Velocity of Venting

Technical Reference Guide // Torricelli's Law

REF-05-EFFLUX

The Science of Efflux

In fluid dynamics, efflux is the flow of a fluid out of a container or through an opening (called an orifice). In 1643, Evangelista Torricelli, a student of Galileo, discovered a remarkable relationship between the height of a fluid in a container and the speed at which it exits.

Torricelli's Theorem

"The speed of efflux of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth \(h\) is the same as the speed that a body would acquire in falling freely from a height \(h\)."

Technical Schematic

h

P₁

P₂

v

The stream follows a parabolic trajectory as it converts potential energy to horizontal kinetic energy.

Deriving the Law

Torricelli's Law is a specific application of Bernoulli’s Principle, which relates pressure, velocity, and height in a moving fluid. By analyzing energy at the surface (P₁) and the orifice (P₂), we find:

Energy Conservation (Bernoulli)

\[ P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2 \]

Simplifying Assumptions

The container is open to the atmosphere at both the top and the orifice (\(P_1 = P_2 = P_{atm}\)).
The surface area of the tank is much larger than the hole, so the surface velocity is negligible (\(v_1 \approx 0\)).

Final Efflux Result

\[ v = \sqrt{2gh} \]

Derived velocity at exit orifice

Core Concepts

1. Height is Power

The velocity of the fluid depends only on the vertical distance between the surface and the hole. It does not depend on the mass of the fluid, the shape of the tank, or the fluid's density (assuming it is inviscid).

2. Free-Fall Analogy

If you dropped a stone from height \(h\), its speed when hitting the ground would be exactly \(\sqrt{2gh}\). Water exiting a tank at depth \(h\) follows this same conversion: all potential energy becomes kinetic energy.

3. Pressure Influence

If the tank is pressurized (sealed with compressed air), the efflux velocity will increase significantly as the internal pressure (\(P_1\)) adds more energy to the system, effectively acting like extra "height".

Engineering Applications

Dam Design

Hydroelectric dams place turbines as low as possible to maximize the fluid height (\(h\)), resulting in higher velocity and more power generation. Doubling height increases velocity by \(\sqrt{2}\).

Industrial Safety

Safety engineers use Torricelli's Law to predict how fast a chemical tank will drain in an emergency. This determines the necessary size of containment basins and drain pipes.

Technical Summary Table

Primary Formula \(v = \sqrt{2gh}\)

Key Factor Height of fluid column (\(h\))

Vena Contracta The stream narrowing effect that occurs just as fluid leaves an orifice, affecting the actual flow rate.

Velocity of Venting Reading Guide Revised V2

Velocity of Venting Reading Guide Revised V2

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