A centrifugal compressor has a constant-width radial diffuser. The diameters at the
diffuser inlet and outlet are 0.2 m and 0.3 m, respectively. The flow at the diffuser
inlet and outlet is assumed to be steady and uniform. The average velocity at the
diffuser inlet and outlet are (60 eĢr + 75 eĢš) m/s and (u eĢr + 50 eĢš) m/s,
respectively. If the flow through the diffuser is treated as steady and
incompressible, the value of u is ________ (rounded off to the nearest integer).
Official Answer Key:
40.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.61)**
**1. Intuitive Physical Setup & Given Parameters**:
Centrifugal compressor constant-width radial diffuser:
- Diffuser inlet: diameter $D_1 = 0.2\text{ m}$ ($r_1 = 0.1\text{ m}$), velocity $\vec{V}_1 = 60\hat{e}_r + 75\hat{e}_\theta\text{ m/s}$.
- Diffuser outlet: diameter $D_2 = 0.3\text{ m}$ ($r_2 = 0.15\text{ m}$), velocity $\vec{V}_2 = u\hat{e}_r + 50\hat{e}_\theta\text{ m/s}$.
- Flow is steady, uniform, and incompressible ($\rho = \text{const}$). Constant channel axial width $b_1 = b_2 = b$.
We need to find the radial velocity component $u$ at the outlet in $\text{m/s}$ (nearest integer).
**2. Step-by-Step Fluid Dynamic Derivation**:
1. **Conservation of Mass for Incompressible Flow**:
Mass flow rate through any cylindrical surface of radius $r$ and axial width $b$ is:
$\dot{m} = \rho A_r V_r = \rho (2\pi r b) V_r$
Since flow is steady, incompressible ($\rho = \text{const}$), and the diffuser width is constant ($b = \text{const}$):
$r_1 V_{r1} = r_2 V_{r2}$
$D_1 V_{r1} = D_2 V_{r2}$
2. **Substitute Known Values**:
- $D_1 = 0.2\text{ m}$
- $V_{r1} = 60\text{ m/s}$
- $D_2 = 0.3\text{ m}$
- $V_{r2} = u$
3. **Solve for $u$**:
$0.2 \times 60 = 0.3 \times u$
$12 = 0.3 u \implies u = \frac{12}{0.3} = 40\text{ m/s}$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $u = 40\text{ m/s}$.
- **Rounding Requirement**: Nearest integer $\implies \mathbf{40}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **40.0** (Acceptable Range: **40 to 40**)
- **Exam Takeaway**: For an incompressible radial diffuser of constant width, radial velocity obeys 1D cylindrical continuity: $V_r \propto 1/r$. Hence $u = V_{r1} (r_1/r_2) = 60 \times (0.2/0.3) = 40\text{ m/s}$.
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