Coefficient of Discharge Calculator

Find an orifice's coefficient of discharge (Cd) from its diameter, the liquid head above it, and the actual measured flow rate, using Torricelli's law for the theoretical flow.

Quick Facts

Definition
Cd = Q_actual / Q_theoretical
Ratio of measured flow to the ideal, frictionless flow predicted by theory.
Theoretical flow (Torricelli)
Q_t = a√(2gh)
a = πd²/4 is orifice area; h is the head of liquid above the orifice center.
Typical Cd values
0.61 orifice · 0.98 nozzle · 0.97 venturi
Sharp-edged orifices lose the most flow to the vena contracta.

Your Results

Calculated
Coefficient of Discharge (Cd)
-
Cd = actual flow ÷ theoretical flow
Theoretical Flow Rate
-
Q_t = a × √(2gh), Torricelli's law
Theoretical Jet Velocity
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V = √(2gh), ideal frictionless velocity
Orifice Area
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a = π × diameter² / 4

Ready

Enter the measured flow rate, orifice diameter, and liquid head, then press Calculate.

Formula and Method for the Coefficient of Discharge

The coefficient of discharge (Cd) measures how closely a real orifice, nozzle, or venturi matches the idealized flow predicted by Bernoulli's equation. It is defined simply as Cd = Qactual / Qtheoretical — the flow rate you actually measure, divided by the flow rate an ideal, frictionless fluid would produce through the same opening under the same driving head. This calculator derives the theoretical flow from the orifice diameter and the liquid head above it, then compares it to your measured (actual) flow rate to solve for Cd.

How the calculation works

For a small orifice discharging from a tank under a liquid head h (the vertical distance from the free surface down to the center of the orifice), Torricelli's law gives the ideal, frictionless exit velocity as V = √(2gh), where g = 9.80665 m/s² is standard gravity. Multiplying this velocity by the orifice's cross-sectional area, a = πd²/4, gives the theoretical volumetric flow rate: Qtheoretical = a√(2gh). Dividing your actual (measured) flow rate by this theoretical value yields the coefficient of discharge: Cd = Qactual / Qtheoretical. Because Cd folds in both the vena contracta (the jet narrows just past the opening) and viscous friction losses, it is also expressed as Cd = Cc × Cv, the product of the coefficient of contraction and the coefficient of velocity.

Common mistakes

  • Measuring head from the wrong point: h must be the vertical distance from the free liquid surface to the center of the orifice, not to its top or bottom edge.
  • Mixing units: convert diameter, head, and flow rate to consistent units before comparing results — this calculator handles the conversions internally, but hand calculations often trip on inches vs. millimeters or gallons vs. liters.
  • Expecting Cd = 1: even a perfectly machined sharp-edged orifice will show Cd ≈ 0.60-0.65 because of the vena contracta; only carefully profiled nozzles or venturis approach Cd ≈ 0.95-0.99.
  • Ignoring Reynolds number effects: Cd is not perfectly constant — it can drift at very low flow velocities where viscous effects dominate, so measurements should be taken at representative operating flow rates.

Real-world applications

  • Flow metering: orifice plates, flow nozzles, and venturi meters all rely on a calibrated Cd to convert a measured pressure drop or head into an accurate flow rate.
  • Tank and reservoir drainage: predicting how fast a tank empties through a drain or spillway orifice requires the real (Cd-corrected) flow rate, not the idealized Torricelli value.
  • Irrigation and drainage design: sizing orifices, weirs, and gates for a target discharge uses Cd to size the opening correctly rather than undersizing it.
  • Nozzle and injector design: fuel injectors, spray nozzles, and fire-hose nozzles are characterized by Cd to predict spray flow rate at a given supply pressure.

Frequently Asked Questions

What is the coefficient of discharge?
The coefficient of discharge (Cd) is the ratio of the actual flow rate through an orifice, nozzle, or venturi to the theoretical flow rate predicted by Bernoulli's equation for an ideal, frictionless fluid: Cd = Q_actual / Q_theoretical. It captures real-world losses such as the vena contracta (jet narrowing) and friction that an idealized calculation ignores.
What is a typical coefficient of discharge value for an orifice?
A standard sharp-edged circular orifice typically has Cd ≈ 0.60-0.65. Well-rounded (bell-mouthed) nozzles reach Cd ≈ 0.95-0.99, and venturi meters typically run Cd ≈ 0.95-0.98 because they minimize flow separation and the vena contracta effect.
How is the theoretical flow rate calculated?
For an orifice discharging under a liquid head h, Torricelli's law gives the theoretical (ideal) jet velocity as V = √(2gh), where g is gravitational acceleration. Multiplying by the orifice's cross-sectional area, a = πd²/4, gives the theoretical volumetric flow rate: Q_theoretical = a√(2gh).
Why is the coefficient of discharge always less than 1?
Cd is less than 1 because the actual jet contracts to an area smaller than the physical orifice opening just downstream of it (the vena contracta) and because viscous friction dissipates some energy. Cd equals the product of the coefficient of contraction (Cc) and the coefficient of velocity (Cv), both of which are less than 1 for a real fluid.

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