Series Pipeline Flow Analysis
Problem Statement
Water flows through series pipelines A-B-C with branches C-D and C-E:
- Pipe AB: d₁ = 50mm
- Pipe BC: d₂ = 75mm, V₂ = 2 m/s
- Pipe CE: d₄ = 30mm
- Pipe CD: V₃ = 1.5 m/s, Q₃ = 2Q₄
- Flow rate Q₁ and velocity V₁ in pipe AB
- Flow rate Q₂ in pipe BC
- Diameter d₃ and velocity V₄
Given Data
| d₁ = 50mm = 0.05m | A₁ = π/4 × (0.05)² = 0.001963 m² |
| d₂ = 75mm = 0.075m | A₂ = π/4 × (0.075)² = 0.004418 m² |
| d₄ = 30mm = 0.03m | A₄ = π/4 × (0.03)² = 0.000707 m² |
| V₂ = 2 m/s | V₃ = 1.5 m/s |
1. Flow Rate in BC (Q₂)
Q₂ = A₂ × V₂ = 0.004418 × 2 = 0.008836 m³/s
2. Flow in AB (Q₁ & V₁)
Q₁ = Q₂ = 0.008836 m³/s (Continuity)
V₁ = Q₁/A₁ = 0.008836/0.001963 = 4.5 m/s
3. Branch Flow Distribution
Q₂ = Q₃ + Q₄ = 3Q₄ (Given Q₃ = 2Q₄)
Q₄ = 0.008836/3 = 0.002945 m³/s
V₄ = Q₄/A₄ = 0.002945/0.000707 = 4.17 m/s
4. Pipe CD Parameters
Q₃ = 2Q₄ = 0.005891 m³/s
A₃ = Q₃/V₃ = 0.005891/1.5 = 0.003927 m²
d₃ = √(4A₃/π) = √(4×0.003927/π) = 0.07m (70mm)
Physical Significance
Key observations:
- Constant flow rate in series pipes (Q₁ = Q₂)
- Velocity reduction in larger pipe BC (4.5 m/s → 2 m/s)
- Smallest pipe CE has highest velocity (4.17 m/s)
- Flow division ratio affects branch pipe sizing
- Velocity increases in smaller diameter pipes (A₄ < A₃ < A₂)
- Demonstrates continuity equation in complex systems
