Hong Life CalculatorsUnderstanding the liquid leak rate calculator — concept and principle
When a tank or pipe develops a hole, hazard assessment first asks how fast the liquid escapes. This calculator uses the classic orifice equation to estimate the initial leak rate through a round hole from vessel pressure, liquid head and density, for spill scenarios and containment sizing.
How it is calculated
ṁ = C_d · A · √(2ρ · (Δp + ρgh))
- A = πd²/4, the hole area, with d converted from mm to m.
- Δp = pressure above the liquid minus outside pressure, in bar (× 10⁵ Pa).
- ρgh = extra pressure from the liquid head h (g = 9.81 m/s²).
- C_d = discharge coefficient; 0.61 is the usual value for a sharp-edged hole.
- The volume rate in L/min is ṁ / ρ × 60,000.
The principle
Bernoulli's equation says that pressure energy turns into kinetic energy as liquid accelerates through the hole: ½ρv² = Δp_total, so the ideal velocity is v = √(2Δp_total / ρ). Multiplying by area and density gives a mass flow. Real jets contract just past a sharp edge (the vena contracta) and lose a little energy to friction, so the real flow is smaller; C_d lumps both effects into one factor. Because flow depends on the square root of pressure, quadrupling the pressure only doubles the leak, while doubling the hole diameter quadruples it.
Worked example
- Inputs: d = 10 mm, Δp = 5 bar, h = 2 m, ρ = 1000 kg/m³, C_d = 0.61.
- A = π × 0.01² / 4 = 7.854 × 10⁻⁵ m².
- Total driving pressure: 500,000 + 1000 × 9.81 × 2 = 519,620 Pa.
- √(2 × 1000 × 519,620) = 32,237.2 kg/(m²·s).
- ṁ = 0.61 × 7.854 × 10⁻⁵ × 32,237.2 = 1.5445 kg/s.
- Volume rate: 1.5445 / 1000 × 60,000 = 92.668 L/min.
| Hole diameter (mm) | Leak rate (kg/s) |
|---|---|
| 2.5 | 0.0965289675 |
| 5 | 0.38611587 |
| 7.5 | 0.8687607074 |
| 10 | 1.54446348 |
| 12.5 | 2.413224187 |
| 15 | 3.47504283 |
| 20 | 6.177853919 |
How to read the result
The value is the initial rate. As the vessel drains and pressure or head falls, the rate drops, so total release time is longer than inventory divided by the initial rate. Hole size dominates: a 20 mm hole leaks four times as much as a 10 mm one. For worst-case screening a C_d of 1 is sometimes used for conservatism; 0.61 represents a sharp-edged hole and rounded or short-pipe openings give higher values.
Common mistakes
- Entering absolute pressure instead of the difference to the outside.
- Forgetting the liquid head for holes near the bottom of tall tanks.
- Applying the equation to liquids that flash to vapour on release, such as liquefied gases.
Frequently asked questions
How long will the tank take to empty?
For an unpressurised open tank the head falls as it drains, so the rate is not constant; integrating gives a drain time about twice the time computed at the initial rate.
Can I use it for gas leaks?
No. Gases are compressible and often reach choked flow, which needs a different equation.
What C_d should I use?
About 0.6 for sharp-edged holes; values closer to 1 for well-rounded nozzles; 1 if you want a conservative upper bound.
Sources: D. A. Crowl and J. F. Louvar, Chemical Process Safety: Fundamentals with Applications, Prentice Hall · Perry's Chemical Engineers' Handbook, McGraw-Hill (flow through orifices)