In a vaneless radial diffuser of an aerospace centrifugal compressor, flow is steady, frictionless, and axisymmetric without heat transfer. The flow enters the diffuser at radius $r_1 = 0.25\text{ m}$ with radial velocity $C_{r1} = 120\text{ m/s}$ and tangential swirl velocity $C_{\theta 1} = 300\text{ m/s}$. Neglecting compressibility (constant density $\rho$), at exit radius $r_2 = 0.50\text{ m}$ ($r_2 = 2 r_1$) of constant axial depth, the exit tangential velocity $C_{\theta 2}$ (in $\text{m/s}$) is:

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In a vaneless radial diffuser with frictionless axisymmetric flow: 1. **Angular Momentum Conservation:** With no external torque acting on the fluid in the vaneless space, angular momentum is strictly conserved: $$r C_\theta = \text{constant} \implies C_{\theta 2} = C_{\theta 1} \left(\frac{r_1}{r_2}\right)$$ 2. **Continuity Equation:** For incompressible flow in a constant-depth channel, radial mass flux conservation requires $r C_r = \text{constant}$, giving a logarithmic spiral streamline. ### 2. Analytical Formulation & Calculation Given: - $r_1 = 0.25\text{ m}$ - $r_2 = 0.50\text{ m} = 2 r_1$ - $C_{\theta 1} = 300\text{ m/s}$ $$C_{\theta 2} = 300 \times \left(\frac{0.25}{0.50}\right) = 300 \times 0.50 = 150.0\text{ m/s}$$ ### 3. Distractor Forensics - Candidate assuming solid body rotation ($C_\theta \propto r$) gets $600\text{ m/s}$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Vaneless space flows obey free-vortex conservation ($r C_\theta = \text{const}$). - **Standard Reference:** Saravanamuttoo, H.I.H., *Gas Turbine Theory*, Chapter 4.

← Prev Question Next Question →