Multi-spectral radiation thermometry based on an Alpha spectrum-LM algorithm under the background of high temperature and intense reflection

Liwei Chen; Xianqi Zhang; Shan Gao; Ying Cui; Can Yang; Xiaokai Wei; Jing Jiang; Yi Niu; Chao Wang; Chao Wang

doi:10.1364/OE.472493

1. Introduction

Multi-spectral temperature measurement is a typical non-contact temperature measurement technology. With its unique advantages of high-speed response, non-contact measurement, and high upper temperature limit, it has been widely used in many industrial processes such as steel smelting, refractory processing, and turbine blade monitoring [1–6]. However, there are currently two key problems that limit the further improvement of the temperature measurement accuracy in the high temperature background. On the one hand, the infrared radiation of the high temperature background will interfere with the target temperature measurement, and the infrared radiation of the high temperature background will form different reflections on the surface of the target to be measured at different positions, and will be superimposed with the radiation of the target to be measured [7,8]. On the other hand, since the true surface temperature and emissivity of the target are unknown, the multi-spectral radiation thermometry target equations are underdetermined equations with more unknown variables than equations, which cannot be solved directly [9].

In recent years, many scholars have studied the temperature measurement method under the high temperature background. Lucia [10] proposed to use the reflection model method to estimate the influence of reflection on the measurement results and to correct the error. Gao et al. [11–13] used the equation method to solve the radiation angle coefficient between regular surface elements, and proposed a reflected radiation analysis model suitable for turbine blades. Then, the errors and correction methods caused by the reflected radiation when the emissivity values of the target to be measured are 0.3 and 0.8, respectively, are discussed. In the follow-up, through simulation experiments and curve fitting techniques, the most suitable emissivity model for multi-spectral radiation temperature measurement of ceramic coatings was determined under the background temperature of 1273K [14]. Zheng et al. [15] established a three-dimensional reflection analysis model of turbine blades by using radiation angle coefficient and discrete coordinate transformation, and proposed a three-wavelength radiation temperature measurement algorithm based on reflection error correction to correct the temperature measurement error caused by reflected radiation.

In addition to the influence of the high temperature background radiation, the target emissivity also affects the interference intensity of the high temperature background reflected radiation. For the unknown emissivity of the target to be measured, Xing et al. [16,17] used constraint algorithms such as penalty functions to process the multi-spectral radiation thermometry data, and subsequently proposed a Generalized inverse matrix normalization (GIM-NOR) algorithm to further improve the calculation speed and accuracy [18]. Yu et al. [19] proposed a multi-wavelength radiometric thermometry data processing algorithm based on the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. Another commonly used approach is to assume that emissivity varies with wavelength, identify a spectral emissivity model, and construct a target equation to solve for temperature. Based on radiation data from three channels, Madura et al. [20] established constant, power, and exponential emissivity models for target radiation temperature measurement. Wen et al. [21] measured the surface temperature of aluminum alloys using linear emissivity models and log-linear emissivity models. Taunay et al. [22] proposed a multi-wavelength thermometry algorithm based on robust statistics and cross-validation of emissivity model. Chen et al. [23,24] relied on Back Propagation (BP) neural network to identify the spectral emissivity model under high temperature background. However, the classification network is a classification network trained under a fixed high temperature background temperature and a fixed position to be measured, which means that whenever the high temperature background temperature or the position of the target to be measured changes, the BP neural network needs to be retrained. Especially in the radiation temperature measurement of complex structures such as turbine blades or combustion chambers, the amount of high temperature background radiation interference changes with the structure and position of the target to be measured. Repeatedly retraining the neural network requires a large amount of valid data and time, otherwise there will be large temperature measurement errors.

Aiming at the problems that reflected radiation interference and spectral emissivity are difficult to obtain in radiation temperature measurement under the background of high temperature and intense reflection, this paper will build an analysis model of high temperature background reflected radiation to obtain the amount of high temperature background radiation interference at different locations to be measured. Further, using the least squares support vector machine optimized by particle swarm optimization (PSO-LSSVM), an emissivity model identification algorithm based on Alpha Spectrum-LM algorithm is proposed, which can accurately identify the emissivity model of the target to be measured and construct a multi-spectral target equation. Finally, the PSO algorithm is used to solve the multi-spectral target equation to realize the error correction of reflected radiation and obtain the actual temperature of the target. At the end of this paper, the accuracy and effectiveness of the proposed radiation temperature measurement method will be further verified by simulation and experimental comparison with the existing radiation temperature measurement methods.

2. Multi-spectral radiation temperature measurement method based on an Alpha spectrum-LM algorithm under the background of high temperature and intense reflection

2.1 Basic principle of multi-spectral radiation thermometry under high temperature background

Planck's law is the basic law of thermal radiation, which describes the relationship between black body radiation and temperature and wavelength, and can be expressed as follows:

(1)$$M(\lambda ,T) = {c_1}{\lambda ^{ - 5}}{({e^{{c_2}/\lambda T}} - 1)^{ - 1}}$$

where ${c_1}{\rm{ = }}3.7418 \times {10^8}{\rm{W}\mathrm{\mu }}{{\rm{m}}^4}{\rm{/}}{{\rm{m}}^2}$ is the first Planck coefficient and ${c_2}{\rm{ = }}1.4388 \times {10^4}{\mathrm{\mu} \rm{m}} \cdot {\rm{K}}$ is the second Planck coefficient.$M(\lambda ,T)$ is the radiant emittance of a blackbody with temperature $T$ at wavelength $\lambda $.

The emissivity is the ratio of the thermal energy radiated by an object at a certain temperature to the radiant energy of a black body at the same temperature, and its value is related to factors such as the temperature, wavelength, and surface state of the object. Equation (2–5) are seven commonly used emissivity models, corresponding to linear, quadratic, sinusoidal and exponential, respectively.

(2)$$\varepsilon (\lambda ,T) = a + b\lambda \;\;\;\;\;\;{\rm{ }}(a \gt 0{\rm{ }}\;\;or\;\;{\rm{ }}a \lt 0)$$

(3)$$\varepsilon (\lambda ,T) = a{\lambda ^2} + b\lambda + c{\rm{ }}\;\;\;(a \gt 0{\rm{ }}\;\;or\;\;{\rm{ }}a \lt 0)$$

(4)$$\varepsilon (\lambda ,T) = \frac{1}{2} + \frac{1}{2}\sin (a + b\lambda )$$

(5)$$\varepsilon (\lambda ,T) = {e^{a + b\lambda }}{\rm{ }}\;\;\;\;\;(a \gt 0{\rm{ }}\;\;or\;\;{\rm{ }}a \lt 0)$$

Because the actual target to be measured is not an ideal black body, there is emissivity, and the high temperature background forms reflected radiation on the target surface. Therefore, the radiation energy received by the radiation pyrometer includes the radiation of the target itself and the reflection of the surrounding high temperature background, which can be expressed as follows:

(6)$$M(\lambda ,T) = {\varepsilon _\lambda }M(\lambda ,{T_b}) + (1 - {\varepsilon _\lambda })\;\;M(\lambda ,{T_{\rm{r}}})$$

where $T$ is the brightness temperature of the target to be measured, ${T_b}$ is the target temperature, ${T_{\rm{r}}}$ is the background temperature, $M(\lambda ,T)$ is the total radiation energy received by the detector, $M(\lambda ,{T_b})$ is the amount of radiation from a black body at the same temperature as the target, $M(\lambda ,{T_{\rm{r}}})$ is the amount of radiation from the high temperature background to the surface of the measured target.${\varepsilon _\lambda }$ is the emissivity of the measured object surface and the value of $(1 - {\varepsilon _\lambda })$ is equal to the surface reflectivity of an opaque object.

For a pyrometer with n channels, n equations can be established according to the radiation transfer relationship as shown in Eq. (7). But there are n+1 unknown parameters in the equation, including the real temperature of the target and the spectral emissivity at n wavelengths.

(7)$$\left\{ \begin{array}{l} M({\lambda_1},{T_b}) = \frac{{M({\lambda_1},T) - (1 - {\varepsilon_1})M({\lambda_1},{T_r})}}{{{\varepsilon_1}}}\\ M({\lambda_2},{T_b}) = \frac{{M({\lambda_2},T) - (1 - {\varepsilon_2})M({\lambda_2},{T_r})}}{{{\varepsilon_2}}}\\ \ldots \\ M({\lambda_n},{T_b}) = \frac{{M({\lambda_n},T) - (1 - {\varepsilon_n})M({\lambda_n},{T_r})}}{{{\varepsilon_n}}} \end{array} \right.$$

Since the number of unknown parameters in Eq. (7) is more than the number of equations, the true temperature cannot be directly solved, so it is necessary to reasonably assume the spectral emissivity model to solve, and convert it into the problem of solving the emissivity model coefficients. In theory, the true temperature of the target to be measured should be the same at each wavelength, so the multi-spectral temperature measurement target equation is constructed as follows:

(8)$$\left\{ \begin{array}{l} \Delta = \min \sum\limits_{i = 1}^{n - 1} {{{\left[ {{M^{ - 1}}\left\{ {{\lambda _{i + 1}},\frac{{M({\lambda _{i + 1}},T) - (1 - {\varepsilon _{i + 1}})M({\lambda _{i + 1}},{T_r})}}{{{\varepsilon _{i + 1}}}}} \right\} - {M^{ - 1}}\left\{ {{\lambda _i},\frac{{M({\lambda _i},T) - (1 - {\varepsilon _i})M({\lambda _i},{T_r})}}{{{\varepsilon _i}}}} \right\}} \right]}^2}} \\ {\varepsilon _i} = f({\lambda _i})\;\;\;\;{\rm{ , }}0\;\;\le {\varepsilon _i} \le 1 \end{array} \right.$$

where $f({\lambda _i})$ is the emissivity model, ${M^{ - 1}}\{{\lambda ,M} \}$ is the inverse transform of Planck's equation used to calculate temperature. From Eq. (8), it can be known that the key to multi-spectral radiation temperature measurement under the background of high temperature and intense reflection is to accurately obtain the high temperature background radiation and the emissivity model of the target to be measured. This paper will focus on the solutions to these two problems.

2.2 High temperature background reflected radiation analysis model

By constructing the reflected radiation analysis model of the high temperature background, the radiation interference of the target to be measured in the high temperature and intense reflection environment can be accurately obtained.

Taking the radiation temperature measurement of the pressure surface of the turbine blade under the background of typical high temperature and intense reflection as an example, as shown in Fig. 1(a) (the example to be measured is the pressure surface of the moving blade, the blade height is 50%, and the relative chord length is 0.5). The specific process is as follows:

Fig. 1. Illustration of high temperature background reflected radiation analysis model: (a) 3D model of turbine blade and background. (b) screening results of ‘visual’ judgment of radiation transfer path. (c) screening results of radiation transfer path occlusion judgment

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Step1: Establish a three-dimensional discrete model of the high temperature background and the target to be measured, as shown in Fig. 1(b). Based on the triangular area coordinate theory, a triangle is selected as the basic unit to discretize the surface of the three-dimensional model (the area of the triangular surface is approximately 2mm²). It is assumed that the temperature distribution and radiant thermal physical parameters of each divided triangular surface are uniform. Compared with the simplified model, this model can better reflect the radiation transfer characteristics between the high temperature background and the target to be measured.

Step2: The ‘visual’ judgment of radiation transfer between the target to be measured and the high temperature background. Without considering the occlusion of the radiation transmission path, the high temperature background surface elements that can transfer thermal radiation to the target to be measured are screened out. If radiation can be transferred between two surface elements, the relationship between their normal vector and the vector represented by the line connecting the center of gravity satisfies Eq. (9).

(9)$$\overrightarrow {{n_1}} \cdot \overrightarrow {{s_{{\kern 1pt} {\kern 1pt} }}} \gt 0{\kern 1pt} {\kern 1pt} {\kern 1pt} \& \& - \overrightarrow {{n_2}} \cdot \overrightarrow {{s_{{\kern 1pt} {\kern 1pt} }}} \gt 0$$

where $\overrightarrow {{n_1}}$ and $\overrightarrow {{n_2}}$ are the normal vectors of surface elements 1 and 2, respectively, and $\overrightarrow {{s_{{\kern 1pt} {\kern 1pt} }}}$ is the vector formed by the line between the center of gravity of the two elements.

Step3: Radiation transfer path occlusion judgment. After Step 2, it is necessary to judge whether the radiation transmission path between the target surface element to be measured and the high temperature background surface element is blocked. The judgment method is to calculate the center of gravity connection between the two surface elements, and determine whether the connection line intersects with other surface elements. The results of path occlusion judgment and screening are shown in Fig. 1(c).

Step4: After the screening of Step3, the remaining background surface elements are all involved in the calculation of the reflected radiation of the target to be measured in the high temperature background, and the radiation angle coefficient between the background surface element and the surface element to be measured needs to be calculated. ${F_{ji}}$ is the radiation angle coefficient from surface j to surface i. For the calculation of the angle coefficient between two finite-size elements, the equation is shown in Eq. (10).

(10)$${F_{ji}} = \frac{1}{{{A_j}}}\int_{{A_i}} {\int_{{A_j}} {\frac{{\cos {\theta _i}\cos {\theta _j}}}{{\pi {R^2}}}} } d{A_i}d{A_j}$$

where ${\theta _i}$ and ${\theta _j}$ are the angles between the surface normal and the line joining two infinitesimal areas $d{A_i}$ and $d{A_j}$ of respective surfaces. ${A_j}$ is the area of surface j, ${A_i}$ is the area of surface i, R is the distance between the two areas.

The radiation angle coefficient combined with the theoretical temperature distribution of the high temperature background can obtain the radiation amount of the high temperature background radiation to the target to be measured. Therefore, when the high temperature radiometer detects the target i to be measured, Eq. (6) can be further expressed as:

(11)$${M_i}(\lambda ,T) = {\varepsilon _\lambda }_{,i}{M_i}(\lambda ,{T_b}) + (1 - {\varepsilon _\lambda }_{,i})\sum\limits_{j = 1}^n {{M_j}_{,i}(\lambda ,{T_r})}$$

(12)$${M_j}_{,i}(\lambda ,{T_r}){\rm{ = }}\frac{{{A_j}}}{{{A_i}}} \cdot {F_{ji}} \cdot {M_j}(\lambda ,{T_r})$$

where ${M_i}(\lambda ,T)$ is the radiation emittance actually received when the pyrometer detects the surface i, ${M_j}_{,i}(\lambda ,{T_r})$ is the radiation amount of the high temperature background element j reaching the surface i to be measured, and ${M_j}(\lambda ,{T_r})$ is the black body radiation emittance of the background element j.

2.3 Emissivity model recognition algorithm based on an Alpha spectrum-LM algorithm

2.3.1 Alpha spectrum-LM algorithm basic principles

Through the high temperature background reflected radiation analysis model, the radiation amount projected by the high temperature background to the target to be measured can be obtained, and the target emissivity determines the superposition of the radiation amount of the target to be measured and the high temperature background radiation, which is also important for the correction of reflected radiation errors.

Planck's equation is expressed using the Wien approximate as:

(13)$$M(\lambda ,T) = {c_1}{\lambda ^{ - 5}}{({e^{{c_2}/\lambda T}} - 1)^{ - 1}} \approx {c_1}{\lambda ^{ - 5}}{e^{ - {c_2}/\lambda T}}$$

The actual radiation received by the pyrometer can be expressed as:

(14)$$\begin{array}{l} {M_i}({\lambda _i},{T_{il}}) = {\varepsilon _i}{M_i}({\lambda _i},{T_b}) + (1 - {\varepsilon _i}){M_i}({\lambda _i},{T_r}) = {\varepsilon _i}\frac{{{c_1}}}{{\lambda _i^5{e^{{c_2}/\lambda {T_b}}}}}{A_i}\\ {A_i} = \frac{{1 - {B_i}}}{{1 - \frac{{{M_i}({\lambda _i},{T_r})}}{{{M_i}({\lambda _i},{T_{il}})}}}}{\rm{ }},{\rm{ }}{B_i} = \frac{{{M_i}({\lambda _i},{T_r})}}{{{M_i}({\lambda _i},{T_b})}} \end{array}$$

where A_i is the correction term of high temperature background reflection. In this study, it is found that replacing the unknown true temperature (T_b) in B_i with the minimum brightness temperature (T_l =min(T_il)) at each wavelength of the pyrometer has little effect on the emissivity shape characteristics obtained by the subsequent solution. Taking logarithms and shifting terms on both sides of Eq. (14), the following relationship can be obtained:

(15)$${\lambda _i}\ln {\varepsilon _i} = {\lambda _i}\ln ({M_i}({\lambda _i},{T_{il}})) - {\lambda _i}\ln {c_1} + 5{\lambda _i}\ln {\lambda _i} + \frac{{{c_2}}}{{{T_b}}} - {\lambda _i}\ln {A_i}$$

(16)$$\frac{1}{N}\sum\limits_{i = 1}^N {{\lambda _i}\ln {\varepsilon _i}} = \frac{1}{N}\sum\limits_{i = 1}^N {{\lambda _i}\ln ({M_i}({\lambda _i},{T_{il}}))} - \frac{1}{N}\sum\limits_{i = 1}^N {{\lambda _i}\ln {c_1}} + \frac{5}{N}\sum\limits_{i = 1}^N {{\lambda _i}\ln {\lambda _i}} + \frac{{{c_2}}}{{{T_b}}} - \frac{1}{N}\sum\limits_{i = 1}^N {{\lambda _i}\ln {A_i}}$$

The alpha spectrum expression is obtained by subtracting Eq. (15) and Eq. (16) as shown in Eq. (17). Alpha spectrum is currently mainly used in remote sensing images, which can better reflect the shape characteristics of the actual emissivity [25,26]. Taking the sinusoidal emissivity model as an example, as shown in Fig. 2, the shape of the Alpha spectrum can already reflect the actual emissivity shape, but there is still a large gap between the numerical value and the actual emissivity.

(17)$${\alpha _i} = {\lambda _i}\ln {\varepsilon _i} - \frac{1}{N}\sum\limits_{i = 1}^N {{\lambda _i}\ln {\varepsilon _i}}$$

In order to further correct the error caused by the Wien approximation and make the alpha spectrum closer to the actual emissivity in value, the alpha spectrum values of adjacent wavelengths are subtracted to obtain the following Eq. (18):

(18)$$\begin{array}{l} d{\alpha _{i - wien}} = {\alpha _{i + 1}} - {\alpha _i} = {\lambda _{i + 1}}\ln {\varepsilon _{i + 1}} - {\lambda _i}\ln {\varepsilon _i}\\ {\rm{ }} = {\lambda _{i + 1}}\ln ({M_{i + 1}}({\lambda _{i + 1}},{T_{i + 1}}_{,l})) - {\lambda _{i + 1}}\ln {c_1} + 5{\lambda _{i + 1}}\ln {\lambda _{i + 1}} + \frac{{{c_2}}}{{{T_b}}} - {\lambda _{i + 1}}\ln {A_{i + 1}}\\ {\rm{ }} - ({\lambda _i}\ln ({M_i}({\lambda _i},{T_i}_{,l})) - {\lambda _i}\ln {c_1} + 5{\lambda _i}\ln {\lambda _i} + \frac{{{c_2}}}{{{T_b}}} - {\lambda _i}\ln {A_i}) \end{array}$$

Fig. 2. Alpha spectrum calculation result

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However, the calculation results under the conditions of Planck's equation should be as follows:

(19)$$\begin{array}{l} d{\alpha _{i - planck}} = {\lambda _{i + 1}}\ln {\varepsilon _{i + 1}} - {\lambda _i}\ln {\varepsilon _i}\\ {\rm{ }} = {\lambda _{i + 1}}\ln ({M_{i + 1}}({\lambda _{i + 1}},{T_{i + 1,l}})) - {\lambda _{i + 1}}\ln {c_1} + 5{\lambda _{i + 1}}\ln {\lambda _{i + 1}} + {\lambda _{i + 1}}\ln ({e^{{c_2}/{\lambda _{i + 1}}{T_b}}} - 1) - {\lambda _{i + 1}}\ln {A_{i + 1}}\\ {\rm{ }} - ({\lambda _i}\ln ({M_i}({\lambda _i},{T_{il}})) - {\lambda _i}\ln {c_1} + 5{\lambda _i}\ln {\lambda _i} + {\lambda _i}\ln ({e^{{c_2}/{\lambda _i}{T_b}}} - 1) - {\lambda _i}\ln {A_i}) \end{array}$$

Therefore, the error caused by the Wien approximation calculation is as follows:

(20)$$\varDelta d{\alpha _i}({T_b}) = d{\alpha _{i - planck}} - d{\alpha _{i - wien}} = {\lambda _{i + 1}}\ln ({e^{{c_2}/{\lambda _{i + 1}}{T_b}}} - 1) - {\lambda _i}\ln ({e^{{c_2}/{\lambda _i}{T_b}}} - 1)$$

Research [19] shows that the relative error caused by the Wien approximation increases with the increase of wavelength and temperature, and the maximum relative error is less than 0.06 when the wavelength is 5 µm and the temperature is 1000.0 K. Therefore, the unknown true temperature (T_b) in Eq. (20) can be replaced by the minimum brightness temperature (T₁) of the pyrometer at each wavelength, and the error caused by the Wien approximation can be further corrected. Equation (18) can be further corrected as follows.

(21)$$d{\alpha _{i - planck}}{\rm{ = }}d{\alpha _{i - wien}} + \varDelta d{\alpha _i}({T_l})$$

After the error correction of the Wien approximation, in order to make the Alpha spectrum closer to the actual emissivity in value, the following spectral radiation equations are constructed:

(22)$$\left\{ \begin{array}{l} {M_i}({\lambda _i},T) = {\varepsilon _i}\frac{{{c_1}}}{{\lambda _i^5{e^{{c_2}/\lambda {T_b}}}}}{A_i}({T_l})\;\;\;\;\;\;\;{\rm{ (}}i = 1,2, \cdot \cdot \cdot N)\\ {\lambda _{i + 1}}\ln {\varepsilon _{i + 1}} - {\lambda _i}\ln {\varepsilon _i} = d{\alpha _{i - wien}} + \Delta d{\alpha _i}({T_l})\;\;{\rm{ (}}i = 1,2, \cdot \cdot \cdot N - 1) \end{array} \right.$$

The emissivity in Eq. (22) is solved by the Levenberg-Marquarelt (LM) algorithm. The LM algorithm is an efficient optimization method for nonlinear least squares problems, combining the excellent local convergence properties close to the minimum of the Gauss-Newton method and the minimum distance provided by the steepest descent method [27]. Taking seven types of emissivity models as an example, the calculated emissivity is shown in Fig. 3. It can be seen that the calculated emissivity characteristics of the seven types are consistent with the real emissivity, which is equivalent to the vertical translation of the actual emissivity, effectively retaining the shape characteristics of the real emissivity. Therefore, the calculated emissivity results can be used as the identification basis for the actual emissivity model.

Fig. 3. Emissivity calculation results of seven emissivity models calculated by Alpha Spectrum-LM algorithm: (a) linear emissivity model. (b) quadratic emissivity model. (c) sinusoidal emissivity model. (d) exponential emissivity model

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2.3.2 Alpha spectrum-LM algorithm performance analysis

In this paper, the spectral angle mapper (SAM) is used to quantitatively describe the similarity between the calculated emissivity and the real emissivity shape. The calculation result can be regarded as the cosine angle between the two spectral emissivities. The smaller the value of the calculation result, the more matched the two emissivities are and the higher the similarity. The calculation equation is as follows:

(23)$$\theta ({\varepsilon _{Alpha - LM}},\varepsilon ) = {\cos ^{ - 1}}(\frac{{{\varepsilon ^T}{\varepsilon _{Alpha - LM}}}}{{{{({\varepsilon ^T}\varepsilon )}^{1/2}}{{({\varepsilon ^T}_{Alpha - LM}{\varepsilon _{Alpha - LM}})}^{1/2}}}})$$

where $\varepsilon$ is the real emissivity, ${\varepsilon _{Alpha - LM}}$ is the emissivity solved by the Alpha spectrum-LM algorithm.

Since the minimum brightness temperature (T_l) is used to replace the true temperature (T_b) of B_i in the Eq. (14) in the derivation process, the emissivity of the wavelength where the minimum brightness temperature is located will be calculated as 1, and if the replacement temperature is closer to the true temperature, then the solved emissivity is closer to the true emissivity. Taking the sinusoidal emissivity model as an example, the true temperature T_b= 1073.2K, the high temperature background temperature T_r= 1373.2K, and the replacement temperature gradually changes from the true temperature to the minimum brightness temperature. The partial results of the emissivity solved under different replacement temperature conditions are shown in Fig. 4, and the SAM value gradually increased from 0.083 to 0.603. It can be seen that using the minimum brightness temperature as a replacement temperature retains the shape characteristics of the true emissivity, which can meet the needs of subsequent emissivity model identification.

Fig. 4. Emissivity solution results at different replacement temperatures

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Under the conditions of the same emissivity value, temperature setting and number of bands, the starting wavelength of the calculation was gradually increased from 1.5 µm to 10.0 µm with a step size of 0.5 µm. The results are shown in Fig. 5.

Fig. 5. Variation curve of SAM value under different wavelengths

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The SAM value gradually decreases from 0.603 to 0.005, which can still meet the needs of subsequent emissivity model identification. It is proved that the method in this paper is less limited by wavelength and can be applied in a wide range of bands.

2.3.3 Particle swarm optimization least squares support vector machine (PSO-LSSVM) emissivity model identification classifier

The emissivity calculated by the Alpha Spectrum-LM algorithm can reflect the shape characteristics of the real emissivity very well, especially it is very similar to the actual emissivity after normalization, so it can be used as the basis for the identification of the emissivity model. LSSVM has achieved very successful results in function estimation and is widely used in the field of nonlinear system identification. The specific principle and derivation process can be found in Ref. [28]. In this paper, LSSVM is selected as the emissivity model identification classifier, and the PSO algorithm is used to select the parameters of LSSVM modeling. Relying on the rapid convergence ability of particle swarm optimization technology, the optimal parameters of the obtained model can be quickly searched, and the reliability of the classifier can be improved.

Given a training dataset {x_k, y_k}, LSSVM represents a nonlinear system in functional regression in the form:

(24)$$y(x) = {\omega ^T} \cdot \varphi (x) + b$$

where x, y are the input and output, $\varphi ({\cdot} )$ is a nonlinear kernel function that maps the input feature space to a higher dimensional feature space. $\omega$ is a coefficient that determines the margin of support vectors, and b is the bias term.

The coefficients ($\omega$, b) are determined by minimizing the following regularized risk function (cost function) and using the equality constraints derived by Eq. (24).

(25)$$\begin{array}{l} \min J(\omega ,e) = \frac{1}{2}{\omega ^T} \cdot \omega + \gamma \frac{1}{2}\sum\limits_{n = 1}^N {e_n^2} {\rm{ (}}\gamma {\rm{ \gt 0)}}\\ S.t.{\rm{ }}{y_n} = {\omega ^T}\varphi (x) + b + {e_n}{\rm{ , }}\;\;\;\;\;n = 1,\ldots ,N \end{array}$$

where e_n is a non-negative error variable and $\gamma$ is the regularization parameter that determines the trade-off between fitting error minimization and smoothness.

This optimization problem is solved by using the following Lagrange function.

(26)$$L(\omega ,b,e,\alpha ) = J(\omega ,e) - \sum\limits_{n = 1}^N {{\alpha _n}} \{{{\omega_n}\varphi ({x_n}) + b + e_n^2 - {y_n}} \}$$

where $\alpha {\rm{ = }}\{{{\alpha_n}} \}_{n = 1}^N$ is the Lagrange multiplier set, and by Calculating the partial derivatives of $L(\omega ,b,e,\alpha )$, we can obtain the optimal condition about Eq. (26) as follows:

(27)$$\left\{ {\begin{array}{{c}} {\frac{{\partial L}}{{\partial \omega }} = 0 \to \omega = \sum\limits_{n = 1}^N {{\alpha_n}\varphi ({x_n}){\rm{ }}} }\\ {\frac{{\partial L}}{{\partial b}} = 0 \to \omega = \sum\limits_{n = 1}^N {{\alpha_n} = 0{\rm{ }}} }\\ {\frac{{\partial L}}{{\partial {e_n}}} = 0 \to {\alpha_n} = \gamma {e_n}{\rm{ }}}\\ {\frac{{\partial L}}{{\partial {\alpha_n}}} = 0 \to {\omega^T}\varphi ({x_n}) + b + e_n^2 - {y_n} = 0} \end{array}} \right.$$

we can transform the above equality into

(28)$$\left[ {\begin{array}{{cc}} 0&{{1^T}}\\ 0&{\Omega + {\gamma ^{ - 1}}I} \end{array}} \right]{\rm{ }}\left[ {\begin{array}{{c}} b\\ \alpha \end{array}} \right] = \left[ {\begin{array}{{c}} 0\\ y \end{array}} \right]$$

(29)$${\varOmega _{nl}} = \varphi {({x_n})^T}\varphi ({x_l}) = K({x_n},{x_l}){\rm{ }}\;\;n,l = 1,\ldots ,N$$

where y = [y₁, …, y_N]^T, 1 = [l,, 1]^T, $\alpha = {[{\alpha _1},\ldots ,{\alpha _N}]^T}$ and $\varOmega$ is a square matrix satisfying the Mercer's condition on the kernel matrix. Finally, the least squares support vector machine model is obtained as shown in Eq. (30).

(30)$$y = \sum\limits_{n = 1}^N {{\alpha _n}} k(x,{x_n}) + b$$

where k(x, x_n) is the kernel function,

Radial basis function (RBF) can reduce the complexity of model selection due to having fewer parameters, and has excellent generalization performance. Therefore, this paper selects the RBF kernel function to construct the LSSVM, and the radial basis kernel function is shown in Eq. (31).

(31)$$k({x_n},{x_l}) = {e^{ - \frac{{{{||{{x_n} - {x_l}} ||}^2}}}{{2{\sigma ^2}}}}}$$

where $\sigma$ is the kernel parameter and controls the LSSVM's regression or classification ability.

It is worth noting that the optimal selection of its regularization parameter γ and kernel parameter σ mentioned above is very difficult, and directly affects the generalization performance and accuracy of the classifier. The traditional parameter selection is obtained through repeated experiments, and there are many inconveniences such as randomness of artificial selection, large workload and long time. Therefore, this paper uses the PSO algorithm to optimize the parameter selection of the LSSVM, and uses the mean square error of the predicted label of the training sample as the fitness function. The particle swarm optimization algorithm updates particle velocities and positions according to the following equations:

(32)$$\begin{array}{l} v_{_{id}}^{k + 1} = \omega v_{_{id}}^k + {c_1}{r_1}(pbes{t_{id}} + x_{id}^k) + {c_2}{r_2}(pbes{t_{gd}} + x_{id}^k)\\ x_{id}^{k + 1} = x_{id}^k + v_{_{id}}^{k + 1} ,{\rm{ }}i = 1,...,n{\rm{ }},{\rm{ }}d = 1,...,D{\rm{ }} \end{array}$$

where n is the number of particles in the swarm and D is the number of swarms. k is the number of iterations, and c₁, c₂ are acceleration constants. r₁, r₂ are uniformly distributed random numbers between 0 and 1. $v_{_{id}}^k$ is the current velocity, and $x_{id}^k$ is the current position of the ith particle in the dth group. $pbes{t_{id}}$ is the best position of the ith particle and $pbes{t_{gd}}$ is the best position in the group.$\omega $ is the inertia weight factor. In this paper, dynamic inertia weight is used, that is, the inertia weight decreases with the number of iterations:

(33)$${\omega _k} = {\omega _{\max }} - ({\omega _{\max }} - {\omega _{\min }})\frac{k}{{{k_{\max }}}}$$

The fitness function of the particle swarm is defined as the root mean square error of the predicted labels of the training samples as follows:

(34)$$fitness = \frac{1}{N}\sum\limits_{i = 1}^N {{{({y_{id}} - y_{id}^{\prime})}^2}}$$

where y_id is the true label of the training sample, and y^'_id is the predicted label of the training sample.

Seven kinds of emissivity samples with values of 0.3 to 1 are theoretically generated in the measurement wavelength range, and the generated emissivity samples are normalized as shown in Eq. (35) as the training samples of the least squares support vector machine classifier.

(35)$$\varepsilon ({\lambda _i}) = \frac{{\varepsilon ({\lambda _i})}}{{\max (\varepsilon ({\lambda _i}))}}$$

After the training of the PSO-LSSVM emissivity model identification classifier is completed, the emissivity obtained by the Alpha Spectrum-LM algorithm is used as the input of the classifier, and the emissivity model can be accurately identified. The flow chart of the multi-spectral radiation temperature measurement method based on the Alpha spectrum-LM algorithm under the background of high temperature and intense reflection is shown in Fig. 6. The part enclosed by the red dotted line in Fig. 6 is the training process of the emissivity model recognition classifier. It only needs to be trained once when the measurement wavelength is unchanged, and it is less affected by changes in the high temperature background temperature and the position of the target to be measured, and has stronger applicability.

Fig. 6. Flow chart of multi-spectral radiation temperature measurement method based on Alpha spectrum-LM algorithm under the background of high temperature and intense reflection

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3. Algorithm simulation comparison and analysis

3.1 Comparative analysis of emissivity model recognition algorithms

In order to compare and verify the accuracy and anti-noise performance of the emissivity model recognition algorithm proposed in this paper, it is compared with the emissivity model recognition algorithm based on BP neural network proposed in the literature [24]. The parameters of the simulation experiment are as follows: The measurement wavelengths were selected as 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0 µm. A total of 7000 groups of seven types of emissivity samples were generated in the range of 0.3 to 1, and each emissivity model was randomly divided into 800 groups of training emissivity samples and 200 groups of test emissivity samples. The high temperature background temperature was set to 1000.0 K and the radiation angle coefficient is 0.9. The target temperature to be measured is to take a temperature point every 10 K between 773.2-963.2 K, a total of 20 temperature points to be measured.

According to Eq. (7), a total of 7000 groups of theoretical radiation data are generated, including 5600 groups of training radiation data samples and 1400 groups of test radiation data samples. According to the process and method shown in Fig. 6, 5600 groups of normalized emissivity training samples are used to train the least squares support vector machine classifier. The number of iterations of PSO algorithm is 30, the population size is 50, and the learning factor c₁= c₂= 1.5. The emissivity results of 1400 groups of spectral radiation test samples solved by the Alpha Spectrum-LM algorithm are used as the test samples of the classifier, and the confusion matrix of the test results is shown in Fig. 7(a). X1-X7 represent linear model (a > 0 and a < 0), quadratic model (a > 0 and a < 0), sinusoidal model and exponential model (a > 0 and a < 0), respectively. The BP neural network selects a typical three-layer BP network structure (the number of hidden layers is 1), in which the number of hidden layers is 20 neurons, and the final network structure is (6-20-7). The BP neural network is trained with 5600 sets of spectral radiation training samples, and the confusion matrix of the test results is shown in Fig. 7(b).

Fig. 7. Confusion matrix for test sample results. (a) Algorithm proposed in this paper (b) BP neural network emissivity model identification algorithm

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The test sample recognition rates of the two types of emissivity model recognition algorithms are shown in Table 1. According to statistics, the average recognition accuracy of the algorithm proposed in this paper is 93.3%. The average recognition accuracy of the emissivity model recognition algorithm based on BP neural network is 83.5%, and the recognition accuracy of sinusoidal model is only 60.5%. In contrast, the emissivity identification algorithm proposed in this paper has a higher accuracy. More notably, the emissivity model recognition algorithm based on BP neural network is trained at a specific ambient temperature. When the amount of high temperature background radiation changes or the temperature of the target to be measured fluctuates in a wide range, the algorithm is no longer applicable, and the recognition accuracy rate will drop significantly.

Table 1. Recognition accuracy of test samples

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In order to further simulate the actual temperature measurement situation, 10% random noise is added to the spectral radiation test samples used by the two algorithms, and the confusion matrices of the test results of the two algorithms are shown in Fig. 8(a) and Fig. 8(b), respectively. The models represented by X1-X7 are consistent with the above definitions.

Fig. 8. Test results of test samples after adding noise (a) Algorithm proposed in this paper (b) BP neural network emissivity model identification algorithm

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The recognition accuracy of the test samples after adding noise to the two types of emissivity model recognition algorithms is shown in Table 2.

Table 2. Recognition accuracy of test samples (after adding noise)

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As shown in Table 2, the average recognition accuracy of the algorithm proposed in this paper is 90.3% and the average recognition accuracy of the emissivity model recognition algorithm based on BP neural network is 75.6%. Compared with the existing BP neural network recognition algorithm, the algorithm proposed in this paper has better accuracy, anti-noise, and applicability.

3.2 Radiation temperature measurement of turbine blades under variable high temperature background

In order to verify the accuracy of the algorithm proposed in this paper in high temperature and variable background conditions, the following simulation comparison experiments are carried out. Taking the turbine blade as the research object of radiation temperature measurement, the reflected radiation amount of the high temperature background varies with the position and structure of the target to be measured during the radiation temperature measurement process of the turbine blade. The target to be measured is 33 points to be measured with the relative chord length of 0-1 located at three blade span heights (25%, 50% and 75%) on the pressure surface of the turbine blade. The comparison algorithm adopts the radiation thermometry algorithm based on the least squares model without the constraints of the emissivity model, the three-wavelength temperature measurement method and the multi-spectral radiation temperature measurement algorithm based on BP neural network emissivity model recognition under the high temperature background. The comparison algorithms all use the PSO algorithm with the same parameter settings to solve the temperature.

The parameters of the simulation experiments are set as follows: the measurement wavelengths are 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0 µm and the bandwidth is 0.02 µm. The temperature distribution of the guide vane is set to 723.2-1253.2K, and the temperature of the moving vane is set to 800.0K. The emissivity of each point to be measured adopts one of seven types of emissivity models randomly generated in the range of 0.6-0.9, and generates spectral radiation data according to Eq. (7). 15% random noise is added to the generated data to simulate actual measurement conditions. The wavelengths of the three-wavelength radiation thermometry algorithm are selected based on the three closest sets of wavelengths in the alpha spectrum. The radiation thermometry algorithm based on the least squares model without the constraints of the emissivity model. the least-squares part in Eq. (8) as the objective function. The parameter settings of the two emissivity model identification and classification algorithms are the same as the above simulation experiments. The population size of the particle swarm algorithm to solve the objective Eq. (8) is set to 50, the number of iterations is 80, the initial population range is -1 to 1, and c₁= c₂= 1.5. The simulation test results of the radiation temperature measurement of the turbine blade under the high temperature variable background are shown in Fig. 9.

Fig. 9. Temperature measurement errors of simulation experiments (a) The radiation temperature measurement algorithm based on the least squares without the constraints of the emissivity model. (b) Three-wavelength radiation thermometry algorithm under high temperature background. (c) Multi-spectral radiation thermometry algorithm based on BP neural network model. (d) The algorithm proposed in this paper.

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Figure 9(a) shows the simulation results of the radiation thermometry algorithm based on the least squares model without the constraints of the emissivity model. Due to the lack of other emissivity constraints other than the emissivity value range of 0 to 1, the calculation results have great fluctuations when dealing with high temperature reflection backgrounds, and the maximum temperature measurement error reaches 18.5K. The simulation results of the three-wavelength temperature measurement algorithm under the high temperature background are shown in Fig. 9(b). Only when the emissivity of the three wavelengths is approximately equal can a good radiation temperature measurement result be achieved. For the actual emissivity, the limit is high, and the maximum temperature measurement error reaches -21.3K. As shown in Fig. 9(c), the BP neural network is used to identify the emissivity model. This method is sensitive to noise, and the emissivity model is incorrectly identified due to changes in the high temperature background radiation, and the temperature measurement error reaches 15.6K. The algorithm proposed in this paper performs better in terms of accuracy and anti-noise performance, as shown in Fig. 9(d), the maximum temperature measurement error is 9.5K, and the average temperature measurement error is 4.4K. In the simulation verification under the high temperature variable background, more accurate temperature measurement results are obtained.

4. Experiment analysis

4.1 Experimental setup

In order to further verify the effectiveness of the proposed algorithm in multi-spectral radiation temperature measurement under the background of actual high temperature and intense reflection, the following experimental verification is carried out. The experimental device is shown in Fig. 10. The Inconel 718 sample to be tested is placed in a high temperature furnace, and the surface temperature of the sample to be tested is changed by setting a cooling chamber to introduce cold air. A pyrometer is used to receive the spectral radiation on the surface of the sample to be measured, and the thermocouple (Type K) is used to determine the true temperature of the surface of the sample to be measured, and the temperature measurement error is calculated according to the thermocouple readings.

Fig. 10. Experimental setup and equipment

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The background temperature of the constant temperature furnace was set to 963.2K, and a total of 40 sets of sample temperature measurement point data were collected. The measurement wavelengths were set to 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0 µm and the bandwidth was 0.02 µm. The comparison algorithms are the radiation thermometry algorithm based on the least squares model without the constraints of the emissivity model, the three-wavelength temperature measurement algorithm under high temperature background, and the multi-spectral radiation temperature measurement algorithm based on BP neural network under high temperature background. Under the condition that the background temperature is set to 963.2K and the target temperature is set to 773.2-963.2K, the training samples of the multi-spectral radiation thermometry algorithm based on BP neural network are generated and the classification network is trained. The parameters of the other two algorithms and the algorithm proposed in this paper are the same as those in the above simulation experiments.

4.2 Experimental results and analysis

The experimental results of multi-spectral radiation temperature measurement under high temperature background are shown in Fig. 11.

Fig. 11. Experimental results and errors: (a) The radiation thermometry algorithm based on the least squares model without the constraints of the emissivity model. (b) Three-wavelength radiation thermometry algorithm under high temperature background. (c) Multi-spectral radiation thermometry algorithm based on BP neural network. (d) Multi-spectral radiation thermometry algorithm based on Alpha Spectrum-LM algorithm.

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Figure 11(a) shows the temperature measurement results of the radiation thermometry algorithm based on the least squares model without the constraints of the emissivity model. Due to the constraints of missing emissivity shape properties, the maximum temperature measurement error is 18.0K and the average temperature measurement error is 7.3K. The temperature measurement results of the three-wavelength radiation thermometry algorithm under the high temperature background are shown in Fig. 11(b). Since the emissivity values of the actual target to be measured at the three wavelengths are not equal, the maximum temperature measurement error is -33.5K, and the average temperature measurement error is 14.8K. The temperature measurement results of the multi-spectral radiation temperature measurement algorithm based on BP neural network are shown in Fig. 11(c). Since the identified emissivity model is not applicable to the true emissivity of the target, the maximum temperature measurement error is 13.0K, and the average temperature measurement error is 5.1K. The temperature measurement results of the multi-spectral radiation thermometry algorithm based on the Alpha spectrum-LM algorithm under the background of high temperature and intense reflection are shown in Fig. 11(d). Since the identified emissivity model is more in line with the emissivity characteristics of the actual target to be tested, the maximum temperature measurement error is 7.8K and the average temperature measurement error is 2.7K. The experimental results verify the effectiveness of the method proposed in this paper for multi-spectral radiation temperature measurement under the background of actual high temperature and intense reflection.

5. Conclusion

The simulation and experimental comparison with the existing radiation temperature measurement algorithm under the high temperature background show that the maximum temperature measurement error of the method proposed in this paper in the high temperature variable background simulation experiment is 9.5K, and the maximum temperature measurement error in the high temperature background experiment is 7.8K. The research results all show that the proposed multi-spectral radiation temperature measurement method based on the Alpha spectrum-LM algorithm under the background of high temperature and intense reflection can effectively correct the reflected radiation error caused by the high temperature background at different target positions to be measured, and get more accurate emissivity model identification results and temperature measurement results. This method can more intuitively describe the spectral shape characteristics of the actual emissivity, provide an important reference for the subsequent research on spectral emissivity, and also help to further improve the accuracy of high temperature measurement in industrial processes.

Funding

National Natural Science Foundation of China (U20A20213); Fundamental Research Funds for the Central Universities (3072021CFT0802).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Types of emissivity models	recognition accuracy
Types of emissivity models	BP neural network	The method of this paper
X1: Linear model (a > 0)	75.0%	97.0%
X2: Linear model (a < 0)	82.5%	98.0%
X3: Quadratic model (a > 0)	87.5%	96.5%
X4: Quadratic model (a < 0)	92.0%	95.5%
X5: Sinusoidal model	60.5%	70.5%
X6: Exponential model (a > 0)	95.0%	99.0%
X7: Exponential model (a < 0)	92.5%	96.5%
Average recognition accuracy	83.5%	93.3%

Types of emissivity models	recognition accuracy
Types of emissivity models	BP neural network	The method of this paper
X1: Linear model (a > 0)	75.0%	96.0%
X2: Linear model (a < 0)	63.5%	92.0%
X3: Quadratic model (a > 0)	86.0%	95.5%
X4: Quadratic model (a < 0)	91.0%	93.0%
X5: Sinusoidal model	47.5%	67.5%
X6: Exponential model (a > 0)	86.5%	93.0%
X7: Exponential model (a < 0)	80.0%	95.5%
Average recognition accuracy	75.6%	90.3%

Types of emissivity models	recognition accuracy
Types of emissivity models	BP neural network	The method of this paper
X1: Linear model (a > 0)	75.0%	97.0%
X2: Linear model (a < 0)	82.5%	98.0%
X3: Quadratic model (a > 0)	87.5%	96.5%
X4: Quadratic model (a < 0)	92.0%	95.5%
X5: Sinusoidal model	60.5%	70.5%
X6: Exponential model (a > 0)	95.0%	99.0%
X7: Exponential model (a < 0)	92.5%	96.5%
Average recognition accuracy	83.5%	93.3%

Types of emissivity models	recognition accuracy
Types of emissivity models	BP neural network	The method of this paper
X1: Linear model (a > 0)	75.0%	96.0%
X2: Linear model (a < 0)	63.5%	92.0%
X3: Quadratic model (a > 0)	86.0%	95.5%
X4: Quadratic model (a < 0)	91.0%	93.0%
X5: Sinusoidal model	47.5%	67.5%
X6: Exponential model (a > 0)	86.5%	93.0%
X7: Exponential model (a < 0)	80.0%	95.5%
Average recognition accuracy	75.6%	90.3%

Multi-spectral radiation thermometry based on an Alpha spectrum-LM algorithm under the background of high temperature and intense reflection

Abstract

1. Introduction

2. Multi-spectral radiation temperature measurement method based on an Alpha spectrum-LM algorithm under the background of high temperature and intense reflection

2.1 Basic principle of multi-spectral radiation thermometry under high temperature background

2.2 High temperature background reflected radiation analysis model

2.3 Emissivity model recognition algorithm based on an Alpha spectrum-LM algorithm

2.3.1 Alpha spectrum-LM algorithm basic principles

2.3.2 Alpha spectrum-LM algorithm performance analysis

2.3.3 Particle swarm optimization least squares support vector machine (PSO-LSSVM) emissivity model identification classifier

3. Algorithm simulation comparison and analysis

3.1 Comparative analysis of emissivity model recognition algorithms

3.2 Radiation temperature measurement of turbine blades under variable high temperature background

4. Experiment analysis

4.1 Experimental setup

4.2 Experimental results and analysis

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (2)

Equations (35)

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