Calculate the discharge velocity and flow rate of water through a circular orifice based on the pressure head and orifice characteristics.
Flow Rate: L/min
Flow Rate: m³/hour
Exit Velocity: m/s
How to Calculate Water Flow Through an Orifice
Calculating the flow rate of water through an orifice is a fundamental task in hydraulic engineering and fluid mechanics. An orifice is simply an opening (usually circular) in a wall or plate through which fluid flows. The calculation is primarily based on Torricelli's Law, which relates the speed of fluid leaving an opening to the height of the fluid above that opening.
The Mathematical Formula
The standard formula for calculating the flow rate (Q) is:
Q = Cd × A × √(2 × g × h)
Q: Flow rate (m³/s)
Cd: Discharge Coefficient (accounts for friction and fluid contraction)
A: Cross-sectional area of the orifice (π × r²)
g: Acceleration due to gravity (approx. 9.81 m/s²)
h: Pressure head (vertical distance from the water surface to the center of the orifice)
Understanding the Discharge Coefficient (Cd)
The discharge coefficient is a dimensionless number that corrects the theoretical flow for real-world losses. It is the product of the coefficient of velocity (Cv) and the coefficient of contraction (Cc). For a standard sharp-edged orifice, a value of 0.62 is commonly used. If the orifice is rounded or streamlined, the Cd can be as high as 0.98.
Practical Example
Imagine a water tank with a 50mm diameter hole located 3 meters below the water surface level. Assuming a standard sharp-edged orifice (Cd = 0.62):
Convert Diameter to Meters: 50mm = 0.05m
Calculate Area: Area = (π × 0.05²) / 4 = 0.0019635 m²
Several variables can change your results in the field:
Fluid Viscosity: While the standard formula works for water, thicker fluids (like oil) require different coefficients.
Orifice Shape: Square or rectangular openings have different contraction characteristics than circular ones.
Approach Velocity: If the water is already moving quickly toward the orifice (rather than being still in a tank), the flow rate will be higher.
function calculateOrificeFlow() {
var d_mm = parseFloat(document.getElementById('orificeDiameter').value);
var h = parseFloat(document.getElementById('headHeight').value);
var cd = parseFloat(document.getElementById('dischargeCoeff').value);
var g = 9.81;
if (isNaN(d_mm) || isNaN(h) || isNaN(cd) || d_mm <= 0 || h < 0 || cd <= 0) {
alert("Please enter valid positive numbers for all fields.");
return;
}
// Convert diameter mm to meters
var d_m = d_mm / 1000;
// Calculate Area (A = pi * d^2 / 4)
var area = (Math.PI * Math.pow(d_m, 2)) / 4;
// Theoretical Velocity (v = sqrt(2gh))
// Note: Actual velocity = Cv * sqrt(2gh). Usually Cv is ~0.97-0.98.
// For simplicity in this tool, we show exit velocity as the theoretical root.
var velocity = Math.sqrt(2 * g * h);
// Discharge Flow Rate Q = Cd * A * sqrt(2gh)
var q_m3s = cd * area * velocity;
// Convert m3/s to other units
var q_lpm = q_m3s * 1000 * 60;
var q_m3h = q_m3s * 3600;
// Actual Exit Velocity (Cv * sqrt(2gh)) – typically Cd/Cc, but we'll show v_theoretical * 0.98 approx
var actual_v = 0.98 * velocity;
// Display results
document.getElementById('resFlowLpm').innerHTML = q_lpm.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 2});
document.getElementById('resFlowM3h').innerHTML = q_m3h.toLocaleString(undefined, {minimumFractionDigits: 3, maximumFractionDigits: 3});
document.getElementById('resVelocity').innerHTML = actual_v.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 2});
document.getElementById('resultsArea').style.display = 'block';
}