Velocity of Venting

Technical Reference Guide // Torricelli's Law

REF-05-EFFLUX-MASTER-FINAL

The Science of Efflux

In fluid dynamics, efflux is the flow of a fluid out of a container through an opening (orifice). In 1643, Evangelista Torricelli discovered that the speed of this flow is directly proportional to the square root of the fluid height above the opening.

Torricelli's Theorem

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

Technical Schematic

Potential Energy \( U_g \) → Kinetic Energy \( K \)

Deriving the Law

Torricelli's Law is a simplified application of Bernoulli’s Principle. By analyzing energy at the surface (\( 1 \)) and the orifice (\( 2 \)):

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

Assumptions

Container open: \( P_1 = P_2 = P_{atm} \).
Large tank: Surface velocity \( v_1 \approx 0 \).

Final Result

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

Core Concepts

1. Height is Velocity

The velocity depends only on vertical distance \( h \). It does not depend on mass or tank shape.

2. Free-Fall Equivalence

A stone dropped from height \( h \) hits at speed \( \sqrt{2gh} \). Water follows the same energy conversion logic.

Engineering Applications

Dam Engineering

Turbines are placed at max depth \( h \). Doubling \( h \) increases velocity by \( \sqrt{2} \approx 1.41 \).

Industrial Safety

Engineers use Torricelli's Law to design tank drainage systems, ensuring proper size of containment basins.

Summary Reference

Velocity Law \( v = \sqrt{2gh} \)

Key Variable Fluid Depth (h)

Check for Understanding

If the depth of water in a tank is quadrupled (\( 4 \times h \)), by what factor does the exit velocity increase? Show reasoning.

© NAVAL ENGINEERING RESEARCH GROUP // FLUID DYNAMICS DIV.

Velocity of Venting Reading Guide Revised V2 Final

Velocity of Venting Reading Guide Revised V2 Final

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