Orifice Flow Calculator
Use this calculator to find the volumetric flow rate through a circular orifice from its diameter, the head of liquid above it, and the discharge coefficient. Switch the mode to solve for the orifice diameter needed to pass a target flow rate, or to find the required head. All results update instantly as you type, and the Show Your Work panel traces every step of the calculation.
Formula
Worked example
A tank has a sharp-edged orifice 50 mm in diameter (d = 0.05 m) with a water head of 2 m above the orifice centre and a discharge coefficient of 0.61. Area: A = pi x 0.05^2 / 4 = 0.001963 m^2. Theoretical velocity: v = sqrt(2 x 9.81 x 2) = 6.264 m/s. Flow rate: Q = 0.61 x 0.001963 x 6.264 = 0.00750 m^3/s = 7.50 L/s = 450 L/min.
What is an orifice and why does it matter?
An orifice is a small opening in the wall or base of a tank, pipe, or vessel through which fluid flows under a pressure difference or under the weight of a liquid column above it. In practice, orifices appear in irrigation nozzles, tank drain valves, hydraulic control circuits, flow-metering plates, fuel injectors, and industrial process control. Because orifice geometry is simple and well-understood, the flow through one can be predicted reliably from just three parameters: the opening size, the driving head (or pressure difference), and a single empirical factor called the discharge coefficient.
The orifice flow equation explained
The classical formula Q = Cd x A x sqrt(2gH) comes from two physical principles. First, Torricelli's theorem (a special case of Bernoulli's equation) says that an ideal frictionless fluid issues from an orifice at a velocity equal to sqrt(2gH), identical to the speed an object would reach falling freely from height H. Second, the real flow is always smaller than this ideal because the fluid contracts and slows at the orifice due to friction and the curvature of streamlines. The discharge coefficient Cd (always between 0 and 1) captures both effects in one number. For a sharp-edged circular orifice in a large tank, Cd is approximately 0.611 at high Reynolds numbers, a result first measured by Torricelli in the 17th century and later confirmed by Weisbach, d'Aubisson, and modern ISO 5167 experiments.
Choosing the right discharge coefficient
Cd depends primarily on the shape of the orifice edge and, to a lesser extent, on Reynolds number and the ratio of orifice diameter to pipe diameter (the beta ratio). A thin, square-edged hole in a flat plate gives Cd around 0.61-0.63. A rounded or bell-mouthed entry removes the vena contracta almost entirely, raising Cd to 0.97-0.99 for a well-designed nozzle. A re-entrant (Borda) mouthpiece, where a tube projects inward into the reservoir, has the lowest Cd at around 0.51. When precise flow metering is needed, use the ISO 5167 Reader-Harris/Gallagher formula or manufacturer tables that relate Cd to Reynolds number and beta ratio. For general engineering estimates, 0.61 is a safe default for a sharp-edged plate.
Reverse solving: finding diameter or required head
Two common design problems run the equation backwards. If you know the flow rate you need and the available head, you can find the required orifice diameter: d = sqrt(4Q / (pi x Cd x sqrt(2gH))). If the diameter is fixed but the flow is still insufficient, you need more head: H = (Q / (Cd x A))^2 / (2g). Both reverse-solve modes are built into this calculator. Select "Orifice diameter" or "Head" from the Solve for menu to access them. After finding the theoretical diameter, always round up to the nearest standard pipe or drill size, then recompute Q to confirm the flow target is still met.
Typical discharge coefficients (Cd) by orifice type
| Orifice type | Typical Cd range | Notes |
|---|---|---|
| Sharp-edged plate (ISO 5167) | 0.60-0.65 | Most common; Cd approx 0.611 at high Re |
| Rounded-entry orifice | 0.80-0.85 | Partial rounding reduces vena contracta |
| Well-rounded nozzle | 0.97-0.99 | Minimal contraction and friction losses |
| Borda mouthpiece (re-entrant) | 0.51-0.52 | Tube projecting into the reservoir |
| Venturi tube | 0.95-0.98 | Gradual convergence, low pressure loss |
| Short cylindrical mouthpiece | 0.80-0.82 | Full bore, turbulent conditions |
Values from ISO 5167 and standard hydraulics references. Actual Cd varies with Reynolds number, pipe-to-orifice diameter ratio, and edge condition.
Frequently asked questions
What is a discharge coefficient and why is it not 1.0?
A discharge coefficient (Cd) is an empirical factor that corrects the theoretical orifice flow for two real-world effects: vena contracta contraction, where the jet narrows to less than the orifice area just after the opening, and friction losses in the fluid. For a perfectly streamlined, frictionless orifice Cd would be 1.0, but in practice sharp-edged orifices lose about 39 percent of the theoretical flow to these effects, giving Cd around 0.61.
What is the difference between orifice flow and weir flow?
An orifice is a fully submerged or free-discharging hole whose entire perimeter is bounded by solid material. A weir is an overflow structure where the fluid spills over the top edge. Orifice flow uses Q = Cd x A x sqrt(2gH); weir flow uses a different formula that involves the breadth of the notch and the head above the crest to a power of 1.5 or 2.5 depending on weir shape. Use this calculator only for orifices.
Does the formula change for pressurised systems?
Yes. The formula above uses gravitational head H (metres or feet of liquid above the orifice). For a pressurised vessel or pipe, convert the gauge pressure difference to an equivalent head using H = delta_P / (rho x g), where delta_P is the pressure difference across the orifice in Pascals and rho is the fluid density in kg/m^3. Then apply the same Q = Cd x A x sqrt(2gH) equation.
What is the vena contracta?
The vena contracta is the point just downstream of an orifice where the jet reaches its minimum cross-sectional area and maximum velocity. Because fluid streamlines converge toward the orifice from all directions, the effective flow area at the vena contracta is smaller than the orifice area - typically 61-64 percent for a sharp-edged plate. The contraction coefficient Cc = A_vc / A_orifice, and the velocity coefficient Cv together make up Cd = Cc x Cv.
Can this formula be used for gas flow?
Only at low pressure ratios. The formula assumes incompressible flow, which is a reasonable approximation for liquids and for gases when the pressure drop across the orifice is less than about 10-20 percent of the upstream absolute pressure. At higher pressure ratios the gas compresses significantly, and you need the ISO 5167 compressibility expansion factor. For precise gas metering use a dedicated compressible-flow orifice calculator.
Why do my actual measurements not match the calculator?
Several factors cause discrepancies: the wrong Cd for your orifice type or Reynolds number, an orifice that is not square-edged (burrs, corrosion, or wear change the effective Cd), partial submergence of the orifice so head is not measured to the right reference point, and swirling or asymmetric upstream flow. Also confirm your head measurement is taken from the free surface (not pipe wall) to the orifice centreline.