Library / Hydraulics, open channels and fluid power / HYD-31
Hydraulics, open channels and fluid power
Time to drain a constant-area tank
Time for the level in a tank of constant plan area At to fall from head h₁ to h₂ above a small outlet of area Ao, with quasi-steady outflow, a constant discharge coefficient and the same gas pressure on both surfaces.
- Ref
- HYD-31
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- t [s]
For a tank of constant plan area At draining through a small outlet Ao, continuity gives At dh/dt = −Cd Ao √(2g h). Integrating from h₁ to h₂ gives t = 2 At(√h₁ − √h₂)/(Cd Ao √(2g)). The time to empty (h₂ = 0) is finite. The formula assumes quasi-steady outflow, a constant discharge coefficient and equal gas pressure on both surfaces.
Time for the level in a tank of constant plan area At to fall from head h₁ to h₂ above a small outlet of area Ao, with quasi-steady outflow, a constant discharge coefficient and the same gas pressure on both surfaces.
Calculator
- Initial outflow, Cd Ao √(2g h₁)0.02746 m³/s
- Final outflow, Cd Ao √(2g h₂)0.01373 m³/s
- Time to empty from h₁72.83 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The level falls from 1 m to 0.25 m in 36.41 s (0.6069 min), while the outflow slows from 0.02746 to 0.01373 m³/s.
- Emptying the tank completely takes 72.83 s, twice the time the initial flow rate would need.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ao Outlet area | 1e-6 to 1 m² | 0.3641 | 364,100 | 364,100 | +10% | −9.09 % |
| At Tank plan area | 0.01 to 1,000 m² | 0.3641 | 36,410 | 36,410 | +10% | +10 % |
| h1 Initial head above the outlet | 0.5 to 50 m | 15.08 | 478.5 | 463.5 | +10% | +9.76 % |
| g Gravitational acceleration | 1 to 30 m/s² | 20.82 | 114 | 93.23 | +10% | −4.65 % |
| h2 Final head above the outlet | 0 to 0.5 m | 21.33 | 72.83 | 51.5 | +10% | −4.88 % |
| Cd Discharge coefficient | 0.4 to 1 | 22.58 | 56.44 | 33.86 | +10% | −9.09 % |
Sweep of Final head above the outlet
| Final head above the outlet (m) | Draining time (s) |
|---|---|
| 0 | 72.83 |
| 0.0625 | 54.62 |
| 0.125 | 47.08 |
| 0.1875 | 41.29 |
| 0.25 | 36.41 |
| 0.3125 | 32.12 |
| 0.375 | 28.23 |
| 0.4375 | 24.66 |
| 0.5 | 21.33 |
Sweep one variable
Variables and how each one contributes
Draining time
What the equation gives you. Everything else feeds it.
Tank plan area
The plan area of the tank, in m². A larger tank takes proportionally longer to drain. Raising it by 10 % (to 1.1) moves the result by +10 %, from 36.41 to 40.05 s.
Outlet area
The outlet area, in m². A larger outlet drains faster, in inverse proportion. Raising it by 10 % (to 0.011) moves the result by −9.09 %, from 36.41 to 33.1 s.
Discharge coefficient
The discharge coefficient of the outlet. A larger coefficient drains faster, in inverse proportion. Raising it by 10 % (to 0.682) moves the result by −9.09 %, from 36.41 to 33.1 s.
Initial head above the outlet
The initial head above the outlet, in m. A higher starting level raises the time, with a square-root dependence. Raising it by 10 % (to 1.1) moves the result by +9.76 %, from 36.41 to 39.97 s.
Final head above the outlet
The final head above the outlet, in m. A higher final level shortens the drain time. Raising it by 10 % (to 0.275) moves the result by −4.88 %, from 36.41 to 34.64 s.
Gravitational acceleration
The gravitational acceleration, in m/s². Lowers the time with a square-root dependence. Raising it by 10 % (to 10.79) moves the result by −4.65 %, from 36.41 to 34.72 s.
Worked example with the values above
- Level term: √h₁ − √h₂ = 1 − 0.5 = 0.5 m^½.
- Outlet term: Cd Ao √(2g) = 0.62 × 0.01 × 4.4294 = 0.027463.
- t = 2 × 1 × 0.5 / 0.027463 = 36.413 s.
Assumptions and limits
- Steady, incompressible, inviscid flow outside the boundary layer, at atmospheric pressure where stated.
- Uniform velocity profiles at the sections used.
Common mistakes
- Using the area of the pipe where the jet area is needed.
- Mixing gauge and absolute pressures in a force balance.
- Forgetting that the force on the support is the reaction to the force on the fluid.
Statistics and uncertainty
Discharge and velocity coefficients vary with the Reynolds number and with the geometry, and measured jet distances carry the reading error of the ruler, so the results are no better than 2 % to 5 %.
Where it is used
- Timing tank drainage and calibrating nozzles and orifices.
- Sizing thrust blocks, deflectors and jet impact plates.
Maths you need
- Torricelli’s law: jet speed √(2gh) from a head h
- Momentum equation: force = rate of change of momentum
- Continuity: Q = A U