Solve liquid orifice flow, diameter, head, or pressure differential using Cd, circular area, fluid density, and pressure, flow, and length conversions.
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Orifice Equation
The orifice equation estimates the volumetric flow rate of a liquid leaving an opening because of gravity head. The head mode suits tanks and reservoirs; the pressure mode uses a net liquid pressure difference and density. Both assume negligible approach velocity and omit the pipe-diameter correction required by an in-pipe orifice meter.
Q = Cd × A × √(2gH); pressure form: Q = Cd × A × √(2ΔP/ρ)
In this relationship, the square-root term gives the ideal exit velocity created by the available head, while the discharge coefficient adjusts that ideal result for real-world effects such as vena contracta formation, edge losses, and minor turbulence at the opening.
Variable Definitions
| Variable | Meaning | Typical Unit |
|---|---|---|
| Q | Volumetric flow rate through the orifice | m³/s, ft³/s, L/s, or gpm |
| C | Discharge coefficient that accounts for non-ideal flow | Dimensionless |
| A | Cross-sectional area of the opening | m², ft², or in² |
| g | Gravitational acceleration | 9.80665 m/s² in this calculator |
| H | Head measured from the liquid surface to the orifice centerline | m, ft, or in; use net driving head |
Rearranged Forms
If you know three of the four main quantities, the same equation can be rearranged to solve for the missing value.
A = Q / [Cd × √(2gH)]
Cd = Q / [A × √(2gH)]
H = [Q / (Cd × A)]² / (2g)
These reference rearrangements include coefficient back-calculation. The calculator directly solves flow from head or pressure, circular diameter, required head, or required pressure difference; it does not directly solve Cd.
How to Use the Calculator
- Enter the discharge coefficient for the orifice geometry.
- Select the solve mode. When the opening is known, enter its circular diameter and select its unit.
- Enter net head, or pressure difference and liquid density, as required by that mode.
- Calculate the missing value and confirm that the output unit matches your application.
The calculator converts each selected input unit through SI and converts the answer to the selected result unit. Flow in gpm uses US gallons; pressure input is a difference, not an absolute pressure.
Finding Orifice Area
The calculator accepts circular diameter and computes area internally. The rectangular-area formula below is a reference only; rectangular geometry is not a calculator input.
Circular orifice
A = πd²/4
Rectangular orifice
A = width × opening height
Be sure to convert diameter, width, and height into the same length unit before calculating area. Small conversion errors can produce large flow-rate errors.
Interpreting the Discharge Coefficient
The discharge coefficient is a correction factor that makes the ideal equation match real flow behavior. It depends on edge shape, surface finish, Reynolds number, and how cleanly the fluid approaches the opening. A perfectly ideal opening would have a coefficient of 1, but actual values are lower. Many sharp-edged orifices use values around 0.60 to 0.65, while better-rounded entries can be higher.
- Higher coefficient: more flow for the same area and head. Cd includes contraction as well as velocity effects; Cd times ideal speed is nominal Q/A, not necessarily the contracted jet velocity.
- Lower coefficient: greater contraction and loss at the opening.
- Uncertain coefficient: test data or manufacturer information should be preferred when accuracy matters.
Important Assumptions
- The fluid is treated as incompressible.
- The flow is driven by the specified net head or equivalent pressure difference. Do not count the same driving energy twice.
- The head is measured to the center of the orifice, not to its top or bottom edge.
- The upstream liquid level is sufficiently stable during the calculation interval.
- Use the net upstream-to-downstream driving difference. This simplified model omits approach velocity, pipe-diameter effects and a viscosity-dependent calibration of Cd.
If the orifice is submerged on both sides, use the net head difference between upstream and downstream liquid levels rather than only the upstream depth.
net H = upstream head − downstream head
Example Calculation
Suppose Cd is0.62, area is0.005 m², and net head is3 m. For the circular-diameter input, that area corresponds to approximately79.788456 mm.
Q =0.62 ×0.005 ×√(2 ×9.80665 ×3) ≈0.0237792 m³/s
This is approximately 23.8 L/s. The example shows two important sensitivities: flow increases directly with area and coefficient, but only with the square root of head.
Quick Design Insights
- If the orifice area doubles, the flow rate doubles.
- If the discharge coefficient increases by 10%, the flow rate also increases by 10%.
- If the head becomes four times larger, the flow rate becomes two times larger.
- If the head is reduced significantly as the tank drains, the flow rate will also fall during discharge.
Common Mistakes
- Using diameter directly where area is required.
- Measuring head to the wrong point on the opening.
- Selecting a unit that does not match the entered number. The calculator allows mixed selected units and converts them internally.
- Assuming the discharge coefficient is always the same for every orifice shape.
- Applying this simple gravity-based form to gas flow or high-backpressure conditions.
Common Questions
Is head the total liquid depth in the tank?
Not always. The head used here is the vertical distance from the liquid surface to the centerline of the orifice.
Can this be used for draining tanks?
Yes, but remember that the head changes as the liquid level falls, so the flow rate is not constant over the full drain period.
Why does the flow rate not increase linearly with head?
The velocity term depends on the square root of head, so larger heads still increase flow, but at a diminishing rate.
