In an axial flow turbine designed according to the free-vortex condition, the tangential velocity component of the gas leaving the nozzle guide vanes satisfies $C_\theta(r) \cdot r = \text{constant}$. If the radius at the blade hub is $r_h = 0.30\text{ m}$ and the radius at the blade tip is $r_t = 0.45\text{ m}$, the ratio of the tangential velocity at the tip to that at the hub $(C_{\theta,t} / C_{\theta,h})$ is:

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### 1. Strategy & Theoretical Principles In turbomachinery radial equilibrium theory, a free-vortex design maintains constant angular momentum across the blade height: $$r C_\theta = K = \text{constant}$$ where $K$ is invariant with radius $r$. This design ensures constant axial velocity and uniform work extraction along the span. ### 2. Analytical Formulation & Calculation Given: - $r_h = 0.30\text{ m}$ - $r_t = 0.45\text{ m}$ From the free-vortex condition: $$r_t C_{\theta,t} = r_h C_{\theta,h} \implies \frac{C_{\theta,t}}{C_{\theta,h}} = \frac{r_h}{r_t}$$ Substitute the values: $$\frac{C_{\theta,t}}{C_{\theta,h}} = \frac{0.30}{0.45} = \frac{2}{3} \approx 0.6667 \approx 0.67$$ ### 3. Distractor Forensics - Candidate inverting the ratio gets $0.45/0.30 = 1.50$ (forced vortex condition $C_\theta \propto r$). ### 4. Exam Trap & Standard Reference - **Trap Alert:** Distinguish free-vortex flow ($C_\theta \propto 1/r$, velocity decreases with radius) from forced vortex solid-body rotation ($C_\theta \propto r$, velocity increases with radius). - **Standard Reference:** Saravanamuttoo, H.I.H., *Gas Turbine Theory*, Chapter 7 (Axial Flow Turbines).

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