Coefficient of Discharge Calculator
Calculate the coefficient of discharge (Cd) for an orifice or nozzle, or use it to find actual flow rate, theoretical flow rate, required hydraulic head, orifice area, or flow velocity. Choose an orifice edge type to pre-fill a typical Cd, switch between metric and imperial units, and follow the step-by-step panel to see exactly how each number is derived.
What is the coefficient of discharge?
The coefficient of discharge (Cd) is a dimensionless number that compares the actual volumetric flow rate through an orifice, nozzle, or valve to the theoretical maximum that would occur in a frictionless, ideal flow. It is defined as Cd = Q_actual / Q_theoretical. Because real fluids experience friction, turbulence, and flow contraction at the vena contracta (the narrowest section of the jet after an orifice), the actual flow is always less than the ideal, so Cd is always less than 1.0. A sharp-edged orifice plate typically achieves Cd of about 0.61 to 0.65, while a well-designed rounded nozzle can reach 0.90 to 0.99.
How to calculate coefficient of discharge
The theoretical flow rate through an orifice is derived from Torricelli's theorem: Q_th = A x sqrt(2gh), where A is the orifice cross-sectional area (m^2), g is gravitational acceleration (9.807 m/s^2), and h is the hydraulic head above the orifice centerline (m). The theoretical exit velocity is V_th = sqrt(2gh). From a physical measurement of actual flow, Cd = Q_actual / Q_th. If Cd is already known, actual flow is Q_actual = Cd x A x sqrt(2gh) and actual velocity is V_actual = Cd x V_th. The flow loss fraction is simply (1 - Cd), so a Cd of 0.62 means 38% of the theoretical capacity is lost.
Orifice edge types and their effect on Cd
The geometry of the orifice edge is the single biggest factor controlling Cd. A sharp square-edged orifice creates a pronounced vena contracta, where the jet narrows to about 64% of the orifice area before expanding again - this contraction explains why sharp-edged plates have Cd near 0.62. A chamfered or beveled downstream edge reduces that contraction, raising Cd to around 0.75. A smoothly rounded entry eliminates it almost entirely, pushing Cd above 0.90 and approaching 0.99 for a well-formed nozzle. A re-entrant (Borda) orifice, where the tube extends inward into the tank, produces the most severe contraction and the lowest Cd, around 0.52. Roughness, inlet approach velocity, and Reynolds number also shift Cd by a few percent, which is why flow meters are individually calibrated.
Relationship between Cd, Cv, and Cc
The coefficient of discharge is often broken into two components. The coefficient of velocity (Cv) accounts for the reduction in speed caused by friction, defined as Cv = V_actual / V_theoretical, and typically ranges from 0.95 to 0.99 for a sharp orifice. The coefficient of contraction (Cc) accounts for the area reduction at the vena contracta, defined as Cc = A_vena / A_orifice, and is typically 0.62 to 0.67 for a sharp edge. Their product is the coefficient of discharge: Cd = Cv x Cc. For most practical engineering calculations, the combined Cd value is used directly, but understanding its components helps diagnose whether losses are mostly frictional (low Cv) or geometric (low Cc).
Typical Cd values by orifice and meter type
| Device type | Cd range | Typical Cd | Notes |
|---|---|---|---|
| Sharp-edged orifice plate | 0.59 - 0.65 | 0.62 | ISO 5167, beta 0.2-0.75 |
| Re-entrant (Borda) orifice | 0.50 - 0.55 | 0.52 | Nozzle recessed into tank wall |
| Chamfered / beveled orifice | 0.72 - 0.78 | 0.75 | Downstream bevel reduces contraction |
| Rounded-entry nozzle | 0.88 - 0.92 | 0.90 | ASME long-radius nozzle |
| Venturi meter | 0.95 - 0.99 | 0.97 | ISO 5167, high-recovery design |
| Flow nozzle (ISA 1932) | 0.94 - 0.99 | 0.96 | High-velocity, low-recovery |
| Gate valve (fully open) | 0.80 - 0.93 | 0.89 | Varies with valve geometry |
| Globe valve (fully open) | 0.60 - 0.75 | 0.68 | High head loss design |
| Broad-crested weir | 0.84 - 0.89 | 0.86 | Hydraulic overflow structure |
| Sharp-crested (rectangular) weir | 0.61 - 0.64 | 0.62 | Standard Francis formula |
Published values from ISO 5167 and ASME MFC standards. Actual values depend on Reynolds number, pipe diameter ratio, and surface finish.
Frequently asked questions
Why is the coefficient of discharge for a sharp-edged orifice about 0.61?
A sharp edge creates a strong vena contracta: the jet continues to contract after leaving the orifice until it reaches a minimum cross-section about half a diameter downstream. The coefficient of contraction Cc is typically 0.63 to 0.64 at this point. Multiplied by a velocity coefficient Cv of around 0.97 to 0.98 (accounting for friction), the product is approximately 0.61 to 0.63. This value is well-established in ISO 5167 and has been confirmed by laboratory measurements across a wide range of Reynolds numbers.
Can the coefficient of discharge ever exceed 1.0?
No. A Cd greater than 1.0 would mean the orifice delivers more flow than the frictionless ideal, which violates conservation of energy. If a calculation produces Cd above 1.0, the most likely causes are measurement error in the actual flow rate, an error in the head measurement (using gauge pressure instead of total head, for example), or an incorrect orifice area. Values very close to 1.0, such as 0.98 for a venturi meter, are real and achievable with smooth, well-designed geometry.
How do I measure Cd experimentally?
The simplest method is the bucket-and-stopwatch test. Let the fluid flow through the orifice under a constant hydraulic head, collect the discharge in a container of known volume, and measure the fill time. Divide volume by time to get Q_actual. Measure the orifice diameter and the head, calculate Q_theoretical = A x sqrt(2gh), then divide: Cd = Q_actual / Q_theoretical. For pipe-flow situations, use a calibrated flow meter upstream and downstream and divide actual by the theoretical flow computed from differential pressure across a flow element.
How does Reynolds number affect Cd?
At very low Reynolds numbers (below about 10,000 for a sharp-edged orifice), Cd drops noticeably because viscous effects are significant. Above Re approximately 30,000, Cd stabilises and becomes nearly constant for a given geometry - this is the regime where ISO 5167 correlations apply. For most industrial water and air flows through orifices larger than about 20 mm, Re is well above 30,000 and the standard Cd values are reliable. For very small orifices or very viscous fluids, use the full Reynolds-number-dependent correction from ISO 5167.
What is the difference between Cd for an orifice plate and for a weir?
The same physical concept applies, but the formulas differ. For an orifice under pressure (submerged or tank-based), Q = Cd x A x sqrt(2gh) from Torricelli's theorem. For an open-channel weir, the formula involves head to the power of 1.5: Q = Cd x (2/3) x sqrt(2g) x L x H^1.5, where L is the weir crest length and H is the head above the crest. The Cd values are also different: a sharp-crested rectangular weir has Cd around 0.61 to 0.63, comparable to a sharp orifice, while a broad-crested weir is typically 0.84 to 0.89.
What is flow resistance (K) and how is it related to Cd?
The flow resistance coefficient K (also called minor loss coefficient or resistance factor) is the reciprocal relationship K = 1 / Cd^2. It is used in head loss calculations: head loss = K x V^2 / (2g). A sharp orifice with Cd = 0.62 has K = 1 / 0.62^2 = 2.60, meaning the head lost across it equals 2.60 velocity heads. A rounded nozzle with Cd = 0.90 has K = 1.23, confirming much lower losses. K is preferred in pipe network calculations, while Cd is preferred when calculating flow rates directly.