump variable is
A rate of change of temperature of
Thus, a critical temperature value expression for thermal runaway can be obtained:
in the step 2, the system steady-state criterion and the critical temperature expression are obtained by deduction according to the step 1, wherein the critical temperature expression is a sharp point mutation model, the sharp point mutation model constructed in the step 1 is selected as a constitutive model of finite element analysis, a geometric model of the microwave heating silicon carbide ceramic is established through a COMSOL Multiphysics software platform, and the temperature jump moment in the heating process can be judged by numerical calculation according to a simulation result and in combination with the steady-state criterion of the microwave heating and the thermal runaway critical temperature expression, and the critical temperature at the moment is obtained;
in the step 3, the critical temperature value of thermal runaway obtained by finite element analysis in the step 2 is used as an expected temperature, and the actual temperature T is compared with the expected temperature T e The temperature error of the material is known as e; the temperature error e and the first derivative thereof of the material
As a state variable of a variable index approximation law sliding mode variable structure control strategy, and designing a sliding mode surface switching function based on the deviation amount of the state into
C is the Hurwitz matrix C = [ C ] 1 ,c 2 ,c 3 …c n-1 ,1] T In sliding mode control, parameter c 1 ,c 2 ,c 3 …c n-1 Are all greater than 0, write the satisfying polynomial p n-1 +c n-1 p n-1 +…+c 2 p+c 1 P is a Laplace operator, and n-1 is a positive integer; and selecting a control law, and dynamically adjusting and transforming parameters of a microwave heating system to change the temperature along an expected track on the premise of not burning the silicon carbide ceramic so as to realize the purpose of rapid heating.
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