An ideal Brayton cycle has a compressor pressure ratio $r_p = 16$. Taking $\gamma = 1.4$, what is the thermal efficiency $\eta_{th}$ of this ideal cycle? (Round off to two decimal places).

Answer Key:

0.55

### Solution: $\eta_{th} = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}} = 1 - \frac{1}{16^{0.4/1.4}} = 1 - \frac{1}{16^{0.285714}}$. $16^{0.285714} = (2^4)^{0.285714} = 2^{1.142857} \approx 2.208$. $\eta_{th} = 1 - \frac{1}{2.208} = 1 - 0.4529 = 0.5471 \approx 0.55$ (or $55\%$). Correct: 0.55 (range 0.53 to 0.57).

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