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Orifice Flow Equation
The orifice equation calculates flow through a sharp-edged opening under gravity. The discharge coefficient (Cd) accounts for real-world losses from friction and flow contraction (vena contracta). Sharp-edged orifices typically have Cd ≈ 0.62; rounded entrances reach 0.95–0.99.
Q = Cd × A × √(2gH)
How It Works
The orifice equation Q = Cd × A × √(2gH) calculates flow through a sharp-edged opening under gravity. The discharge coefficient (Cd) accounts for real-world losses from friction and flow contraction (vena contracta). Sharp-edged orifices typically have Cd ≈ 0.62; rounded entrances reach 0.95–0.99.
Example Problem
A sharp-edged orifice (Cd = 0.62) has a diameter of 50 mm and a head of 3 m. What is the flow rate?
- Area: A = π/4 × 0.05² = 0.001963 m²
- Q = 0.62 × 0.001963 × √(2 × 9.81 × 3)
- Q = 0.001217 × 7.672 = 0.00934 m³/s (9.3 L/s)
When to Use Each Variable
- Solve for Flow Rate — when you know the orifice geometry, discharge coefficient, and head.
- Solve for Discharge Coefficient — when you have measured flow rate and need to calibrate the orifice.
- Solve for Orifice Area — when you need to size an orifice to achieve a target flow rate.
- Solve for Head — when you need to determine the required water level for a target flow.
Key Concepts
The orifice flow equation derives from Bernoulli's principle and Torricelli's theorem. The discharge coefficient (Cd) corrects for real-world effects: flow contraction at the vena contracta and frictional losses. Sharp-edged orifices have Cd around 0.62 because the jet contracts to about 62% of the orifice area. Well-rounded or bell-mouth entrances approach Cd = 1.0 by minimizing contraction.
Applications
- Flow measurement: orifice plates in pipelines create a measurable pressure drop proportional to flow rate squared
- Dam and reservoir engineering: sizing outlet works and spillway openings for controlled discharge
- Irrigation: designing tank and canal outlet structures for gravity-fed water delivery
- Industrial processes: controlling flow through nozzles, valves, and pressure-relief devices
Common Mistakes
- Using the wrong discharge coefficient — Cd varies significantly between sharp-edged (0.62), short-tube (0.80), and rounded (0.95+) orifice geometries
- Measuring head to the center of the orifice instead of to the centerline of the jet — for large orifices, the head should be measured to the centroid of the opening
- Neglecting approach velocity — the simple formula assumes the upstream reservoir is large; for small tanks where approach velocity is significant, a velocity-of-approach correction is needed
Frequently Asked Questions
What is the discharge coefficient for an orifice?
Cd accounts for energy losses as fluid passes through the opening. A sharp-edged orifice has Cd ≈ 0.61–0.65. Well-rounded entrances approach 0.95–0.99.
What is the difference between an orifice and a nozzle?
An orifice is a thin plate with a hole; a nozzle has a converging profile that guides flow smoothly. Nozzles have higher discharge coefficients (0.95+) because they reduce turbulence and flow separation.
How is orifice flow used for flow measurement?
Orifice plates installed in pipelines create a measurable pressure drop proportional to flow rate squared. By measuring the differential pressure, engineers can accurately determine the flow rate using a calibrated Cd.
Related Calculators
- Venturi Meter Calculator — a low-loss alternative to orifice plates for flow measurement.
- Bernoulli Theorem Calculator — the energy principle behind the orifice equation.
- Fluid Pressure Calculator — calculate hydrostatic head driving orifice flow.
- Continuity Equation Calculator — relate flow rate, area, and velocity at the orifice.
- Reynolds Number Calculator — determine the flow regime affecting the discharge coefficient.
- Pressure Unit Converter — convert between head and pressure units for orifice calculations.
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