An axial compressor stage has a symmetric velocity diagram with a degree of reaction $R = 0.50$. The rotor blade mean radius is $r_m = 0.40\text{ m}$ and rotates at $N = 9{,}000\text{ rpm}$. The axial velocity is uniform and constant throughout the stage at $C_a = 150\text{ m/s}$. If the absolute air flow enter the rotor with an inlet swirl angle $\alpha_1 = 15^\circ$, the rotor relative air exit flow angle $\beta_2$ (in degrees) is:

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In axial turbomachinery, the degree of reaction $R$ is defined as the ratio of static enthalpy rise across the rotor to the total stage stagnation enthalpy rise: $$R = \frac{h_2 - h_1}{h_{03} - h_{01}} = \frac{C_a}{2 U} (\tan \beta_1 + \tan \beta_2)$$ For the special and widely used case of symmetric blading ($50\%$ reaction stage, $R = 0.50$): $$\alpha_1 = \beta_2 \quad \text{and} \quad \beta_1 = \alpha_2$$ where $\alpha$ denotes absolute flow angles and $\beta$ denotes relative flow angles measured from the axial direction. ### 2. Analytical Formulation & Symmetry Proof The velocity triangles at rotor entry and exit have: $$C_{\theta 1} = C_a \tan \alpha_1, \quad W_{\theta 1} = U - C_{\theta 1} = C_a \tan \beta_1$$ $$C_{\theta 2} = U - W_{\theta 2} = U - C_a \tan \beta_2, \quad W_{\theta 2} = C_a \tan \beta_2$$ Reaction is: $$R = 1 - \frac{C_{\theta 1} + C_{\theta 2}}{2 U} = 1 - \frac{C_a \tan \alpha_1 + U - C_a \tan \beta_2}{2 U} = \frac{1}{2} - \frac{C_a}{2U}(\tan \alpha_1 - \tan \beta_2)$$ Setting $R = 0.50$ requires: $$\frac{C_a}{2U} (\tan \alpha_1 - \tan \beta_2) = 0 \implies \tan \beta_2 = \tan \alpha_1$$ Therefore: $$\beta_2 = \alpha_1$$ ### 3. Step-by-Step Calculation Given that $\alpha_1 = 15^\circ$: $$\beta_2 = \alpha_1 = 15^\circ$$ ### 4. Distractor Forensics - Candidates frequently perform tedious, error-prone trigonometry computing blade peripheral speed $U = \frac{2\pi \times 9000 \times 0.40}{60} = 376.99\text{ m/s}$ and stage work without realizing that $R = 0.50$ identically dictates symmetric velocity triangles ($\beta_2 = \alpha_1 = 15^\circ$). ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always recall the primary consequence of $50\%$ degree of reaction in axial flow compressors: $\alpha_1 = \beta_2$ and $\beta_1 = \alpha_2$. - **Standard Reference:** Saravanamuttoo, H.I.H., *Gas Turbine Theory*, 6th Edition, Pearson, Chapter 5 (Axial Flow Compressors).

← Prev Question Next Question →