Rectangular & V-Notch Weir Flow Rate Discharge formulas (H^1.5 & H^2.5)

Use a simplified discharge relation with declared assumptions

The rectangular model uses Q=(2/3) Cd L sqrt(2g) H^(3/2). The triangular model uses Q=(8/15) Cd tan(theta/2) sqrt(2g) H^(5/2). With lengths in metres and g in m/s², Q is in m³/s. Theta is supplied in degrees and converted to radians.

H is upstream head above the crest (or V-notch vertex), not the channel depth measured from its bed. The model assumes free discharge with an aerated nappe and negligible approach velocity. Submergence, end contractions, head corrections and actual installation geometry require an appropriate calibrated relation.

The example assumes Cd=0.62 separately for each shape, using B2 for the rectangle and B7 for the V-notch. This is not an exact flow meter, a Francis contracted-weir implementation, or a spillway-design calculation. The simple coefficient range 0<Cd<=1 is a guard for this illustrative model, not a universal calibration rule.

The zero-head result is a mathematical limit, not a statement that the formula is calibrated for arbitrarily small heads. Similarly, accepting an angle between 0 and 180 degrees only prevents invalid arithmetic; it does not establish an acceptable physical weir geometry.

Rectangular model in D2

For fixed geometry and coefficient, multiplying H by four multiplies this model output by eight. Retain the unrounded value for conversions or comparisons.

=IF(COUNT(B2:B5)<>4,"Enter numeric rectangular inputs",IF(OR(B2<=0,B2>1,B3<=0,B4<0,B5<=0),"Invalid rectangular model inputs",(2/3)*B2*B3*SQRT(2*B5)*B4^1.5))

V-notch model in D3

For fixed angle and coefficient, multiplying H by four multiplies this model output by 32. Use a calibrated coefficient for the actual notch if applying a measured relation.

=IF(COUNT(B4:B7)<>4,"Enter numeric V-notch inputs",IF(OR(B7<=0,B7>1,B4<0,B5<=0,B6<=0,B6>=180),"Invalid V-notch model inputs",(8/15)*B7*TAN(RADIANS(B6/2))*SQRT(2*B5)*B4^2.5))

Convert rectangular discharge to litres per second

A cubic metre contains 1000 litres. D2 retains full precision; this result is rounded only for display.

=IF(ISNUMBER(D2),ROUND(1000*D2,1),D2)

Further reading

• Microsoft Support — RADIANS function
• USBR — Weir calibration background and approach-velocity limitations
• USBR — V-notch calibration and installation conditions