Weir Flow Calculator
Calculate the volumetric flow rate of water passing over a weir based on weir width, upstream head, and weir type. Used by civil engineers and hydrologists to measure or control open-channel flow in rivers, canals, and water treatment plants.
About this calculator
A weir is a structure placed across an open channel that forces water to flow over its crest, creating a measurable relationship between upstream head and flow rate. For a broad-crested or sharp-crested rectangular weir, the standard formula is Q = C_d × L × H^1.5, where Q is the volumetric flow rate (m³/s), C_d is the discharge coefficient (typically 1.84 for SI units with a rectangular sharp-crested weir), L is the weir crest length (m), and H is the head of water above the weir crest (m). The H^1.5 (or H^(3/2)) exponent arises from integrating the velocity profile across the overflowing nappe. Different weir shapes — rectangular, V-notch (triangular), and trapezoidal (Cipolletti) — have distinct formulas and C_d values. The V-notch weir (Q = C_d × tan(θ/2) × H^2.5) is more sensitive at low flows, making it preferred for accurate low-flow measurement.
How to use
A sharp-crested rectangular weir is 2.5 m wide with a measured head of 0.35 m above the crest. Using a discharge coefficient of 1.84, apply the formula: Q = 1.84 × 2.5 × (0.35)^1.5. First calculate (0.35)^1.5: √0.35 = 0.5916; 0.35 × 0.5916 = 0.2071. Then Q = 1.84 × 2.5 × 0.2071 = 1.84 × 0.5178 = 0.953 m³/s, or approximately 953 litres per second. This flow rate can then be used to assess channel capacity or calibrate downstream control structures.
Frequently asked questions
What discharge coefficient should I use for a rectangular sharp-crested weir?
For a rectangular sharp-crested weir in SI units, the combined discharge coefficient C_d is commonly taken as 1.84 when the formula Q = C_d × L × H^(3/2) is used — this value already incorporates the 2/3 factor and √(2g) from the theoretical derivation. If you are using the form Q = (2/3) × C_d × L × √(2g) × H^(3/2) with g = 9.81 m/s², then C_d ≈ 0.611 for a sharp-crested weir. The exact value depends on weir geometry, approach velocity, and submergence conditions. For broad-crested weirs, C_d is typically 0.848 in the same framework. Always confirm which formula form your reference or standard uses before selecting a coefficient.
How does head height above a weir affect the flow rate measurement accuracy?
Accuracy of weir flow measurement is strongly dependent on precise head measurement, because flow rate is proportional to H^1.5 for rectangular weirs and H^2.5 for V-notch weirs. A 5% error in measuring H produces approximately a 7.5% error in Q for a rectangular weir. The head should be measured upstream at a distance of at least 4–5 times the maximum expected head to avoid the drawdown effect near the weir. Staff gauges or pressure transducers should be zeroed to the exact weir crest elevation. Very low heads (below about 60 mm) are unreliable due to surface tension effects and viscosity, and should be avoided in accurate measurement applications.
When should I use a V-notch weir instead of a rectangular weir for flow measurement?
A V-notch (triangular) weir is the preferred choice for measuring low flow rates accurately, typically flows below 0.1 m³/s. Because the flow area in a V-notch grows with both width and depth as head increases, it maintains a strong head-to-flow relationship (Q ∝ H^2.5) even at very small flows. A rectangular weir becomes insensitive at low heads because the wetted length stays constant while head drops. V-notch weirs are common in laboratory flumes, small streams, and irrigation ditches. For large flow rates or wide channels, a rectangular or trapezoidal weir is more practical because the V-notch would require an impractically large head to pass the required flow.