Flame temperature distribution reconstruction method based on complex permittivity model

Abstract

The present application relates to a kind of flame temperature distribution reconstruction methods based on complex dielectric constant model, comprising the following steps: first, the relationship model between relative dielectric constant ε r And conductivity σ With flame thermodynamic temperature T It is constructed;Second, a plurality of groups of flame different positions relative dielectric constant, conductivity and temperature data are collected, and the coefficient in the relationship model is determined;Third, the relative dielectric constant and conductivity distribution of flame are obtained using complex value capacitance tomography system, and the temperature of flame is calculated based on the relationship model to reconstruct the temperature distribution of flame.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of flame detection, and relates to a flame temperature distribution reconstruction method based on a positive ion and electron complex dielectric constant model. BACKGROUND

[0002] According to statistics, more than 80% of energy conversion is realized by combustion methods, and combustion is ubiquitous in industrial production, aerospace and environmental engineering. Combustion is usually accompanied by flame, so flame is closely related to the combustion state. Flame temperature is an important parameter reflecting the combustion process. Accurate measurement of flame temperature is of great significance to improving combustion efficiency, studying combustion reaction mechanism and reducing pollutant emissions in the combustion process. At present, the measurement of flame temperature mainly includes contact and non-contact methods. The contact temperature measurement methods mainly include thermocouple method and optical fiber method. Non-contact temperature measurement methods are particularly concerned due to little disturbance to the flame combustion state, and common non-contact temperature measurement methods include acoustic method, optical method and radiation method.

[0003] At present, most flame temperature measurement methods are based on optical and thermal parameters of flame. In addition, the electrical parameter of flame, such as complex dielectric constant (conductivity and relative dielectric constant), has also been used for flame temperature measurement in recent years. Hu et al. [1] Based on experimental data, the limit gradient enhancement (XGBoost) regression modeling method is used to fit the mapping model between the flame complex dielectric constant and the temperature, the mapping relationship between the flame relative dielectric constant and the conductivity and the temperature is established, and the flame temperature field distribution is reconstructed based on the mapping relationship. However, the mapping relationship is completely based on experimental data and is not considered from the mechanism, so its applicability is limited.

[0004] Jia Yunhao et al. [2] Based on the mechanism model of flame complex dielectric constant, the dielectric constant is used to calibrate the flame temperature, but the error is relatively large. The mechanism model used is the expression of flame complex dielectric constant related to flame temperature, electron temperature and electron density obtained by considering only electrons. However, while considering the effect of electrons on the flame complex dielectric constant, the contribution of ions to the flame complex dielectric constant cannot be ignored [3] , Wang et al. [4] derived a flame complex dielectric constant expression that comprehensively reflects the effects of positive ions and electrons. In this flame complex dielectric constant model, the complex dielectric constant is related to temperature, pressure, electron number density and positive ion number density, so the model can be used for flame temperature estimation. However, there is no analysis and research on flame temperature estimation and temperature distribution reconstruction based on the complex dielectric constant model.

[0005] REFERENCES

[0006] [1]. Hu D, Tian Y, Chang L, et al. Estimation of combustion temperature field from the electrical admittivity distribution obtained by electrical tomography[J]. IEEE Transactions on Instrumentation and Measurement, 2020, 69(9): 6271-6280.

[0007] [2]. Jia Y, Chen Q, Mao X, Liu J, Liu S, et al. Theoretical analysis of plasma flame dielectric properties and measurement research by electrical capacitance tomography[J]. Combustion Science and Technology, 2015, 21(04): 370-377.

[0008] [3]. Wang C, Jin S, Cao X, et al. Improvement of Flame Complex Permittivity Model Considering Positive Ions and Electrons[C]. 2022 IEEE International Instrumentation and Measurement Technology Conference (I2MTC). IEEE, 2022: 1-5.

[0009] [4]. Wang C, Jin S, Zhen Z, et al. Flame Complex Permittivity Model Considering Electrons and Positive Ions[J]. IEEE Transactions on Instrumentation and Measurement, 2023, 72: 1-10. SUMMARY

[0010] The application is based on a flame complex permittivity model considering positive ions and electrons, and proposes a method for obtaining flame relative permittivity and conductivity and reconstructing flame temperature distribution using a complex value electrical capacitance tomography system. The technical solution is as follows:

[0011] A flame temperature distribution reconstruction method based on a complex permittivity model, comprising the following steps:

[0012] First, construct the relative permittivity εr The relationship between electrical conductivity σ and flame thermodynamic temperature T is represented by the following model:

[0013]

[0014] The second step is to collect several sets of relative permittivity, conductivity and temperature data at different locations of the flame, and determine the coefficients a, b, c and d in the relationship model.

[0015] The third step involves using a complex-valued capacitance tomography system to obtain the relative permittivity and conductivity distribution of the flame, and then calculating the flame temperature based on a relational model to reconstruct the flame temperature distribution.

[0016] Furthermore, in step (3), the problem of solving for the flame thermodynamic temperature T is considered as solving for the numerical solution of the equation. Solving for the numerical solution of the equation is transformed into finding the zero point of f(T). In f(T), only the flame thermodynamic temperature T is unknown. The expression for f(T) is:

[0017]

[0018] Furthermore, it includes the following steps:

[0019] (1) Set the initial value of flame thermodynamic temperature T0, and determine the search interval and step size;

[0020] (2) Expand the intervals in both positive and negative directions of the initial value of flame thermodynamic temperature T0. According to the zero point theorem, determine the range of intervals containing a unique zero point. Perform interpolation within this range to further narrow the range containing the zero point. Repeat the interpolation process and iterate the intervals to obtain the zero point, i.e., the estimated value of flame thermodynamic temperature T.

[0021] (3) For the relative permittivity and conductivity at different positions of the flame, the corresponding temperature estimates are obtained by solving them one by one according to the above process, thereby reconstructing the temperature distribution of the flame.

[0022] Furthermore, a model relating the relative permittivity, conductivity, and flame temperature was constructed, using the expression for the complex permittivity of the flame that considers both positive ions and electrons:

[0023]

[0024] Wherein, the real part of the complex permittivity is the relative permittivity, and the imaginary part is the conductivity; therefore, the relative permittivity ε r The expression for conductivity σ is:

[0025]

[0026] Where, n e For electron number density, n iγ is the positive ion number density, m is the electron mass, M is the positive ion mass, ε0 is the vacuum permittivity, q is the unit charge, ω is the excitation frequency, and γ is the positive ion number density. e It is the frequency of electron-neutral particle collisions, γ i It is the collision frequency of ions and neutral particles.

[0027] γ e With γ i The expression is

[0028]

[0029] Where p is pressure, T is thermodynamic temperature, d is the diameter of the neutral particle, and k is Boltzmann's constant.

[0030] Furthermore, based on the electron-neutral particle collision frequency γ e ion-neutral particle collision frequency γ i The numerical relationship with the excitation frequency ω is obtained.

[0031] ε r With γ e and γ i By combining these factors, the relationship between relative permittivity, positive ion number density, electron number density, and temperature can be constructed as follows:

[0032] ε r =1+C0(n) e +n i )T

[0033] in,

[0034] σ and γ e and γ i Combined, the relationship between conductivity, positive ion number density, electron number density, and temperature is constructed as follows:

[0035]

[0036] in,

[0037] Furthermore,

[0038] according to n e +0.005n i ≈n e The similarity relationship is used to construct the relationship between electron number density and temperature, expressed as follows:

[0039]

[0040] in,

[0041] The relationship between the positive ion number density and temperature is expressed as

[0042]

[0043] Wherein, A0, A1 and b are constants related to ionization reaction;

[0044] Further, the relationship model between the relative dielectric constant, the conductivity and the temperature is

[0045]

[0046] By merging the constant coefficient, the relationship model between the relative dielectric constant, the conductivity and the temperature is

[0047]

[0048] Wherein, a = C0A0, c = A1,

[0049] The method has the following advantages:

[0050] (1) When estimating the temperature by using the relative dielectric constant and the conductivity, the expression based on the flame complex dielectric constant has a theoretical basis;

[0051] (2) The flame complex dielectric constant model used is a complete expression considering the comprehensive action of positive ions and electrons, and the accuracy of estimating the temperature is higher. BRIEF DESCRIPTION OF DRAWINGS

[0052] Figure 1 The flow chart of the flame temperature measurement method based on the complex dielectric constant model considering the positive ions and electrons involved in the present application.

[0053] Figure 2 It is an embodiment of the present application. DETAILED DESCRIPTION

[0054] The theoretical basis of the present application is the complex dielectric constant expression of Wang et al. [4] The derived complex flame dielectric constant expression considering the comprehensive action of positive ions and electrons fully reflects the complex dielectric characteristics of the flame. Based on the complex dielectric constant model considering the positive ions and electrons, the present application proposes a method for reconstructing the flame temperature distribution by using the relative dielectric constant and the conductivity. The technical scheme of the present application is as follows:

[0055] Step one, the complex flame dielectric constant expression considering the positive ions and electrons is

[0056]

[0057] Wherein, the real part of the complex dielectric constant is the relative dielectric constant, and the imaginary part is the conductivity. The expression of the relative dielectric constant ε r and the conductivity σ is

[0058]

[0059] Wherein, n e is the electron number density, n i is the positive ion number density, m is the electron mass, M is the positive ion mass, ε0 is the vacuum dielectric constant, q is the unit charge, ω is the excitation frequency, γ e is the electron-neutral particle collision frequency, γ i is the ion-neutral particle collision frequency.

[0060] The expression of γ e and γ i is

[0061]

[0062] Wherein, p is the pressure, T is the thermodynamic temperature, d is the neutral particle diameter, and k is the Boltzmann constant.

[0063] The collision frequency is usually around 1 GH, and the excitation frequency is selected as 100 kHz, so

[0064] Combine ε r with γ e and γ i to build the relationship of the relative dielectric constant, the positive ion number density, the electron number density and the temperature as

[0065] ε r =1+C0(n e +n i )T

[0066] Wherein,

[0067] Combine σ with γ e and γ i to build the relationship of the conductivity, the positive ion number density, the electron number density and the temperature as

[0068]

[0069] Wherein,

[0070] Step two, the mass of the positive ion in the flame is much larger than that of the electron, In the local area of the flame where n e ≥n i , n e +0.005ni ≈n e And in the flame n e <n i The local area, because the positive ion number density and the number density of electrons generally will not more than one order of difference, so also can get n e +0.005n i ≈n e Thus, the relationship between the number density of electrons and temperature is expressed as

[0071]

[0072] The relationship between the number density of positive ions and temperature is expressed as

[0073]

[0074] Wherein, A0, A1 and b are constants related to ionization reaction.

[0075] Further, the relationship model between the relative dielectric constant, conductivity and temperature is obtained as

[0076]

[0077] By merging the constant coefficients, the relationship model between the relative dielectric constant, conductivity and temperature is expressed as

[0078]

[0079] Wherein, a = C0A0, c = A1,

[0080] Step three, collect several groups of relative dielectric constant, conductivity and temperature data of the flame at different positions, and determine the coefficients a, b, c, d in the relationship model.

[0081] Step four, obtain the relative dielectric constant and conductivity distribution of the flame by using the complex value capacitance tomography system, calculate the flame temperature based on the relationship model, and thus reconstruct the flame temperature distribution.

[0082] The present application will be described below in conjunction with the drawings and examples.

[0083] This example uses the premixed flame generated by the Bunsen burner, and the implementation diagram is as Figure 2The relative permittivity and conductivity of the flame are obtained by detecting the impedance value of the electric probe. In the article "Verification for Electrical Tomography in Flame Monitoring by Ion Probe" published by Hu et al. in IEEE International Instrumentation and Measurement Technology Conference in 2019, the relative permittivity and conductivity of the flame are obtained by detecting the impedance value of the electric probe. This embodiment uses an LCR meter to connect the electric probe to detect the impedance value of the flame, thereby obtaining the relative permittivity and conductivity of the flame. At the same time, a thermocouple is used to collect the temperature at the same position. By moving the displacement table, 10 groups of data are obtained, each group of data containing the temperature and the corresponding relative permittivity and conductivity.

[0084] The determination of a, b, c, d uses a fitting method based on the least squares principle. The fitting relationship is defined as the relationship model between the relative permittivity, conductivity and temperature

[0085]

[0086] The 10 groups of collected relative permittivity, conductivity and temperature are input into the relationship model. Based on the least squares method, when a = 0.3988, b = 0.1182, c = 2457, d = 2816, the sum of squared errors SSE is 4.4147 x 10 -4 The square value of the correlation coefficient R-Square between the actual data and the data obtained based on the fitting relationship model is 0.9972, which achieves a good fitting effect, and a, b, c, d obtained at this time are the optimal solution. Thus, the relationship model based on the relative permittivity and conductivity is

[0087]

[0088] In the article "Dual-modality electrical tomography for flame monitoring" published by Hu et al. in IEEE Sensors Journal, Vol. 18, No. 21, pp. 8847-8854 in 2018, a dual-modality electrical tomography system was built to reconstruct the distribution of the relative permittivity and conductivity of the flame. This embodiment uses a complex-valued electrical capacitance tomography system to reconstruct the distribution of the relative permittivity and conductivity of the flame, as shown in Figure 2The sensor electrode array is arranged around the flame, and the flame impedance data detected by the sensor electrode array is collected by a data acquisition system, and the relative permittivity and conductivity distribution of the flame is reconstructed by combining an image reconstruction algorithm. The reconstructed relative permittivity and conductivity of the flame are substituted into the obtained relationship model to solve the temperature. For example, at a certain position in the flame, ε r = 1.1012, σ = 2.455 x 10 -7 S / m into the relationship model, we get

[0089]

[0090] At this time, there is only one unknown quantity of temperature in the relationship model, and it is a relatively complex nonlinear equation, and the analytical solution of temperature is difficult to obtain, and the numerical solution of the equation is considered.

[0091] The numerical solution of the equation can be transformed into the zero point problem of f(T), and the expression of f(T) is

[0092]

[0093] First, set the initial value T0 = 800, and search around 800. Expand the interval in both positive and negative directions of 800 by a certain step, find that f(544)·f(1056)<0, according to the zero point theorem, there is a zero point of f(T) in the interval [544,1056]. And since f'(T) is always greater than 0 in [0,+∞], f(T) is monotonic in [0,+∞], so the zero point in the interval [544,1056] is the only zero point.

[0094] Then, interpolation is carried out in the interval [544,1056] containing the zero point, and the value f(x) at the interpolation x is calculated. Based on the zero point theorem, the sign of f(x) and f(T) at the endpoints of the interval [544,1056] is judged, and the interval containing the zero point is further narrowed. Repeat the interpolation process and iterate the interval, find that f(1021.21) = 0. Therefore, 1021.21 is the zero point, that is, the temperature value estimated by ε r = 1.1012 and σ = 2.455 x 10 -7 S / m.

[0095] For the relative permittivity and conductivity of different positions of the flame, the corresponding temperature estimation value is obtained by solving one by one according to the above process, so as to reconstruct the temperature distribution of the flame.

Claims

1. A method for reconstructing flame temperature distribution based on complex permittivity model, comprising the following steps: First, the model of the relationship between the relative dielectric constant ε r and the electrical conductivity σ and the flame thermodynamic temperature T is expressed as follows: Secondly, collecting the relative permittivity, conductivity and temperature data of several groups of flame at different positions, and determining the coefficients a, b, c and d in the relationship model; Thirdly, obtaining the relative permittivity and conductivity distribution of the flame by using the complex value electrical capacitance tomography system, and calculating the flame temperature based on the relationship model to reconstruct the flame temperature distribution.

2. The method of claim 1, wherein, In step (3), the problem of solving the flame thermodynamic temperature T is considered as the numerical solution of the equation, and the numerical solution of the equation is converted into the zero point problem of f(T), wherein only the flame thermodynamic temperature T is unknown, and the expression of f(T) is:

3. The method of claim 2, wherein, comprising the following steps: (1) setting the initial value of the flame thermodynamic temperature T0, determining the search interval and the step size; (2) expanding the interval in the positive and negative directions of the initial value of the flame thermodynamic temperature T0, respectively, determining the interval range containing a unique zero point according to the zero point theorem, performing interpolation in the interval range, further narrowing the interval containing the zero point, repeating the interpolation process and iterating the interval, and obtaining the zero point, i.e. the estimated value of the flame thermodynamic temperature T; (3) for the relative permittivity and conductivity of the flame at different positions, the corresponding temperature estimate value is obtained by solving according to the above process, so as to reconstruct the temperature distribution of the flame.

4. The method of claim 1, wherein, The relationship model between the relative permittivity, conductivity and flame temperature is constructed, and the expression of the flame complex permittivity considering positive ions and electrons is used Wherein, the real part of the complex dielectric constant is the relative dielectric constant, and the imaginary part is the conductivity; the expression of the relative dielectric constant ε r and the conductivity σ is where n e is the electron number density, n i is the positive ion number density, m is the electron mass, M is the positive ion mass, ε0is the vacuum permittivity, q is the unit charge, ω is the excitation frequency, γ e is the electron-neutral particle collision frequency, and γ i is the ion-neutral particle collision frequency. gamma e with gamma i the expression is Wherein, p is the pressure, T is the thermodynamic temperature, d is the diameter of neutral particles, and k is the Boltzmann constant.

5. The method of claim 4, wherein, According to the value relation between the electron-neutral particle collision frequency γ e , ion-neutral particle collision frequency γ i and the excitation frequency ω, we get ε r and γ e and γ i The relationship of relative dielectric constant, positive ion number density, electron number density and temperature is constructed as ε r = 1 + C0(n e + n i )T wherein Combining σ and γ e and γ i The relationship of conductivity, positive ion number density, electron number density and temperature is constructed as wherein 6. The method for reconstructing flame temperature distribution based on complex permittivity model according to claim 5, wherein According to n e +0.005n i ≈n e The relationship between the electron number density and temperature is expressed as wherein The relationship between the number density of positive ions and the temperature is expressed as Wherein, A0, A1 and b are constants related to ionization reaction.

7. The method of claim 6, wherein the complex permittivity model-based flame temperature profile reconstruction method further comprises: The relationship model between the relative permittivity, conductivity and temperature is obtained as By merging the constant coefficients, the relationship model between the relative permittivity, conductivity and temperature is expressed as wherein a = CoAo, c = A1,

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