Rectangular Weir Flow Calculator (Francis Formula)
Flow over a sharp-crested rectangular weir from the crest length and the head of water above it, with the effect of end contractions.
Results
The dashed line marks the value you entered.
What this tool does
A weir turns a depth you can measure into a flow you cannot: the flow rises with the head to the power 1.5, so doubling the head gives almost three times the flow. The Francis formula is an old empirical fit for a sharp-edged crest with a free-falling nappe; each side contraction narrows the effective crest by one tenth of the head. Measure the head with a gauge set well upstream of the crest, several times the head away, where the water surface has not yet dropped towards the weir, and keep the nappe ventilated.
Formula
Q = 1.84 (L − 0.1 n H) H^(3/2) (Francis, SI: Q in m³/s, L and H in m, n = 0, 1 or 2 end contractions)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
l | Length of the crest | m |
h | Head over the crest (measured upstream) | m |
nc | End contractions | — |
QL | Flow over the weir | L/s |
QS | Flow over the weir | m³/s |
QH | Flow per hour | m³/h |
LE | Effective crest length | m |
QU | Flow per metre of crest | L/(s·m) |
Worked example
- Length of the crest1 m
- Head over the crest (measured upstream)0.15 m
- End contractions2
- Flow over the weir103.7 L/s
- Flow over the weir0.1037 m³/s
- Flow per hour373.3 m³/h
- Effective crest length0.970 m
- Flow per metre of crest103.7 L/(s·m)
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.