Discharge Analysis of a Rectangular Orifice
Problem Statement
A rectangular orifice 1.0 m wide and 1.5 m deep is discharging water from a vessel. The top edge of the orifice is 0.8 m below the water surface in the vessel. Calculate the discharge through the orifice if Cd = 0.6. Also, calculate the percentage error if the orifice is treated as a small orifice.
Given Data
| Width of orifice (b) | 1.0 m |
| Depth of orifice (d) | 1.5 m |
| Top edge below water (H1) | 0.8 m |
| Bottom edge below water (H2) | H1 + d = 0.8 + 1.5 = 2.3 m |
| Coefficient of Discharge (Cd) | 0.6 |
Calculations
Step 1: Discharge through Large Orifice
The discharge (Q) through a large orifice is given by:
Q = (2/3) × Cd × b × √(2g) × (H23/2 - H13/2)Substituting the given values (with g = 9.81 m/s²):
Q = (2/3) × 0.6 × 1.0 × √(2×9.81) × (2.33/2 - 0.83/2)Evaluating the expression yields:
Q ≈ 4.90 m³/sStep 2: Discharge through Small Orifice Approximation
For a small orifice, a representative head is taken as the average of the top and bottom edges:
h = H1 + d/2 = 0.8 + (1.5/2) = 1.55 mThe area of the orifice is:
A = b × d = 1.0 × 1.5 = 1.5 m²The discharge (Q₁) using the small orifice approximation is:
Q₁ = Cd × A × √(2g × h) Q₁ = 0.6 × 1.5 × √(2×9.81×1.55)Evaluating the expression yields:
Q₁ ≈ 4.96 m³/sStep 3: Percentage Error Calculation
The percentage error when treating the orifice as a small orifice is given by:
% error = ((Q₁ - Q) / Q) × 100 % error = ((4.96 - 4.90) / 4.90) × 100 ≈ 1.22%Q ≈ 4.90 m³/s
Q₁ ≈ 4.96 m³/s
% error ≈ 1.22%
Conclusion
The discharge through the rectangular orifice is approximately 4.90 m³/s. When approximated as a small orifice, the discharge is about 4.96 m³/s, leading to a percentage error of roughly 1.22%. This indicates that, for this orifice geometry, the small orifice approximation introduces only a minor error.