Torricelli's Law Calculator

Torricelli's law (Evangelista Torricelli, 1643) states that the speed of liquid exiting a small orifice in a large tank is v = √(2gh), where h is the vertical height from the free surface to the center of the hole and g ≈ 9.81 m/s². It follows from Bernoulli's equation along a streamline from the free surface (v₁ ≈ 0, P₁ = P_atm) to the jet (P₂ = P_atm, height z₂ ≈ 0): ½ρv² = ρgh.

The result is remarkable: efflux speed depends only on head h, not on tank diameter, fluid total volume, or hole area (for an ideal small orifice in a large tank). Doubling h increases v by √2, not by 2. At h = 5 m, v ≈ √(2 × 9.81 × 5) ≈ 9.9 m/s — the same speed a dropped object would have after falling 5 m.

Volume flow rate Q = A_orifice × v. A hole of diameter 20 mm (A ≈ 3.14×10⁻⁴ m²) at h = 4 m gives Q ≈ 3.14×10⁻⁴ × 8.86 ≈ 2.8×10⁻³ m³/s (about 2.8 L/s). Real flows use v = C_v√(2gh) with discharge coefficient C_v ≈ 0.60–0.98 (sharp-edged hole lower, rounded higher).

Torricelli assumes: inviscid flow, small orifice, large tank (surface velocity negligible), unvented or vented to atmosphere equally. Viscosity, vena contracta (jet area smaller than hole), and surface depression lower real speed. Tank must be vented — a sealed tank cannot sustain flow.

As the tank drains, h drops and v decreases — flow is transient. Torricelli gives the instantaneous exit speed at the current head. Draining time requires integrating Q = −A_tank dh/dt.

Horizontal range of a side orifice: treat as projectile motion with initial horizontal speed v = √(2gh). Same √(2gh) appears in Pitot tubes and Torricelli — all are Bernoulli energy conversions.