Venturimeter Discharge and Pressure Difference Calculation
Problem Statement
A 20 cm x 10 cm venturimeter is mounted in a vertical pipeline carrying oil of specific gravity 0.8 flowing upward. The throat section is located 20 cm above the inlet of the venturimeter. A differential U-tube manometer indicates a gauge deflection of 25 cm. With a discharge coefficient (Cd) of 0.96, calculate:
- The discharge of oil (Q)
- The pressure difference between the inlet and the throat (P₁ – P₂)
Given Data
| Inlet Diameter (d₁) | 20 cm = 0.2 m |
| Inlet Area (A₁) | (π/4) × (0.2)² ≈ 0.0314 m² |
| Throat Diameter (d₂) | 10 cm = 0.1 m |
| Throat Area (A₂) | (π/4) × (0.1)² ≈ 0.00785 m² |
| Specific Gravity of Oil (S₀) | 0.8 |
| Density of Oil (ρ) | 0.8 × 1000 = 800 kg/m³ |
| Elevation Difference (Z₂ – Z₁) | 20 cm = 0.2 m |
| Manometer Deflection (x) | 25 cm = 0.25 m |
| Specific Gravity of Mercury (S) | 13.6 |
| Discharge Coefficient (Cd) | 0.96 |
| Acceleration due to Gravity (g) | 9.81 m/s² |
1. Determining the Head (h)
The head is calculated using the manometer reading:
Substituting the values:
h = 0.25 m × (13.6/0.8 – 1) = 0.25 × (17 – 1) = 0.25 × 16 = 4 m
2. Calculating the Discharge (Q)
The discharge is given by:
Substituting the values:
Q = 0.96 × √(2 × 9.81 × 4) × (0.0314 × 0.00785) / √(0.0314² – 0.00785²)
Evaluating the above expression gives approximately Q ≈ 0.0689 m³/s.
3. Calculating the Pressure Difference (P₁ – P₂)
The head can also be expressed as:
Rearranging for (P₁ – P₂):
(P₁ – P₂) = ρg × (h – (Z₁ – Z₂))
Note: Since the throat is 20 cm above the inlet, Z₁ – Z₂ = -0.2 m (or considering the magnitude, 0.2 m added to h). Thus, the effective head for pressure difference becomes: h + 0.2 = 4 + 0.2 = 4.2 m.
Now, substituting the values:
(P₁ – P₂) = 800 kg/m³ × 9.81 m/s² × 4.2 m ≈ 32962 N/m²
Conclusion
The venturimeter analysis shows that the discharge of oil is approximately 0.0689 m³/s, and the pressure difference between the inlet and the throat sections is about 32962 N/m². The calculation uses the manometer head corrected for the difference in specific gravities, the geometric areas of the inlet and throat, and the discharge coefficient.
