Broad-Crested Weir Discharge Calculator
Broad-Crested Weir Discharge Calculator — Q = Cd · b · √(g) · H3/2
Compute Discharge / Head / Crest Width
A broad-crested weir is a weir with a crest that is long in the flow direction. This calculator uses the general formula Q = Cd · b · √(g) · H3/2 — for free flow only, not submerged flow.
Verified example
| Parameter | Value |
|---|---|
| Crest width, b | 2 m |
| Head, H | 0.5 m |
| Discharge coefficient, Cd | 0.544 |
| Discharge, Q | 1.2048 m³/s (= 1,204.8096 L/s) |
Calculation: Q = 0.544 × 2 × √9.81 × 0.51.5 = 1.2048 m³/s. Imperial units: L = 10 ft, H = 1 ft, Cd = 0.544 → Q = 30.857 ft³/s (cfs).
Formulas used
- Q = discharge (m³/s or ft³/s), b = crest width, H = head above crest, g = 9.81 m/s² (SI) or 32.174 ft/s² (imperial).
- Practical SI form: Q = 1.70·b·H1.5 (with Cd = 0.544). Practical imperial form: Q = 3.09·L·H1.5 (Q in ft³/s).
- Inverse: H = [Q / (Cd·b·√g)]2/3 and b = Q / (Cd·√g·H1.5).
- Honest limitation: free flow only. If downstream is submerged (h₂/H > ~0.67), actual discharge is smaller and the formula does not apply. Not for sharp-crested weirs (different formula: Francis).
Frequently asked questions (FAQ)
What is the discharge formula for a broad-crested weir?
Q = C_d · b · √(g) · H^(3/2), where Q is discharge (m³/s), C_d the discharge coefficient, b the crest width (m), g = 9.81 m/s², and H the head above the crest (m). With C_d = 0.544 the practical form is Q = 1.70·b·H^1.5. Example: b = 2 m, H = 0.5 m, C_d = 0.544 → Q = 1.2048 m³/s.
Which discharge coefficient (C_d) value should I use?
The theoretical value from critical-flow theory is (2/3)^1.5 = 0.5443, so 0.544 is a conservative default choice. Laboratory calibrations typically fall in the range 0.544–0.577 depending on crest geometry and the H/L ratio. This calculator defaults to 0.544 and allows custom values 0.30–0.70.
For imperial units, is the practical constant 3.33?
No. With C_d = 0.544, the imperial practical constant is 0.544·√32.174 = 3.09, so Q = 3.09·L·H^1.5 (Q in ft³/s, L and H in ft). The number 3.33 is the Francis formula constant for sharp-crested weirs — a different formula for a different weir type. Example: L = 10 ft, H = 1 ft, C_d = 0.544 → Q = 30.857 cfs.
Does this calculator apply to submerged flow?
No. This formula is only valid for free flow, i.e., when the flow over the crest reaches critical conditions. If the downstream water level rises so that the downstream-depth to upstream-head ratio exceeds about 0.67 (general approximation; literature range 0.67–0.80 depending on geometry), the weir becomes submerged and actual discharge is smaller than the formula result. Use the downstream depth field in the calculator to check this indication.
What is the difference between a broad-crested weir and a sharp-crested weir?
A broad-crested weir has a crest that is long in the flow direction, so streamlines run parallel to the crest and full hydrostatic pressure develops; its behavior follows Q = C_d·b·√(g)·H^(3/2) with C_d ≈ 0.544–0.577, valid over the approximate H/L range 0.08–0.50 (L = crest length in flow direction). A sharp-crested weir has a sharp crest and uses the Francis formula Q = 3.33·(L − 0.2H)·H^(3/2) in imperial units. The two are not interchangeable.
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