Flow Rate of Water Through a Hole Calculator

Calculate the discharge rate of water based on Torricelli's Law.

Please enter valid positive numbers for diameter and height.

Understanding Water Flow Through an Orifice

Calculating the flow rate of water through a hole (often called an orifice) is a critical task in hydraulic engineering, plumbing, and fluid mechanics. Whether you are designing a drainage tank, assessing a leak in a pipe, or engineering an irrigation system, understanding how gravity and pressure drive fluid through an opening is essential.

This calculator utilizes Torricelli's Law combined with the Bernoulli Principle to estimate the discharge rate of water from a tank or container due to gravity.

The Physics: Torricelli's Law

Torricelli's Law states that the speed of a fluid flowing out of an orifice under the force of gravity is proportional to the square root of the vertical distance between the water surface and the center of the hole.

v = √(2 · g · h)

Where:

• v = Theoretical velocity of the fluid (m/s)
• g = Acceleration due to gravity (approx. 9.81 m/s²)
• h = Head of the liquid (height of water above the hole in meters)

Calculating Volume Flow Rate (Q)

While Torricelli's Law gives us the velocity, to find the actual volume of water escaping, we must account for the area of the hole and the real-world efficiency of the flow. In reality, water does not exit at the full theoretical speed or occupy the full area of the hole due to flow contraction (vena contracta) and friction.

We use a Coefficient of Discharge (Cd) to correct for these factors. The formula for Flow Rate (Q) is:

Q = C_d × A × √(2 · g · h)

Where:

• Q = Flow rate (m³/s)
• C_d = Discharge Coefficient (typically 0.60 to 0.62 for sharp-edged holes)
• A = Cross-sectional area of the hole (m²)
• h = Head height (m)

Key Input Variables

1. Hole Diameter: The physical width of the opening. Doubling the diameter quadruples the area, significantly increasing the flow rate. Our calculator accepts this in millimeters (mm) for precision.

2. Water Head (Height): This is the driving force. It is the vertical distance from the top of the water surface down to the center of the hole. As the tank drains and the height decreases, the pressure drops, and the flow rate slows down.

3. Discharge Coefficient (Cd): This is a dimensionless number that characterizes the efficiency of the hole.
– 0.60 – 0.62: Sharp-edged orifice (like a drilled hole in a thin tank wall).
– 0.95 – 0.98: Smooth, rounded nozzle (highly efficient flow).
– 0.50: Projecting pipe (Borda mouthpiece).

Example Calculation

Imagine a water tank with a 20 mm diameter hole located 3 meters below the water surface. Assuming a standard sharp-edged hole (Cd = 0.61):

1. Convert Diameter to Area: 20mm = 0.02m. Radius = 0.01m. Area = π × 0.01² ≈ 0.000314 m².
2. Calculate Theoretical Velocity: √(2 × 9.81 × 3) ≈ 7.67 m/s.
3. Calculate Flow Rate (Q): 0.61 × 0.000314 × 7.67 ≈ 0.00147 m³/s.
4. Convert Units: This equals approximately 1.47 Liters per second or 88 Liters per minute.