Efflux (Torricelli’s Law)
When you poke a hole near the bottom of a full water bottle, water shoots out horizontally rather than just dribbling down. This everyday observation is governed by Torricelli's Law, which tells us exactly how fast that jet of fluid travels. Torricelli's Law states that the speed of fluid exiting a small orifice (hole) in a large container equals the speed an object would reach if it fell freely from the height of the fluid surface above the hole. In equation form: v = √(2gh), where h is the vertical distance from the free surface down to the orifice and g is gravitational acceleration. The law was discovered by the Italian physicist Evangelista Torricelli in 1643, just a year after the death of his mentor Galileo. It is a beautiful special case of Bernoulli's principle — energy conservation applied to fluids. Imagine the fluid at the top surface as a vast, lazy reservoir barely moving, while at the orifice the same fluid has been converted from pressure energy (due to the weight of the column above it) entirely into kinetic energy. This energy exchange is exactly analogous to a ball rolling down a ramp: gravitational potential energy converts to kinetic energy, and the math turns out identical. Torricelli's Law has far-reaching applications: engineers use it to size outlet pipes in dams and tanks, doctors apply it to model blood flow through arterial openings, and it appears in the design of hourglasses and water clocks. Understanding it requires combining hydrostatics (pressure in still fluids) with the dynamic energy equation of moving fluids, making it a perfect bridge concept between the two major branches of fluid mechanics.
- Set Up the System
- Apply Bernoulli's Equation
- Apply the Large-Tank Approximation
- Solve for the Efflux Speed
- Volume Flow Rate and the Vena Contracta
Set Up the System
Consider a large open tank filled with liquid to a height H above an orifice located near the base. Choose two points: Point 1 is on the free surface of the liquid at the top, and Point 2 is at the orifice where the fluid exits. Both points are exposed to atmospheric pressure (P₁ = P₂ = Pₐₜₘ), since the top is open and the jet exits into open air.
Apply Bernoulli's Equation
Bernoulli's equation along a streamline connecting Points 1 and 2 reads: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂. Since P₁ = P₂, the pressure terms cancel. Set the reference height at the orifice, so h₂ = 0 and h₁ = h (the depth of fluid above the hole). This gives: ½ρv₁² + ρgh = ½ρv₂².
Apply the Large-Tank Approximation
If the tank's cross-sectional area A_tank is much larger than the orifice area A_orifice, then by the continuity equation (A_tank · v₁ = A_orifice · v₂), the surface velocity v₁ ≈ 0. This is the key approximation — the surface drops so slowly it is effectively stationary. Discarding the ½ρv₁² term gives: ρgh = ½ρv₂².
Solve for the Efflux Speed
Dividing both sides by ρ and multiplying by 2: v₂² = 2gh. Taking the square root yields Torricelli's result: v₂ = √(2gh). Notice this is identical to the speed of a body dropped from rest through height h under gravity (from kinematics: v² = 2gh). The fluid 'falls' energetically through height h, converting its pressure-potential energy into kinetic energy — just as a stone falls physically through the same height.
Volume Flow Rate and the Vena Contracta
The ideal volume flow rate through the orifice is Q = A · v = A√(2gh). In practice, the actual jet narrows after exiting — a phenomenon called the vena contracta — because streamlines converge and inertia causes the jet to contract slightly beyond the hole. To account for this, engineers multiply by a discharge coefficient Cₐ (typically 0.61–0.65 for a sharp-edged circular hole): Q_real = Cₐ · A · √(2gh). As the tank drains, h decreases and so does the flow rate, meaning the tank empties more slowly over time — following a square-root decay law rather than a linear one.
The Key Equation
- v — Efflux speed (m s⁻¹): Speed of the fluid jet at the orifice exit
- g — Gravitational acceleration (m s⁻²): Approximately 9.81 m s⁻² near Earth's surface
- h — Head height (m): Vertical distance from the free surface of the liquid down to the centre of the orifice
Worked Example
Problem. A cylindrical water tank is filled to a depth of 1.80 m above a small circular orifice of diameter 2.0 cm drilled in its side. (a) Calculate the ideal efflux speed of the water. (b) Calculate the volume flow rate through the orifice, using a discharge coefficient Cₐ = 0.62. (g = 9.81 m s⁻²)
Answer. Ideal efflux speed ≈ 5.94 m/s; Volume flow rate ≈ 1.16 × 10⁻³ m³/s (1.16 L/s)
Origins · 1 minEvangelista Torricelli and how this was discovered
Torricelli was a student and secretary of Galileo in the last months of Galileo's life. In 1643 he formulated the efflux law bearing his name, derived from his analysis of fluid pressure and motion. In the same year he invented the mercury barometer, demonstrating atmospheric pressure for the first time and creating the first sustained vacuum in a laboratory — what became known as the 'Torricellian vacuum'. His work laid crucial groundwork for Bernoulli's later energy-based treatment of fluid motion. Torricelli died of typhoid fever at just 39 years old, only four years after his greatest discoveries. Despite his short career, the unit of pressure called the 'Torr' (1 Torr ≈ 133.3 Pa, roughly 1 mmHg) is named in his honour and is still used in vacuum engineering today.
Related Concepts
- Bernoulli's Principle — Torricelli's Law is a direct special case of Bernoulli's equation applied between the free surface and the orifice; understanding Bernoulli's Principle is essential for deriving and extending Torricelli's result.
- Continuity Equation (Fluids) — The large-tank approximation (v_surface ≈ 0) used in Torricelli's derivation comes directly from applying the continuity equation A₁v₁ = A₂v₂ when A_tank ≫ A_orifice.
- Hydrostatic Pressure — The gauge pressure ρgh at the orifice depth, which drives the efflux, is the fundamental result of hydrostatics — the pressure in a static fluid due to the weight of the column above.
- Projectile Motion — Once the fluid jet exits the orifice horizontally with speed v = √(2gh), it travels as a projectile under gravity, following a parabolic path — connecting fluid dynamics to kinematics.