## 2 Theoretical Analysis of Reflection Error Correction for Ambient Radiation of Turbine Blade  
### Ambient Radiation Analysis of Turbine Blades  
Figure 1 shows a schematic cross-sectional view of an arrangement of turbine engine blades. When the engine is started, the high-pressure gas generated in the combustion  
chamber passes backwards through the various guide vanes and rotor blades in turn. The guide vanes are fixed, and the rotor blades rotate at high speed around the axis of the engine driven by the gas. Among them, the first-stage rotor blades are most likely to be damaged because of the high temperature and high-speed rotation. This paper studies the reflection and temperature inversion of the trailing edge of the suction surface of the rotor blade measured by the radiation pyrometer probe in Figure 1. The main reflection sources are the adjacent guide vanes and the upper rotor blade. Radiation from these adjacent blades to the target blade was calculated. The low radiation from the casing wall was ignored due to the low temperature.  
According to [PERSON]'s law, the radiant emittance of an object whose temperature is higher than 0 K to the surrounding space is expressed as  
\[M_{B}(\lambda,T)=c_{1}\lambda^{-5}\left(e^{2/\lambda T}-1\right)^{-1} \tag{1}\]  
where \(M_{B}(\lambda,T)\) is the radiant emittance of a blackbody with temperature \(T\) at wavelength \(\lambda\), \(c_{1}=3.7418\times 10^{8}\) W \(\mu\)m\({}^{4}\)/m\({}^{2}\) is the first radiation constant, and \(c_{2}=1.4388\times 10^{4}\)\(\mu\)m K is the second radiation constant.  
Under the influence of reflection, the total radiant emittance of a point on the surface of the target blade, \(M_{total}\), should include the heat radiation of the target point itself and the reflection of ambient radiation and is expressed by  
\[M_{total}=\varepsilon_{b}M_{B}(T)+(1-\varepsilon_{b})E_{e} \tag{2}\]  
where \(E_{e}\) is the irradiance of the target point radiated by the surrounding environment and \(\varepsilon_{b}\) is the emissivity of the target blade. For opaque surfaces, the sum of the emissivity and the reflectivity is 1.  
A reflection model was established to analyze the radiation propagating from the adjacent blades to the target blade. Figure 2 shows the principle of the reflection model of a discretized blade. The adjacent blade is divided into many small panels, and the radiation angle factor of each panel is calculated. When considering the effect of reflection (or ambient radiation), \(M_{total}\) can also be expressed as [20]  
\[M_{total}=\varepsilon_{b}M_{B}(T)+(1-\varepsilon_{b})\sum_{i=1}^{n}M_{i}(x_{i },y_{i},z_{i})F_{i} \tag{3}\]  
Figure 1: Schematic diagram of engine turbine blade distribution and infrared pyrometer probe temperature measurement.  
In Equation (3), \(M_{i}\) is the total radiantitance of the \(i\)-th panel on the adjacent blade, and \(F_{i}\) is the radiation angle factor between panel \(i\) and the target point. Combined with the total radiant emittance of the adjacent blade surface, the radiation illuminance of the adjacent blade to the target blade can be calculated to correct for the reflection error of the target blade. To accurately calculate the radiation angle factor, it is necessary to establish a geometric model of the turbine blade in a three-dimensional space according to the actual shape of the turbine blade.