The V-notch weir is one of the most accurate devices available for measuring small volume flow rates in open channels, streams, and industrial discharge lines. By funneling water through a precise triangular opening (the "notch"), engineers can determine the rate of flow (discharge) simply by measuring the height of the water surface upstream from the weir crest.
[Diagram Representation: A triangular notch with angle θ, water flowing over it, and Head (H) measured from the vertex up to the water surface level]
The Physics and Formula
The calculation of discharge over a V-notch weir is governed by the principles of fluid mechanics. The most common formula used for this calculation is derived from the Bernoulli equation, adjusted with a coefficient of discharge to account for energy losses and contraction of the flow stream.
The standard equation used in this calculator is:
Q = Cd × (8/15) × √(2g) × tan(θ/2) × H2.5
Where:
Q = Volumetric flow rate (m³/s)
Cd = Coefficient of discharge (typically between 0.58 and 0.62)
g = Acceleration due to gravity (9.81 m/s²)
θ = The total included angle of the V-notch (degrees)
H = The hydraulic head (height of fluid above the vertex of the notch)
Input Variables Explained
Hydraulic Head (H)
This is the most critical measurement. It represents the height of the water above the bottom point (vertex) of the V. Crucial Tip: Do not measure the height directly at the weir face, as the water level curves downward as it approaches the drop (drawdown). Measure the head at a distance of at least 3 to 4 times the maximum head upstream from the weir.
Notch Angle (θ)
The angle of the V-shape determines the weir's capacity and sensitivity.
90° Notch: The most common configuration, suitable for general flow measurement.
45° and 60° Notches: Better for measuring lower flow rates, as they create a higher head for the same volume of liquid, making measurement easier and more accurate.
30° Notch: Used for very small flows.
Discharge Coefficient (Cd)
This is an experimentally determined factor that corrects theoretical flow to actual flow. For a clean, sharp-edged V-notch weir, a value of 0.585 is a standard engineering approximation suitable for most angles between 20° and 100°. Rougher edges or submerged conditions will alter this coefficient.
Common Applications
V-notch weirs are extensively used in:
Wastewater Treatment Plants: To measure effluent discharge accurately.
Hydrology: Monitoring stream flows in small catchment areas.
Irrigation: Measuring water delivery to fields.
Industrial Processes: Monitoring cooling water or wash water usage.
Accuracy and Limitations
While highly accurate, V-notch weirs require specific conditions to function correctly:
The upstream channel should be straight and free of turbulence.
The edge of the weir (the crest) must be sharp (1-2mm thickness recommended).
The flow must discharge freely into the atmosphere (free flow condition), not into a submerged pool that backs up water against the weir.
Debris must be cleared regularly, as sediments accumulating behind the weir plate can alter the approach velocity and affect readings.
function calculateFlowRate() {
// 1. Get DOM elements
var headInput = document.getElementById("headHeight");
var unitSelect = document.getElementById("headUnit");
var angleInput = document.getElementById("notchAngle");
var cdInput = document.getElementById("dischargeCoeff");
var resultBox = document.getElementById("resultsArea");
var elM3Hr = document.getElementById("resM3Hr");
var elLps = document.getElementById("resLps");
var elGpm = document.getElementById("resGpm");
var elCfs = document.getElementById("resCfs");
// 2. Parse values
var hVal = parseFloat(headInput.value);
var unit = unitSelect.value;
var angleDeg = parseFloat(angleInput.value);
var cd = parseFloat(cdInput.value);
// 3. Validation
if (isNaN(hVal) || hVal < 0) {
alert("Please enter a valid positive number for the Hydraulic Head.");
return;
}
if (isNaN(angleDeg) || angleDeg = 180) {
alert("Please enter a valid Notch Angle (typically between 20 and 120 degrees).");
return;
}
if (isNaN(cd) || cd <= 0) {
alert("Please enter a valid Discharge Coefficient.");
return;
}
// 4. Normalize Head to Meters (Base SI unit)
var hMeters = 0;
if (unit === "m") {
hMeters = hVal;
} else if (unit === "cm") {
hMeters = hVal / 100;
} else if (unit === "ft") {
hMeters = hVal * 0.3048;
} else if (unit === "in") {
hMeters = hVal * 0.0254;
}
// 5. Calculation Logic
// Formula: Q (m3/s) = Cd * (8/15) * sqrt(2*g) * tan(theta/2) * H^2.5
// g = 9.81 m/s^2
var g = 9.81;
var thetaRadians = angleDeg * (Math.PI / 180);
var tanThetaOver2 = Math.tan(thetaRadians / 2);
// Constant part: 8/15 * sqrt(2 * 9.81)
// sqrt(19.62) approx 4.429
// (8/15) * 4.429 approx 2.362
var constantFactor = (8.0 / 15.0) * Math.sqrt(2 * g);
// Q in cubic meters per second
var q_m3s = cd * constantFactor * tanThetaOver2 * Math.pow(hMeters, 2.5);
// 6. Unit Conversions
// m3/hr = m3/s * 3600
var val_m3hr = q_m3s * 3600;
// Liters/sec = m3/s * 1000
var val_lps = q_m3s * 1000;
// CFS (Cubic Feet per Second) = m3/s * 35.3147
var val_cfs = q_m3s * 35.3147;
// GPM (US Gallons per Minute) = m3/s * 15850.3
var val_gpm = q_m3s * 15850.32;
// 7. Update UI
// Helper to format numbers nicely
function formatNum(num) {
if (num 0) return num.toExponential(3);
return num.toLocaleString('en-US', { minimumFractionDigits: 3, maximumFractionDigits: 3 });
}
elM3Hr.innerHTML = formatNum(val_m3hr) + " m³/hr";
elLps.innerHTML = formatNum(val_lps) + " L/s";
elGpm.innerHTML = formatNum(val_gpm) + " gpm";
elCfs.innerHTML = formatNum(val_cfs) + " ft³/s";
resultBox.style.display = "block";
}