Torricelli's Law in Action

Torricelli's law provides a remarkably simple estimate for the speed of a fluid jet emerging from an orifice in the side or bottom of a tank. Evangelista Torricelli reasoned in 1643 that the efflux speed equals that acquired by a particle falling freely under gravity through the same vertical distance. When applied to a draining reservoir of constant cross-section, the law leads to a differential equation that predicts how quickly the liquid level drops over time. Engineers and physicists employ this relationship when sizing water towers, designing emergency drains, or performing introductory laboratory demonstrations where the change of height is measurable with a ruler and stopwatch.

The instantaneous speed of fluid leaving a hole a distance $h$ below the free surface is $v=\sqrt{2gh}$. For a small circular opening of area $\mathrm{A_o}$ connected to a large tank of cross-sectional area $\mathrm{A_t}$, the volumetric flow rate is $Q=\mathrm{A_o}v$. Mass conservation dictates that the downward change in fluid height is related to this flow rate by $\mathrm{A_t}\frac{\mathrm{dh}}{\mathrm{dt}}=-Q$. Combining these expressions yields a separable differential equation:

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

Integrating from an initial height $\mathrm{h₀}$ to a final height $\mathrm{h₁}$ produces the general solution:

$$t=\frac{2\mathrm{A_t}(\sqrt{\mathrm{h₀}}-\sqrt{\mathrm{h₁}})}{\mathrm{A_o}\sqrt{2g}}$$

Because the square root of height appears, the liquid level drops rapidly at first and more slowly as the tank empties. The calculator implements this formula and rearrangements of it to solve for any single variable given the others. Users may adjust the gravitational acceleration to explore how draining would proceed on planets with different surface gravities.

How to Use the Form

Enter known values for the starting and ending heights, the tank's cross-sectional area, the orifice area, gravitational acceleration, and the drain time. Leave one of these fields blank depending on the quantity of interest. For example, to determine how long it takes for a cylindrical tank with a 0.5 m² cross-section to drain from 1.2 m to 0.2 m through a 0.01 m² valve under Earth's gravity, input all numbers except the time field. Pressing the Compute button yields the predicted duration. If instead you wish to size the orifice to achieve draining within a fixed time, leave the orifice area empty and supply the desired drain time.

The script checks that exactly one field is empty and then applies the appropriate algebraic manipulation. Solving for the initial height requires adding the time-dependent term to the square root of the final height and squaring the result. Determining the final height involves subtracting the time term from the initial square root and squaring. Calculating the gravitational acceleration from timing data can reveal deviations due to experimental error or be used in educational activities to estimate $g$ by measuring the rate at which a container drains.

Worked Example

Suppose a rectangular tank with cross-sectional area 0.3 m² is filled with water to a height of 0.8 m. It drains through a small hole of area 0.005 m² located at the bottom until only 0.1 m of water remains. Assuming $g=9.81$ m/s², the drain time is

$$t=\frac{2\times 0.3(\sqrt{0.8}-\sqrt{0.1})}{0.005\sqrt{2\times 9.81}}$$

Substituting the numbers gives $t\approx 132$ seconds, or a little over two minutes. The calculator performs this computation instantly and can be used to examine how doubling the orifice area or halving the initial height would affect the drain time.

Interpreting the Variables

The tank area represents the horizontal cross-section through which the liquid level drops. For cylindrical or rectangular tanks the area is straightforward, but for irregular shapes the formula becomes approximate. The orifice area corresponds to the opening through which fluid exits; for circular holes this is $\mathrm{A_o}=\frac{\pi }{4}{d}^2$. The initial and final heights are measured from the centerline of the orifice to the free surface. Because the derivation assumes the tank cross-section is large compared with the hole, the velocity of the free surface is neglected, an assumption valid for small or moderate openings.

Table of Example Calculations

The table below summarizes how changing the area ratio affects the drain time for a tank with $\mathrm{h₀}=1$ m, $\mathrm{h₁}=0$, $g=9.81$ m/s², and $\mathrm{A_t}=0.5$ m².

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

Doubling the orifice area halves the drain time, demonstrating the inverse relationship predicted by Torricelli's formula. Users can exploit this sensitivity to design drainage systems that meet specific timing requirements, such as ensuring a water tower empties slowly enough to maintain pressure or a laboratory experiment completes within a class period.

Assumptions and Limitations

The calculator assumes ideal fluid behavior: incompressible, non-viscous, and with laminar flow at the orifice. In reality viscosity and turbulence reduce the flow rate. Engineers introduce a discharge coefficient to account for such effects, often around 0.6 for sharp-edged orifices. Our calculator omits this correction for simplicity but users can approximate viscosity effects by reducing the effective orifice area. Additionally, the derivation presumes the tank cross-section remains constant; tapered or irregular vessels require more sophisticated integration. Atmospheric pressure is assumed to act equally on both sides of the orifice, so submerged outlets or significant downstream pressure would invalidate the formula.

Beyond Simple Tanks

Torricelli's framework extends to irrigation canals, dam spillways, and emergency fuel dump systems in aircraft. In such cases the basic square-root relation is embedded within more complex hydraulic models that account for entrance losses, pipe friction, and turbulent discharge. Still, the core principle—that gravitational potential energy converts to kinetic energy of outflowing fluid—remains central. Educators can adapt the calculator to explore how planetary gravity would change the emptying time of a water tank on Mars versus Earth, or to demonstrate the mathematics of separable differential equations.

By providing a clear mapping between measurable quantities and predicted drain times, this tool serves as both a teaching aid and a preliminary design check. Whether timing a backyard water feature or setting up a classroom experiment, Torricelli's four-century-old insight continues to prove practical and engaging.