An ideal Joule-Brayton gas turbine cycle operates between maximum cycle temperature $T_3 = 1{,}600\text{ K}$ and minimum cycle temperature $T_1 = 300\text{ K}$ with air ($\\gamma = 1.4$). The cycle pressure ratio $r_p = p_2 / p_1$ that produces the **maximum net specific work output** per unit mass of air is closest to:

(A) 18.7

(B) 11.4

(C) 28.5

(D) 6.2

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In an ideal Brayton cycle, net specific work is: $$w_{\text{net}} = c_p (T_3 - T_4) - c_p (T_2 - T_1) = c_p \left[ T_3 \left(1 - \frac{1}{r_p^{(\gamma-1)/\gamma}}\right) - T_1 \left(r_p^{(\gamma-1)/\gamma} - 1\right) \right]$$ Differentiating $w_{\text{net}}$ with respect to temperature ratio $\tau_r = r_p^{(\gamma-1)/\gamma}$ and setting to zero yields the optimum temperature ratio: $$\tau_{r,\text{opt}} = \sqrt{\frac{T_3}{T_1}}$$ Therefore, the pressure ratio for maximum specific work is: $$r_{p,\text{opt}} = \left(\frac{T_3}{T_1}\right)^{\frac{\gamma}{2(\gamma - 1)}}$$ ### 2. Analytical Formulation & Calculation Given: - $T_3 = 1{,}600\text{ K}$ - $T_1 = 300\text{ K}$ - $\gamma = 1.4 \implies \frac{\gamma}{2(\gamma - 1)} = \frac{1.4}{2(0.4)} = \frac{1.4}{0.8} = 1.75$ $$\frac{T_3}{T_1} = \frac{1600}{300} = \frac{16}{3} \approx 5.3333$$ ### 3. Step-by-Step Calculation $$r_{p,\text{opt}} = (5.3333)^{1.75}$$ Taking logarithms: $$\ln(r_{p,\text{opt}}) = 1.75 \times \ln(5.3333) = 1.75 \times 1.67398 = 2.92946$$ $$r_{p,\text{opt}} = e^{2.92946} \approx 18.717 \approx 18.7$$ ### 4. Distractor Forensics - **Option A (18.7):** Correct optimum pressure ratio for maximum specific work. - **Option B (11.4):** Used exponent $\gamma / (\gamma - 1) = 3.5$ with square root inside ($5.333^{0.875} = 4.3$). - **Option C (28.5):** Maximum pressure ratio where work drops to zero ($r_{p,\max} = (T_3/T_1)^{3.5} = 5.333^{3.5} \approx 350$). ### 5. Exam Trap & Standard Reference - **Trap Alert:** Distinguish pressure ratio for maximum work output ($r_p = (T_{\max}/T_{\min})^{1.75}$) from pressure ratio for maximum thermal efficiency (which increases monotonically with $r_p$). - **Standard Reference:** Mattingly, J.D., *Elements of Propulsion*, Chapter 4.

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