Orifice Flow Calculator

What is Orifice Flow?

An orifice is an opening or a restriction (typically a circular hole in a plate or tank wall) through which fluid passes. In fluid mechanics, measuring and calculating the flow rate through an orifice is crucial for hydraulic design, tank drainage, flow metering, and process control. The rate of fluid discharge through an orifice is determined by the head pressure above the opening, the area of the orifice, and a discharge coefficient that accounts for energy losses. For related fluid dynamics calculations, see the Bernoulli Equation Calculator.

The Orifice Flow Equation

The discharge rate through a submerged or free-discharge orifice under steady flow conditions is calculated using Torricelli's theorem combined with a correction factor:

$$Q = C_d A_o \sqrt{2gH}$$

Where:

• $Q$ is the volumetric flow rate or discharge (measured in $m^3/s$ or gallons per minute).
• $C_d$ is the discharge coefficient (dimensionless), which accounts for losses due to friction and the contraction of the fluid jet (known as the vena contracta).
• $A_o$ is the cross-sectional area of the orifice (in $m^2$ or $ft^2$).
• $g$ is the acceleration due to gravity ($9.80665\text{ m/s}^2$ in metric or $32.174\text{ ft/s}^2$ in imperial).
• $H$ is the centerline head of fluid above the orifice (in meters or feet).

Discharge Coefficients ($C_d$)

The discharge coefficient is a ratio of actual flow to theoretical flow. It depends heavily on the geometry of the orifice entrance:

• Sharp-edged Orifices ($C_d \approx 0.62$): The fluid streamlines contract significantly downstream of the opening (vena contracta), reducing the effective area to about $62\%$ of the physical hole.
• Short Tube / Orifice ($C_d \approx 0.80$): Streamlines re-attach to the tube walls, reducing flow contraction and increasing discharge.
• Rounded / Bell-mouth Orifices ($C_d \approx 0.95$ to $0.99$): The smooth converging profile guides the flow without separation or contraction, achieving near-perfect efficiency.

Worked Example

Problem: A sharp-edged circular orifice ($C_d = 0.62$) with a diameter of $50\text{ mm}$ is located on the side of a tank. The water level is maintained at a height of $3\text{ m}$ above the center of the orifice. What is the discharge flow rate in liters per second?

Solution:

1. Calculate orifice area ($A_o$): Diameter $d = 50\text{ mm} = 0.05\text{ m}$. $$A_o = \frac{\pi}{4} d^2 = \frac{\pi}{4} (0.05)^2 \approx 0.001963\text{ m}^2$$
2. Apply the orifice flow equation: $$Q = C_d A_o \sqrt{2gH}$$ $$Q = 0.62 \times 0.001963 \times \sqrt{2 \times 9.80665 \times 3}$$
3. Calculate velocity: $$v_t = \sqrt{58.84} \approx 7.67\text{ m/s}$$
4. Compute Q: $$Q \approx 0.62 \times 0.001963 \times 7.67 \approx 0.00934\text{ m}^3\text{/s}$$
5. Convert to liters per second: $$0.00934\text{ m}^3\text{/s} \times 1,000 \approx 9.34\text{ L/s}$$