Torricelli Tank Draining Time Calculator

Reviewed by: JJ Ben-Joseph

Initial height h₀ (m) Final height h₁ (m) Tank area A_t (m²) Orifice area A_o (m²) Gravitational acceleration g (m/s²) Drain time t (s) Compute

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Why Torricelli's Law Matters

Torricelli's 17th‑century insight connects fluid height to efflux speed. It remains a staple of hydraulics because it converts a messy flow problem into manageable algebra. Whether emptying a rain barrel, sizing an emergency drain, or demonstrating physics principles, estimating how fast a tank empties is surprisingly useful. Understanding the underlying math also illuminates broader concepts like energy conservation and differential equations.

Deriving the Draining Formula

The instantaneous speed of fluid leaving a hole a distance $h$ below the surface is $v=\sqrt{2gh}$. For a small opening of area $A_o$ connected to a reservoir of cross‑section $A_t$, the volumetric flow rate is $Q=A_{o}v$. Mass conservation links this outflow to the drop in fluid height:

$$A_t\frac{dh}{dt}=-Q$$

Substituting $v$ and separating variables yields:

$$\frac{dh}{\sqrt{h}}=-\frac{A_o}{A_t}\sqrt{2g}dt$$

Integrating from an initial height $h_0$ to a final height $h_1$ produces the general solution:

$$t=\frac{2A_t(\sqrt{h_0}-\sqrt{h_1})}{A_o\sqrt{2g}}$$

Step-by-Step Logic

1. Measure or estimate the initial and final fluid heights relative to the orifice.
2. Enter the cross‑sectional area of the tank and the opening.
3. Supply the local gravitational acceleration. On Earth use 9.81 m/s².
4. Leave the quantity you want to solve for blank and press compute.
5. The calculator rearranges Torricelli's formula algebraically to isolate the missing variable.

Worked Example

A rectangular tank with $A_t=0.3$ m² drains from $h_0=0.8$ m to $h_1=0.1$ m through an orifice of $A_o=0.005$ m². Plugging the numbers into the formula with $g=9.81$ m/s² gives a drain time of roughly 132 s.

Comparison Table

The table below summarizes how changing the area ratio affects drain time for a tank with $h_0=1$ m, $h_1=0$, $g=9.81$ m/s², and $A_t=0.5$ m².

Ao (m²)	Drain Time (s)
0.0025	201
0.0050	100
0.0100	50
0.0200	25

Assumptions and Limitations

The calculator assumes incompressible, non‑viscous flow with the tank cross‑section much larger than the orifice. Viscosity and turbulence reduce actual drain speed; engineers account for this with a discharge coefficient typically around 0.6 for sharp‑edged holes. We also neglect any back pressure or submerged outlets. For tall, narrow containers the falling surface velocity may not be negligible, introducing error.

Practical Applications

Torricelli's relation guides the design of rain barrels, water towers, and even emergency fuel dump systems. Educators use the equation to demonstrate separable differential equations, while hobbyists employ it to plan decorative fountains. Because it scales with the square root of height, the law also illustrates why drain rates slow dramatically as tanks empty.

Related Calculators

Explore other hydraulic and fluid dynamics calculators for more complex flow scenarios.

Conclusion

Torricelli's simple formula, now nearly four centuries old, continues to offer quick estimates of how fast liquids escape. While real-world factors may require correction, starting with this model reveals the key relationships among height, gravity, and orifice size. Use the calculator as a springboard for deeper hydraulic analysis or as a teaching aid in fluid dynamics classes.