About

The coefficient of discharge C_d quantifies the ratio of a device's actual flow rate to its theoretical flow rate predicted by Bernoulli's equation. Real flow through an orifice, nozzle, or weir is always less than ideal due to vena contracta formation, viscous friction, and turbulent losses. Ignoring C_d when sizing a drainage orifice or calibrating a flow meter leads to systematic under- or over-estimation of discharge, which can cause tank overflow, pump cavitation, or failed pressure relief. This calculator uses the standard formulation C_d = Q_actual ÷ Q_theoretical, where theoretical flow derives from Q_th = A ⋅ √2gΔh for gravity-driven heads or Q_th = A ⋅ √2ΔP ÷ ρ for pressure-driven systems. The tool approximates ideal conditions: incompressible, steady-state flow with uniform velocity distribution across the orifice cross-section.

Formulas

The coefficient of discharge is defined as the dimensionless ratio of measured (actual) volumetric flow to ideal (theoretical) volumetric flow through an opening.

C_d = Q_actualQ_theoretical

For a gravity-driven system (tank draining through an orifice under head Δh), the theoretical flow rate is derived from Torricelli's theorem (a special case of Bernoulli's equation):

Q_th = A ⋅ √2 ⋅ g ⋅ Δh

For a pressure-driven system (flow meter in a pressurised pipe), the theoretical velocity uses the pressure differential:

Q_th = A ⋅ √2 ⋅ ΔPρ

The orifice cross-sectional area for a circular opening of diameter d:

A = π4 ⋅ d²

Where: C_d = coefficient of discharge (dimensionless, typically 0 < C_d ≤ 1); Q_actual = measured volumetric flow rate m³/s; Q_th = ideal volumetric flow rate m³/s; A = cross-sectional area of orifice m²; g = gravitational acceleration 9.80665 m/s²; Δh = head difference m; ΔP = pressure differential Pa; ρ = fluid density kg/m³; d = orifice diameter m.

Reference Data

Device Type	Typical C_d	Reynolds Number Range	Notes
Sharp-edged orifice (thin plate)	0.61 - 0.65	> 10⁴	Most common; ISO 5167 standard
Round-edged orifice	0.97 - 0.99	> 10⁴	Reduced vena contracta effect
Borda (re-entrant) orifice	0.50 - 0.53	> 10³	Pipe projects inward into tank
Short tube (L/d ≈ 2-3)	0.80 - 0.85	> 10⁴	Flow reattaches inside tube
Convergent nozzle	0.94 - 0.98	> 10⁴	Smooth contraction minimises losses
Venturi meter (machined)	0.95 - 0.99	2×10⁵ - 10⁶	ISO 5167-4; gradual divergence
Venturi meter (rough cast)	0.92 - 0.96	2×10⁵ - 10⁶	Surface roughness increases losses
ISA 1932 nozzle	0.93 - 0.98	2×10⁴ - 10⁷	Standard industrial flow nozzle
Long orifice (L/d > 10)	0.72 - 0.82	> 10⁴	Pipe-like friction dominates
Rectangular sharp-crested weir	0.60 - 0.62	-	Francis formula correction needed
V-notch weir (90°)	0.58 - 0.62	-	Low-flow measurement
Broad-crested weir	0.85 - 0.95	-	Critical depth forms on crest
Sluice gate (free flow)	0.55 - 0.61	-	Depends on gate opening ratio
Submerged orifice	0.60 - 0.65	> 10⁴	Downstream submergence reduces C_d
Fire hose nozzle	0.96 - 0.99	> 10⁵	Precision-machined smooth bore
Sprinkler head	0.70 - 0.85	> 10⁴	Internal deflector creates spray pattern

Frequently Asked Questions

In real flow, the streamlines contract downstream of the orifice forming a vena contracta where the effective cross-sectional area is smaller than the geometric area. Viscous friction along the orifice edges and turbulent mixing further reduce kinetic energy. These losses make Q_actual < Q_theoretical, so C_d < 1. A well-designed convergent nozzle can approach 0.99 by virtually eliminating the contraction, but never reaches unity due to boundary-layer effects.

At low Reynolds numbers (< 10⁴), viscous forces dominate and C_d drops significantly because friction losses consume a larger fraction of the driving energy. Above 10⁴, C_d stabilises as inertial forces dominate and the flow separation pattern becomes Reynolds-number-independent. ISO 5167 specifies minimum Reynolds numbers for each device type precisely because C_d correlations assume fully turbulent flow.

The coefficient of discharge C_d is the product of two independent coefficients: C_d = C_v × C_c. The coefficient of velocity C_v (typically 0.95 - 0.99) accounts for energy loss due to friction. The coefficient of contraction C_c (typically 0.61 - 0.66 for sharp-edged orifices) accounts for the reduced jet area at the vena contracta. A rounded nozzle has C_c ≈ 1 because no contraction occurs.

This tool assumes incompressible flow, which is valid when the Mach number at the orifice is below approximately 0.3. For compressible gases at higher velocities, you must apply an expansibility (expansion) factor ε to account for density changes across the restriction. ISO 5167-2 provides ε correlations. For choked flow (sonic conditions), the entire Bernoulli framework is inapplicable and isentropic relations must be used instead.

The volumetric method is the most direct approach. Collect the discharged fluid in a calibrated tank over a measured time interval: Q_actual = V ÷ t. Ensure steady-state conditions by maintaining a constant head (overflow weir) or constant upstream pressure during the collection period. Repeat at least three times and average the results. For pipe systems, an ultrasonic or electromagnetic flow meter serves as the reference instrument.

Yes. The beta ratio β = d ÷ D (orifice diameter to pipe diameter) significantly influences C_d. For sharp-edged orifice plates, C_d increases as β increases from 0.2 to 0.75 because the approach velocity becomes a larger fraction of the jet velocity, reducing the relative contraction. ISO 5167 limits β to the range 0.10 - 0.75 for standard correlations to remain valid.