## 2 Theoretical Analysis of Reflection Error Correction for Ambient Radiation of Turbine Blade
### Correction Method of Reflected Energy Based on Total Radiation Exitance of Adjacent Surface  
This paper proposes a method of collecting radiation signals of the adjacent surface to correct for the reflection error in the process of radiation temperature measurement. Equation (3) shows that when ambient radiation is considered, the total radiant exitance of the target surface can be expressed as  
\[M_{total}=\varepsilon_{b}M_{B}(T)+(1-\varepsilon_{b})\sum_{i=1}^{n}M_{i}(x_{i },y_{i},z_{i})F_{i} \tag{16}\]  
In the narrow working wavelength range of the radiation pyrometer, the emissivity and radiation emission are considered, to a high approximation, not to change with the wavelength. Therefore, the actual signal \(S_{total}\) output by the radiation pyrometer can be expressed as  
\[S_{total}=M_{total}A_{b}\Omega_{b}g(\lambda)\Delta\lambda/2\pi \tag{17}\]  
where \(A_{b}\) is the area of the object surface of the radiation pyrometer; \(\Omega_{b}\) is the solid angle of the entrance pupil of the optical system of the radiation pyrometer to a point on the object surface; \(g(\lambda)\) is the product of the optical system's transmittance, detector response coefficient, and other functions related to the wavelength; and \(\Delta\lambda\) is the operating bandwidth of the radiation pyrometer. Then, from the perspective of ambient radiation, according to Equations (16) and (17), the actual output signal of the radiation pyrometer can be expressed as  
\[S_{total}=\varepsilon_{b}M_{B}(T)A_{b}\Omega_{b}g(\lambda)\Delta\lambda/2\pi+ (1-\varepsilon_{b})\times\sum_{i=1}^{i=n}M_{i}(x_{i},y_{i},z_{i})A_{b}\Omega_{ b}g(\lambda)\Delta\lambda/2\pi\;F_{i} \tag{18}\]  
For the surface of the adjacent blades, \(M_{e_{i}}\) is the total radiant exitance on the \(i\)-th panel of the adjacent blade surface, which includes heat radiation and reflection. As the target surface interrogated by the probe is very small (r = 1 mm), the temperatures of several surface elements contained on the probe target surface are considered equal. The total number of panels of the adjacent blades becomes \(n^{\prime}\), and the actual output signal of the detector can be expressed as follows:  
\[S_{total}=S_{b}+(1-\varepsilon_{b})\sum_{i^{\prime}=1}^{n^{\prime}=n^{\prime}} S_{e_{i^{\prime}}}\;F^{\prime}_{i^{\prime}} \tag{19}\]  
\(S_{total}\) is the actual voltage signal output of the detector, \(S_{b}\) is the voltage signal of thermal radiation on the target surface, and \(S_{e_{i^{\prime}}}\) is the voltage signal obtained using the same pyrometer to measure the \(i^{\prime}\)-th panel of the adjacent blades' surface; the last item includes the signal caused by the thermal radiation of adjacent blades themselves and reflection of other radiation sources on the blades' surface. \(F^{\prime}_{i^{\prime}}\) is the sum of the angle factors of several panels contained in the object plane of the radiation pyrometer. The radius of the object plane of a radiation pyrometer is generally on the order of millimeters, so it can be considered that the angular factors of the adjacent blade radiation at each point in the object plane are equal. The second term at the right end of Equation (19) is the signal response to the reflected energy, which can be calculated by the angle factor and the actual measured radiation signal without referring to the actual temperature and emissivity of the adjacent blade surface. The temperature measurement error can thus be reduced accordingly. As a versatile correction method, it benefits the temperature measurement performance of both single-wavelength and multi-wavelength frames.