AJ Designer

Orifice Flow Design Calculator

Flow rate equals discharge coefficient times orifice area times square root of two times gravity times head

Solution

—
Share:

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?

  1. Identify the known values: discharge coefficient Cd = 0.62, orifice diameter = 50 mm, head H = 3 m, gravitational acceleration g = 9.81 m/s².
  2. Calculate the orifice area from the diameter: A = π/4 × 0.05² = 0.001963 m².
  3. Write the orifice flow equation: Q = Cd × A × √(2gH).
  4. Calculate the term under the square root: 2gH = 2 × 9.81 × 3 = 58.86 m²/s², so √(58.86) = 7.672 m/s.
  5. Substitute all known values: Q = 0.62 × 0.001963 × 7.672.
  6. Compute the result: Q = 0.00934 m³/s (9.34 L/s). This assumes a large upstream reservoir with negligible approach velocity.

A simpler example: Cd = 0.62, A = 0.01 m², H = 2 m → Q = 0.62 × 0.01 × √(39.24) = 0.0388 m³/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

How does orifice size affect the flow rate through it?

Flow rate is directly proportional to orifice area. Doubling the orifice area doubles the flow rate (at the same head and Cd). Since area scales with the square of diameter, doubling the orifice diameter quadruples the flow rate.

What is a typical discharge coefficient for a sharp-edged orifice?

A sharp-edged circular orifice in a large tank has Cd ≈ 0.61–0.65, with 0.62 being the most commonly used design value. The exact value depends on the Reynolds number and the ratio of orifice size to tank size. Well-rounded entrances can reach Cd = 0.95–0.99.

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.

What is the vena contracta in orifice flow?

The vena contracta is the narrowest point of the fluid jet downstream of the orifice, where the streamlines are most contracted. For a sharp-edged orifice, the jet area at the vena contracta is about 62% of the orifice area, which is why Cd ≈ 0.62.

Does water temperature affect orifice flow rate?

Temperature affects viscosity, which in turn affects the discharge coefficient slightly. For most engineering calculations with water between 5–40 °C, the change in Cd is small (< 2%) and the standard value of 0.62 is adequate. For very viscous fluids or extreme temperatures, apply a Reynolds number correction.

Orifice Flow Formula

The orifice equation derives from Bernoulli's principle and Torricelli's theorem:

Q = Cd × Ao × √(2gH)

Where:

  • Q — volumetric flow rate, measured in m³/s
  • Cd — discharge coefficient (dimensionless), accounts for contraction and friction losses
  • Ao — orifice area, measured in m²
  • g — gravitational acceleration, typically 9.81 m/s²
  • H — centerline head above the orifice, measured in meters

The discharge coefficient corrects for real-world effects: flow contraction at the vena contracta (where the jet narrows downstream of the orifice) and frictional losses. Sharp-edged orifices have Cd around 0.62; well-rounded entrances approach 1.0.

Worked Examples

Dam Outlets

What is the discharge from a reservoir through a circular outlet?

A reservoir has a 300 mm diameter circular outlet (Cd = 0.62) with 5 m of head above the centerline. What is the flow rate?

  • Area: A = π/4 × 0.3² = 0.0707 m²
  • Q = 0.62 × 0.0707 × √(2 × 9.81 × 5)
  • Q = 0.0438 × 9.905
  • Q = 0.434 m³/s (434 L/s)

This simple estimate ignores approach velocity and head losses in the conduit. For precise reservoir design, apply a velocity-of-approach correction.

Flow Measurement

What orifice size is needed to pass 50 L/s at 3 m of head?

An orifice plate (Cd = 0.62) must pass 0.05 m³/s with an available head of 3 m. What area is required?

  • Rearrange: A = Q / (Cd × √(2gH))
  • A = 0.05 / (0.62 × √(58.86))
  • A = 0.05 / (0.62 × 7.672)
  • A = 0.01051 m² (diameter ≈ 116 mm)

In practice, round up to the nearest standard pipe size. Recalculate the actual flow rate with the chosen area.

Tank Drainage

What head drives a known flow rate through a tank drain orifice?

A tank drains through a 50 mm sharp-edged orifice (A = 0.001963 m², Cd = 0.62) at 0.005 m³/s. What head is required?

  • Rearrange: H = Q² / (2g × Cd² × A²)
  • H = 0.000025 / (2 × 9.81 × 0.3844 × 0.000003853)
  • H = 0.000025 / 0.00002916
  • H = 0.857 m

As the tank level drops, the head decreases and so does the flow rate. For time-to-drain calculations, integrate the flow rate over the changing head.

Related Calculators

Related Sites