Discharge Calculation Using an Orificemeter in a Vertical Pipeline
Problem Statement
An orificemeter is installed in a vertical pipeline of 250 mm diameter carrying oil with a specific gravity of 0.9. The oil flows upward. The elevation difference between the upstream and downstream manometer taps is 350 mm, and the differential U-tube manometer shows a gauge deflection of 200 mm. Given that the orifice diameter is 150 mm and the discharge coefficient (Cd) is 0.65, calculate the discharge of oil.
Given Data
| Pipe Diameter (d₁) | 250 mm = 0.25 m |
| Pipe Area (A₁) | (π/4) × (0.25)² = 0.049 m² |
| Orifice Diameter (d₂) | 150 mm = 0.15 m |
| Orifice Area (A₂) | (π/4) × (0.15)² = 0.0176 m² |
| Specific Gravity of Oil (S₀) | 0.9 |
| Elevation Difference (Z₁ – Z₂) | 350 mm = 0.35 m |
| Manometer Deflection (x) | 200 mm = 0.2 m |
| Specific Gravity of Mercury (S) | 13.6 |
| Discharge Coefficient (Cd) | 0.65 |
| Acceleration due to Gravity (g) | 9.81 m/s² |
1. Calculating Cross-sectional Areas
Pipe Area (A₁):
A₁ = (π/4) × (0.25)² ≈ 0.049 m²
Orifice Area (A₂):
A₂ = (π/4) × (0.15)² ≈ 0.0176 m²
2. Determining the Head (h)
The head due to the manometer reading is calculated as:
Substituting the given values:
h = 0.2 m × (13.6/0.9 – 1)
h ≈ 0.2 × (15.11 – 1) = 0.2 × 14.11 = 2.82 m
3. Calculating the Discharge (Q)
The discharge through the orifice is given by:
Substituting the values:
Evaluating the above expression gives Q ≈ 0.091 m³/s.
Conclusion
The orificemeter analysis for the vertical pipeline shows that the discharge of oil is approximately 0.091 m³/s. This calculation incorporates the manometer reading adjusted for the difference in specific gravities and the cross-sectional areas of both the pipe and the orifice, along with the discharge coefficient.
