Flow Measurement Using a Horizontal Venturimeter
Problem Statement
A horizontal venturimeter is used to measure the flow of water in a 200 mm diameter pipe. The throat diameter of the venturimeter is 80 mm and the discharge coefficient is 0.85. If the pressure difference between the two measurement points is 10 cm of mercury, calculate the average velocity in the pipe.
Given Data
| Diameter at Inlet (d₁) | 200 mm = 0.2 m |
| Cross-sectional Area at Inlet (A₁) | (π/4) × (0.2)² = 0.0314 m² |
| Diameter at Throat (d₂) | 80 mm = 0.08 m |
| Cross-sectional Area at Throat (A₂) | (π/4) × (0.08)² = 0.00503 m² |
| Pressure Difference (P₁ – P₂) | 10 cm Hg = 13.6 × 9810 × 0.1 Pa |
| Head (h) | 1.36 m |
| Discharge Coefficient (C_d) | 0.85 |
| Acceleration due to Gravity (g) | 9.81 m/s² |
1. Calculating Cross-sectional Areas
At the inlet:
A₁ = (π/4) × (d₁)² = (π/4) × (0.2)² = 0.0314 m²
At the throat:
A₂ = (π/4) × (d₂)² = (π/4) × (0.08)² = 0.00503 m²
2. Converting Pressure Difference to Head
The pressure difference is given as 10 cm of Hg. Using the specific weight of mercury (γ_Hg = 13.6 × 9810 N/m³):
(P₁ – P₂) = 13.6 × 9810 × 0.1 Pa
The head is computed as:
h = (P₁ – P₂) / (ρg) = (13.6 × 9810 × 0.1) / 9810 = 1.36 m
(Note: The inlet and throat are at the same elevation since the pipe is horizontal.)
3. Calculating the Flow Rate (Q)
The flow rate using a venturimeter is given by:
Substituting the given values:
Evaluating this expression gives Q ≈ 0.0223 m³/s.
4. Determining the Average Velocity in the Pipe
The average velocity (V) is calculated as:
Substituting the values:
V = 0.0223 m³/s / 0.0314 m² ≈ 0.71 m/s
Conclusion
In this analysis using a horizontal venturimeter, the flow rate was determined to be 0.0223 m³/s, which results in an average velocity in the 200 mm diameter pipe of approximately 0.71 m/s.
The solution combined the continuity equation and Bernoulli’s principle along with the discharge coefficient to accurately determine the flow characteristics.
