## 3 Three-Wavelength Radiation Temperature Measurement Based on Reflection Correction  
It is difficult to measure the emissivity of running turbine blades. For targets with unknown emissivity, the colorimetric temperature measurement algorithm is practical to measure the radiation temperature. However, suffering from both the ambient radiation and the emissivity difference with the wavelength, the relative temperature measurement error using the colorimetric algorithm could be higher than 15% [21]. Based on the aboveanalysis of reflection error correction, the proposed algorithm has proven feasible in the radiation temperature measurement of a surface with unknown surface emissivity. To eliminate the influence of blade surface emissivity, it is necessary to select multiple wavelengths for radiation measurement. When performing radiation temperature measurement of turbine blades, according to Equation (2), the response of the detector at a certain wavelength can be expressed as  
\[S=\epsilon_{b}S_{B}(T)+(1-\epsilon_{b})S_{e}+S_{D} \tag{20}\]  
where \(S\) is the detector's voltage signal output, \(\epsilon_{b}\) is the surface emissivity, \(S_{B}(T)\) is the voltage signal of a blackbody measured by a radiation pyrometer with the same temperature as the target surface, \(S_{e}\) is the voltage signal of the ambient radiation (obtained by combining the simulation results and measurement results of the reflection model), and \(S_{D}\) is the detector noise. By deriving Equation (20), we obtain  
\[S-S_{D}-S_{e}=\epsilon_{b}(S_{B}(T)-S_{e}) \tag{21}\]  
Substituting wavelength 1 and wavelength 2 into Equation (21), we can obtain  
\[S_{1}-S_{e1}-S_{D1}=\epsilon_{b1}(S_{B1}(T)-S_{e1}) \tag{22}\]  
\[S_{2}-S_{e2}-S_{D2}=\epsilon_{b2}(S_{B2}(T)-S_{e2}) \tag{23}\]  
We can divide Equation (22) and Equation (23) to obtain  
\[\frac{S_{2}-S_{e2}-S_{D2}}{S_{1}-S_{e1}-S_{D1}}=\frac{\epsilon_{b2}}{\epsilon_ {b1}}\times\frac{S_{B2}(T)-S_{e2}}{S_{B1}(T)-S_{e1}} \tag{24}\]  
Ignoring the difference in emissivity between the two wavelengths, we define the error function \(\epsilon_{1}(T)\):  
\[\epsilon_{1}(T)=\frac{S_{2}-S_{e2}-S_{D2}}{S_{1}-S_{e1}-S_{D1}}-\frac{S_{B2}(T )-S_{e2}}{S_{B1}(T)-S_{e1}} \tag{25}\]  
Then, the solution when \(e_{1}(T)=0\) is the true temperature of the target point. Combined with the mathematical properties of the Planck function, function \(e_{1}(T)\) satisfies the following conditions:  
(1) The values of \(e_{1}(0)\) and the limit \(\lim\limits_{T\rightarrow\infty}e_{1}(T)\) exist and are finite real numbers;  
(2) In the domain \(\left[0,{S_{B1}}^{-1}(S_{e1})\right)\), the function \(e_{1}(T)\) has a discontinuity point only at \(T={S_{B1}}^{-1}(S_{e1})\), and this point is the infinite discontinuity point of the function \(e_{1}(T)\). The left limit and the right limit of the function \(e_{1}(T)\) at this point have opposite signs. In the defined intervals \(\left[0,{S_{B1}}^{-1}(S_{e1})\right)\) and \(\left({S_{B1}}^{-1}(S_{e1}),\infty\right)\), the function \(e_{1}(T)\) is continuous. According to the zero-point theorem, if the function \(e_{1}(T)\) satisfies the following two conditions at the same time:  
\[e_{1}(0)\times\lim\limits_{T\rightarrow{S_{B1}}^{-1}(S_{e1})}e_{1}(T)<0 \tag{26}\]  
\[\lim\limits_{T\rightarrow{S_{B1}}^{-1}(S_{e1})}e_{1}(T)\times\lim\limits_{T \rightarrow\infty}e_{1}(T)<0 \tag{27}\]  
Function \(e_{1}(T)\) will have two zeros in the domain. Therefore, using only two wavelengths will result in two temperature solutions for temperature inversion. When the target blade temperature is 1150 K, the guide vane temperature is 100 K lower, the upper rotor blade temperature is the same as the target blade temperature, the target blade emissivity is 0.58, detector noise is ignored, and two adjacent wavelengths of 1600 nm and 1500 nm are selected.