A high-pressure gas turbine stage uses internal convection cooling for its rotor blades. The mainstream combustion gas temperature is $T_g = 1{,}500\text{ K}$ and the cooling air supply temperature inside the blade passage is $T_c = 750\text{ K}$. If the blade cooling thermal effectiveness is $\epsilon_{\text{cool}} = \frac{T_g - T_b}{T_g - T_c} = 0.60$, the external blade surface metal temperature $T_b$ (in $\text{K}$) is:

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### 1. Strategy & Theoretical Principles Blade cooling effectiveness $\epsilon_{\text{cool}}$ is defined as the actual temperature reduction of the blade metal relative to the maximum theoretical cooling potential: $$\epsilon_{\text{cool}} = \frac{T_g - T_b}{T_g - T_c}$$ where: - $T_g$ is the hot gas total temperature - $T_b$ is the average blade metal surface temperature - $T_c$ is the coolant air temperature. ### 2. Analytical Formulation & Calculation Rearranging directly for blade temperature $T_b$: $$T_g - T_b = \epsilon_{\text{cool}} (T_g - T_c)$$ $$T_b = T_g - \epsilon_{\text{cool}} (T_g - T_c)$$ Substitute the given values: - $T_g = 1{,}500\text{ K}$ - $T_c = 750\text{ K}$ - $\epsilon_{\text{cool}} = 0.60$ $$T_g - T_c = 1500 - 750 = 750\text{ K}$$ $$T_b = 1500 - (0.60 \times 750) = 1500 - 450 = 1050\text{ K}$$ ### 3. Distractor Forensics - Candidate evaluating $(T_b - T_c)/(T_g - T_c) = 0.60$ gets $T_b = 1200\text{ K}$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Be sure of which temperature difference appears in the numerator of thermal cooling effectiveness. - **Standard Reference:** Mattingly, J.D., *Elements of Propulsion*, Chapter 6 (Turbines).

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