Understanding the Relationship Between Pressure and Flow
Calculating the flow rate from pressure and diameter is a fundamental task in fluid mechanics, civil engineering, and irrigation. The relationship is governed by the principles of energy conservation, specifically Bernoulli's equation and Torricelli's law.
The Flow Rate Formula
For a typical discharge into the atmosphere, we use a simplified version of the orifice equation:
Q = 29.84 × Cd × d² × √P
Q: Flow rate in Gallons Per Minute (GPM).
Cd: Discharge coefficient (dimensionless).
d: Diameter of the pipe or opening in inches.
P: Pressure at the source in Pounds per Square Inch (PSI).
How to Calculate Step-by-Step
Measure Diameter: Determine the inside diameter of the pipe or the opening of the nozzle in inches.
Measure Pressure: Use a pressure gauge to find the static or residual pressure in PSI.
Select Coefficient: Determine the shape of the opening. A sharp-edged hole loses more energy (Cd ≈ 0.61) than a smooth, rounded nozzle (Cd ≈ 0.98).
Apply the Formula: Square the diameter, multiply it by 29.84 and the coefficient, then multiply by the square root of the pressure.
Common Discharge Coefficients (Cd)
Type of Opening
Coefficient (Cd)
Sharp-Edged Orifice
0.61 – 0.65
Short Flush-Mount Tube
0.80
Well-Rounded Nozzle
0.97 – 0.99
Smooth Pipe Exit
0.90
Practical Example
Suppose you have a 2-inch diameter fire hydrant outlet with a pressure reading of 40 PSI. Assuming a standard smooth-bore nozzle (Cd = 0.90):
d² = 2 × 2 = 4
√P = √40 ≈ 6.32
Calculation: 29.84 × 0.90 × 4 × 6.32 = 679.35 GPM
Limitations
This calculation assumes the fluid is water. It does not account for pipe friction losses over long distances (which require the Hazen-Williams or Darcy-Weisbach equations). It also assumes the pressure is measured immediately upstream of the discharge point.
function calculateFlow() {
var d = parseFloat(document.getElementById('pipeDiameter').value);
var p = parseFloat(document.getElementById('pressurePsi').value);
var c = parseFloat(document.getElementById('dischargeCoeff').value);
var resCont = document.getElementById('resultContainer');
var resGPM = document.getElementById('flowGPM');
var resLPM = document.getElementById('flowLPM');
var resCFS = document.getElementById('flowCFS');
if (isNaN(d) || d <= 0) {
alert("Please enter a valid diameter greater than zero.");
return;
}
if (isNaN(p) || p < 0) {
alert("Please enter a valid pressure (0 or greater).");
return;
}
// The formula for GPM using PSI and Inches
// Q = 29.84 * C * d^2 * sqrt(p)
var gpm = 29.84 * c * Math.pow(d, 2) * Math.sqrt(p);
// Conversions
var lpm = gpm * 3.78541; // Liters per minute
var cfs = gpm * 0.002228; // Cubic feet per second
// Display results
resGPM.innerHTML = gpm.toFixed(2) + " Gallons Per Minute (GPM)";
resLPM.innerHTML = "Metric Flow: " + lpm.toFixed(2) + " Liters Per Minute (L/min)";
resCFS.innerHTML = "Velocity Flow: " + cfs.toFixed(4) + " Cubic Feet Per Second (CFS)";
resCont.style.display = 'block';
// Scroll to result smoothly
resCont.scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}