Torricelli’s Law Calculator
Calculate the efflux velocity of a fluid from an opening based on the height of the liquid head.
Understanding Torricelli’s Law: The Physics of Efflux Velocity
Have you ever noticed how water shoots out faster from a hole near the bottom of a bucket than from a hole near the top? This isn’t just an observation; it is a fundamental principle of fluid dynamics known as Torricelli’s Law. Named after the Italian physicist Evangelista Torricelli in 1643, this theorem provides a bridge between the potential energy of a liquid at rest and its kinetic energy as it escapes through an orifice.
What is Torricelli’s Law?
Torricelli’s Law (also known as Torricelli’s Theorem) states that the speed of efflux, v, of a fluid under the force of gravity through an opening is equivalent to the speed that a single drop of fluid would acquire falling freely from the height of the liquid surface to the level of the opening.
The Mathematical Formula
The beauty of Torricelli’s Law lies in its simplicity. If we assume the fluid is ideal (incompressible and non-viscous), the formula is:
Where:
- v: The efflux velocity (m/s).
- g: The acceleration due to gravity (9.81 m/s² on Earth).
- h: The depth of the hole below the liquid surface (m).
Derivation from Bernoulli’s Principle
Torricelli’s Law is actually a specific case of Bernoulli’s Equation. Bernoulli’s equation states that for an incompressible, frictionless fluid, the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline.
By applying Bernoulli’s equation to the surface of the liquid (Point 1) and the exit hole (Point 2), and assuming the surface area is much larger than the hole (meaning the surface moves down at a negligible speed), the pressure at both points is atmospheric. The potential energy at the surface converts entirely into kinetic energy at the exit, leading us directly to the Torricelli formula.
Real-World Applications
Torricelli’s Law isn’t just a classroom theory; it has massive implications in engineering and everyday life:
- Hydroelectric Dams: Engineers use these principles to calculate the speed and force of water hitting turbines at the base of a dam.
- Tank Drainage: Determining how long it will take for a chemical tank or a water reservoir to empty through a valve.
- Medical Equipment: Understanding flow rates in IV drips based on the height of the bag relative to the patient.
- Firefighting: Estimating the pressure required to reach certain heights with fire hoses.
Factors Affecting Real-World Accuracy
While our Torricelli’s Law Calculator provides the theoretical maximum velocity, real-world results are often slightly lower (usually 2-5% less). This is due to:
- Viscosity: Internal friction within the fluid slows it down.
- Orifice Shape: The shape of the opening affects flow efficiency. This is often corrected using a Coefficient of Discharge (Cd).
- Vena Contracta: As the fluid exits, the jet diameter actually shrinks slightly to a size smaller than the hole itself, reducing the effective flow rate.
Frequently Asked Questions
Does the density of the liquid matter?
Surprisingly, no. In the theoretical formula, density cancels out. Whether you are using water, oil, or mercury, the speed of exit depends only on the depth and gravity. However, in practice, a thicker (more viscous) liquid like honey will flow slower due to friction, which the basic law doesn’t account for.
How do you calculate flow rate?
To find the volume of fluid leaving the tank per second (Flow Rate Q), you multiply the velocity (v) by the cross-sectional area (A) of the hole: Q = A × v. Our calculator provides this if you input the diameter of the hole.
Is the velocity constant?
No. As the liquid drains, the height (h) decreases. As h decreases, the velocity v also decreases. This is why a tank drains faster when it’s full and slower as it nears empty.
Example Calculation
Imagine a water tank with a small leak located 5 meters below the water line. Using the formula:
v = √(2 * 9.81 * 5) = √98.1 ≈ 9.90 m/s.
If the hole has a diameter of 2 cm, the area is approx 0.000314 m². The flow rate would be approximately 0.0031 liters per second.