An ideal air-standard Brayton cycle operates with a compressor pressure ratio $r_p = 8.0$. Taking the ratio of specific heats for air as $\gamma = 1.4$, what is the thermal efficiency (in %) of the cycle?

Answer Key:

44.8

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Brayton Efficiency Equation:** $\eta = 1 - r_p^{-(\gamma-1)/\gamma}$. 2. **Stage 2: Exponent Calculation:** $(\gamma - 1)/\gamma = 0.4 / 1.4 = 2/7 \approx 0.285714$. 3. **Stage 3: Pressure Ratio Factor:** $8^{0.285714} = (2^3)^{2/7} = 2^{6/7} \approx 1.811447$. 4. **Stage 4: Thermal Efficiency:** $\eta = 1 - (1 / 1.811447) = 1 - 0.55204 = 0.44796 \approx 44.80\%$. 5. **Stage 5: Acceptance Range:** 44.5 to 45.1.

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