Pipe Flow Analysis: Branching System
Problem Statement
A 40cm diameter water pipe branches into two pipes (30cm and 20cm diameter). Given:
- Main pipe discharge: 0.38 m³/s
- 30cm pipe velocity: 2 m/s
- Velocity in main pipe
- Discharge in 20cm pipe
- Velocity in 20cm pipe
Given Data
| d₁ = 40cm = 0.4m | A₁ = π/4 × (0.4)² = 0.1256 m² |
| d₂ = 30cm = 0.3m | A₂ = π/4 × (0.3)² = 0.07068 m² |
| d₃ = 20cm = 0.2m | A₃ = π/4 × (0.2)² = 0.0314 m² |
1. Velocity in Main Pipe (V₁)
Q₁ = A₁ × V₁
V₁ = Q₁/A₁ = 0.38/0.1256 = 3.025 m/s
2. Discharge in 30cm Pipe (Q₂)
Q₂ = A₂ × V₂ = 0.07068 × 2 = 0.1413 m³/s
3. Continuity Equation Application
Q₁ = Q₂ + Q₃
Q₃ = 0.38 – 0.1413 = 0.2387 m³/s
4. Velocity in 20cm Pipe (V₃)
V₃ = Q₃/A₃ = 0.2387/0.0314 = 7.6 m/s
Physical Significance
Key observations from the solution:
- The velocity increases significantly (3.025 → 7.6 m/s) in smaller diameter pipe due to area reduction
- Continuity equation maintains mass conservation: Q₁ = Q₂ + Q₃
- Velocity in 20cm pipe is 3.75× faster than in 30cm pipe despite similar diameter ratio
- High velocities in smaller pipes may cause erosion issues
- Proper pipe sizing crucial for system pressure management
- Demonstrates inverse square relationship between diameter and velocity
