A single-stage axial flow gas turbine operates with zero exit swirl (axial exit flow, $\alpha_3 = 0^\circ$). The mean rotor blade speed is $U = 320\text{ m/s}$ and the axial flow velocity is constant at $C_a = 240\text{ m/s}$. If the absolute gas velocity entering the rotor has a swirl component $C_{\theta 2} = 500\text{ m/s}$, the specific power output $w$ extracted by the turbine (in $\text{kJ/kg}$) is:

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### 1. Strategy & Theoretical Principles From Euler's turbine work equation: $$w = U (C_{\theta 2} - C_{\theta 3})$$ where: - $U$ is the mean peripheral blade speed - $C_{\theta 2}$ is the absolute swirl velocity at rotor inlet - $C_{\theta 3}$ is the absolute swirl velocity at rotor exit. ### 2. Analytical Formulation & Calculation Given: - Axial discharge with zero exit swirl: $C_{\theta 3} = 0$ - Rotor blade speed: $U = 320\text{ m/s}$ - Inlet swirl velocity: $C_{\theta 2} = 500\text{ m/s}$ $$w = U C_{\theta 2} = 320 \times 500 = 160{,}000\text{ J/kg} = 160.0\text{ kJ/kg}$$ ### 3. Distractor Forensics - Candidate multiplying by axial velocity $C_a$ instead of blade speed $U$ gets $240 \times 500 = 120\text{ kJ/kg}$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** In Euler's work equation, the multiplier is the blade speed $U$, not axial velocity $C_a$. - **Standard Reference:** Saravanamuttoo, H.I.H., *Gas Turbine Theory*, Chapter 7.

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