Formula & Calculator
Torricelli's Law (Efflux Velocity)
Calculates the velocity of fluid exiting an opening at the bottom of a tank from the height of fluid above the opening.
Interpretation
Torricelli's law gives the exit velocity of a fluid from a hole in a tank, assuming ideal inviscid flow. v = √(2·g·h). This is derived from Bernoulli's equation and is useful for draining tanks and orifice flow calculations.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v | Efflux (exit) velocity | m/s |
| g | Gravitational acceleration | 9.81 m/s2 |
| h | Height of fluid above the opening | m |
What it means
Torricelli's law describes the speed of fluid flowing out of an opening in a container, under the influence of gravity. It states that the exit velocity v is equal to √(2gh), where g is the acceleration due to gravity and h is the height of the fluid above the opening (the head). This assumes that the tank is large enough so that the velocity of the fluid surface is negligible (quasi‑steady), and that the fluid is ideal (inviscid and incompressible). The law is derived from Bernoulli's equation between the free surface and the orifice. It ignores losses due to friction and contraction of the jet. In practice, a discharge coefficient C_d is introduced to account for these losses: v_actual = C_d √(2gh). Torricelli's law is used in civil engineering for designing spillways, in hydraulics for determining flow rates from tanks, and in irrigation systems. It is also a classic example of the conversion of potential energy to kinetic energy.
Worked example
Torricelli's Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| h | 2 m |
| Parameter | Value |
|---|---|
| h | 10 m |
Common mistakes
- Height h: The depth of the hole below the free surface of the fluid.
- Assumptions: Inviscid, steady, incompressible flow – real fluids have losses, so actual velocity is lower.
- Vena contracta: The effective area may be smaller than the hole area; use a discharge coefficient.
- Units: g in m/s², h in m → v in m/s.
- Pressure at surface: Assumes atmospheric pressure at the free surface; if pressurised, add that term.
Applications
Torricelli's law gives the efflux velocity of a fluid from an opening in a tank, assuming ideal inviscid flow. It is commonly used to calculate the discharge rate from storage tanks, reservoirs, and vessels in chemical and hydraulic engineering. The formula is applied in the design of weirs, spillways, and orifice meters. In firefighting, it helps determine the flow rate from hydrants. In irrigation and drainage, it guides the design of culverts and channels. Although it neglects viscosity and friction, it provides a good first approximation for many practical scenarios. Engineers use it to size openings and predict emptying times, ensuring that fluid systems operate as intended.
- Design of storage tank outlets and drains
- Orifice flow measurement devices
- Spillway and weir design in civil engineering
- Fire hose and hydrant flow calculations
- Irrigation and drainage system design
Frequently Asked Questions
Torricelli's law gives the velocity of fluid exiting an opening (orifice) at a depth h below the free surface. It is v = √(2·g·h). It assumes the tank is large (so the surface velocity is negligible), the flow is steady, and the fluid is inviscid (no friction). It is derived from Bernoulli's equation.
- v = efflux velocity (m/s, ft/s)
- g = gravitational acceleration (9.81 m/s², 32.2 ft/s²)
- h = vertical distance from the free surface to the centre of the opening (m, ft)
- Ignoring the discharge coefficient (C_d) – for real fluids, the actual velocity is v_actual = C_v · √(2gh), where C_v is a velocity coefficient (≈0.97 for sharp‑edged orifices).
- Using h as the height of the opening from the bottom – h must be measured from the free surface to the opening, not from the bottom.
- Applying it when the tank is not large – if the surface area is small, the surface velocity is not negligible, and the formula must be corrected.
- Using it for compressible fluids – Torricelli's law is for incompressible liquids.
Apply Bernoulli between the free surface (point 1) and the outlet (point 2). At the surface, v₁ ≈ 0, P₁ = P_atm, and z₁ = h. At the outlet, P₂ = P_atm, z₂ = 0. Bernoulli: P₁/ρg + v₁²/2g + z₁ = P₂/ρg + v₂²/2g + z₂. Cancelling P_atm/ρg and v₁, we get h = v₂²/2g, so v₂ = √(2gh).
For an ideal fluid, the velocity is independent of the orifice size. In practice, for very small orifices, viscous losses become significant, and the velocity is lower. The coefficient of discharge C_d accounts for contraction (vena contracta) and friction: Q_actual = C_d · A · √(2gh). C_d is typically 0.6‑0.8 for sharp orifices.
As the liquid level drops, h decreases, so the velocity decreases. The flow is unsteady. The time to empty a tank can be found by integrating the continuity equation: dV/dt = −A_orifice·√(2gh). For a tank of constant cross‑section A_t, the time to empty from h₀ to 0 is t = (2·A_t/(C_d·A_orifice·√(2g)))·√h₀.
The vena contracta is the point where the jet's cross‑sectional area is minimum, just downstream of the orifice. The actual area of the jet is smaller than the orifice area due to fluid inertia. The ratio of jet area to orifice area is the contraction coefficient C_c ≈ 0.62 for a sharp orifice. The discharge coefficient C_d = C_v × C_c.
For gases, the density changes with pressure, so the incompressible Bernoulli equation is not valid. Instead, for isentropic flow of an ideal gas, the mass flow rate is given by the orifice flow equations (using pressure ratios and the specific heat ratio). Torricelli's law is only for liquids.
If the tank is pressurised (P_surface > P_atm), then the effective head is increased: h_eff = h + (P_surface − P_atm)/(ρ·g). Then v = √(2·g·h_eff). This is used in pressurised water systems and fire extinguishers.
A siphon uses an inverted U‑tube to transfer liquid over a barrier. The flow is driven by the difference in height between the free surfaces. The maximum velocity is limited by the height difference, and Torricelli's law gives the theoretical velocity at the outlet. However, siphons also require that the pressure at the crest does not drop below vapour pressure (cavitation limit).